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Word (group theory) In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G,[1] or even in every group.[2] Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory. Definitions Let G be a group, and let S be a subset of G. A word in S is any expression of the form $s_{1}^{\varepsilon _{1}}s_{2}^{\varepsilon _{2}}\cdots s_{n}^{\varepsilon _{n}}$ where s1,...,sn are elements of S, called generators, and each εi is ±1. The number n is known as the length of the word. Each word in S represents an element of G, namely the product of the expression. By convention, the unique[3] identity element can be represented by the empty word, which is the unique word of length zero. Notation When writing words, it is common to use exponential notation as an abbreviation. For example, the word $xxy^{-1}zyzzzx^{-1}x^{-1}\,$ could be written as $x^{2}y^{-1}zyz^{3}x^{-2}.\,$ This latter expression is not a word itself—it is simply a shorter notation for the original. When dealing with long words, it can be helpful to use an overline to denote inverses of elements of S. Using overline notation, the above word would be written as follows: $x^{2}{\overline {y}}zyz^{3}{\overline {x}}^{2}.\,$ Reduced words Any word in which a generator appears next to its own inverse (xx−1 or x−1x) can be simplified by omitting the redundant pair: $y^{-1}zxx^{-1}y\;\;\longrightarrow \;\;y^{-1}zy.$ This operation is known as reduction, and it does not change the group element represented by the word. Reductions can be thought of as relations (defined below) that follow from the group axioms. A reduced word is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions: $xzy^{-1}xx^{-1}yz^{-1}zz^{-1}yz\;\;\longrightarrow \;\;xyz.$ The result does not depend on the order in which the reductions are performed. A word is cyclically reduced if and only if every cyclic permutation of the word is reduced. Operations on words The product of two words is obtained by concatenation: $\left(xzyz^{-1}\right)\left(zy^{-1}x^{-1}y\right)=xzyz^{-1}zy^{-1}x^{-1}y.$ Even if the two words are reduced, the product may not be. The inverse of a word is obtained by inverting each generator, and reversing the order of the elements: $\left(zy^{-1}x^{-1}y\right)^{-1}=y^{-1}xyz^{-1}.$ The product of a word with its inverse can be reduced to the empty word: $zy^{-1}x^{-1}y\;y^{-1}xyz^{-1}=1.$ You can move a generator from the beginning to the end of a word by conjugation: $x^{-1}\left(xy^{-1}z^{-1}yz\right)x=y^{-1}z^{-1}yzx.$ Generating set of a group Main article: Generating set of a group A subset S of a group G is called a generating set if every element of G can be represented by a word in S. When S is not a generating set for G, the set of elements represented by words in S is a subgroup of G, known as the subgroup of G generated by S and usually denoted $\langle S\rangle $. It is the smallest subgroup of G that contains the elements of S. Normal forms A normal form for a group G with generating set S is a choice of one reduced word in S for each element of G. For example: • The words 1, i, j, ij are a normal form for the Klein four-group with S = {i, j} and 1 representing the empty word (the identity element for the group). • The words 1, r, r2, ..., rn-1, s, sr, ..., srn-1 are a normal form for the dihedral group Dihn with S = {s, r} and 1 as above. • The set of words of the form xmyn for m,n ∈ Z are a normal form for the direct product of the cyclic groups ⟨x⟩ and ⟨y⟩ with S = {x, y}. • The set of reduced words in S are the unique normal form for the free group over S. Relations and presentations Main article: Presentation of a group If S is a generating set for a group G, a relation is a pair of words in S that represent the same element of G. These are usually written as equations, e.g. $x^{-1}yx=y^{2}.\,$ A set ${\mathcal {R}}$ of relations defines G if every relation in G follows logically from those in ${\mathcal {R}}$ using the axioms for a group. A presentation for G is a pair $\langle S\mid {\mathcal {R}}\rangle $, where S is a generating set for G and ${\mathcal {R}}$ is a defining set of relations. For example, the Klein four-group can be defined by the presentation $\langle i,j\mid i^{2}=1,\,j^{2}=1,\,ij=ji\rangle .$ Here 1 denotes the empty word, which represents the identity element. Free groups Main article: Free group If S is any set, the free group over S is the group with presentation $\langle S\mid \;\rangle $. That is, the free group over S is the group generated by the elements of S, with no extra relations. Every element of the free group can be written uniquely as a reduced word in S. See also • Word problem (mathematics) • Word problem for groups Notes 1. for example, fdr1 and r1fc in the group of square symmetries 2. for example, xy and xzz−1y 3. Uniqueness of identity element and inverses References • Epstein, David; Cannon, J. W.; Holt, D. F.; Levy, S. V. F.; Paterson, M. S.; Thurston, W. P. (1992). Word Processing in Groups. AK Peters. ISBN 0-86720-244-0.. • Novikov, P. S. (1955). "On the algorithmic unsolvability of the word problem in group theory". Trudy Mat. Inst. Steklov (in Russian). 44: 1–143. • Robinson, Derek John Scott (1996). A course in the theory of groups. Berlin: Springer-Verlag. ISBN 0-387-94461-3. • Rotman, Joseph J. (1995). An introduction to the theory of groups. Berlin: Springer-Verlag. ISBN 0-387-94285-8. • Schupp, Paul E; Lyndon, Roger C. (2001). Combinatorial group theory. Berlin: Springer. ISBN 3-540-41158-5. • Solitar, Donald; Magnus, Wilhelm; Karrass, Abraham (2004). Combinatorial group theory: presentations of groups in terms of generators and relations. New York: Dover. ISBN 0-486-43830-9. • Stillwell, John (1993). Classical topology and combinatorial group theory. Berlin: Springer-Verlag. ISBN 0-387-97970-0.
Word Processing in Groups Word Processing in Groups is a monograph in mathematics on the theory of automatic groups; these are a type of abstract algebra whose operations are defined by the behavior of finite automata. The book's authors are David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Mike Paterson, and William Thurston. Widely circulated in preprint form, it formed the foundation of the study of automatic groups even before its 1992 publication by Jones and Bartlett Publishers (ISBN 0-86720-244-0).[1][2][3] Topics The book is divided into two parts, one on the basic theory of these structures and another on recent research, connections to geometry and topology, and other related topics.[1] The first part has eight chapters. They cover automata theory and regular languages, and the closure properties of regular languages under logical combinations; the definition of automatic groups and biautomatic groups; examples from topology and "combable" structure in the Cayley graphs of automatic groups; abelian groups and the automaticity of Euclidean groups; the theory of determining whether a group is automatic, and its practical implementation by Epstein, Holt, and Sarah Rees; extensions to asynchronous automata; and nilpotent groups.[1][2][4] The second part has four chapters, on braid groups, isoperimetric inequalities, geometric finiteness, and the fundamental groups of three-dimensional manifolds.[1][4] Audience and reception Although not primarily a textbook, the first part of the book could be used as the basis for a graduate course.[1][4] More generally, reviewer Gilbert Baumslag recommends it "very strongly to everyone who is interested in either group theory or topology, as well as to computer scientists." Baumslag was an expert in a related but older area of study, groups defined by finite presentations, in which research was eventually stymied by the phenomenon that many basic problems are undecidable. Despite tracing the origins of automatic groups to early 20th-century mathematician Max Dehn, he writes that the book studies "a strikingly new class of groups" that "conjures up the fascinating possibility that some of the exploration of these automatic groups can be carried out by means of high-speed computers" and that the book is "very likely to have a great impact".[2] Reviewer Daniel E. Cohen adds that two features of the book are unusual, and welcome: First, that the mathematical results that it presents all have names, not just numbers, and second, that the cost of the book is low.[3] Years later, in 2009, mathematician Mark V. Lawson wrote that despite its "odd title" the book made automata theory, once the domain of computer scientists, respectable among mathematicians, and that it became part of "a quiet revolution in the diplomatic relations between mathematics and computer science".[5] References 1. Apanasov, B. N., "Review of Word Processing in Groups", zbMATH, Zbl 0764.20017 2. Baumslag, Gilbert (1994), "Review of Word Processing in Groups", Bulletin of the American Mathematical Society, New Series, 31 (1): 86–91, doi:10.1090/S0273-0979-1994-00481-1, MR 1568123 3. Cohen, D. E. (November 1993), "Review of Word Processing in Groups", Bulletin of the London Mathematical Society, 25 (6): 614–616, doi:10.1112/blms/25.6.614 4. Thomas, Richard M. (1993), "Review of Word Processing in Groups", Mathematical Reviews, MR 1161694 5. Lawson, Mark V. (December 2009), "Review of A Second Course in Formal Languages and Automata Theory by Jeffrey Shallit", SIAM Review, 51 (4): 797–799, JSTOR 25662348
Combinatorics on words Combinatorics on words is a fairly new field of mathematics, branching from combinatorics, which focuses on the study of words and formal languages. The subject looks at letters or symbols, and the sequences they form. Combinatorics on words affects various areas of mathematical study, including algebra and computer science. There have been a wide range of contributions to the field. Some of the first work was on square-free words by Axel Thue in the early 1900s. He and colleagues observed patterns within words and tried to explain them. As time went on, combinatorics on words became useful in the study of algorithms and coding. It led to developments in abstract algebra and answering open questions. Definition Combinatorics is an area of discrete mathematics. Discrete mathematics is the study of countable structures. These objects have a definite beginning and end. The study of enumerable objects is the opposite of disciplines such as analysis, where calculus and infinite structures are studied. Combinatorics studies how to count these objects using various representations. Combinatorics on words is a recent development in this field that focuses on the study of words and formal languages. A formal language is any set of symbols and combinations of symbols that people use to communicate information.[1] Some terminology relevant to the study of words should first be explained. First and foremost, a word is basically a sequence of symbols, or letters, in a finite set.[1] One of these sets is known by the general public as the alphabet. For example, the word "encyclopedia" is a sequence of symbols in the English alphabet, a finite set of twenty-six letters. Since a word can be described as a sequence, other basic mathematical descriptions can be applied. The alphabet is a set, so as one would expect, the empty set is a subset. In other words, there exists a unique word of length zero. The length of the word is defined by the number of symbols that make up the sequence, and is denoted by \|w\|.[1] Again looking at the example "encyclopedia", \|w\| = 12, since encyclopedia has twelve letters. The idea of factoring of large numbers can be applied to words, where a factor of a word is a block of consecutive symbols.[1] Thus, "cyclop" is a factor of "encyclopedia". In addition to examining sequences in themselves, another area to consider of combinatorics on words is how they can be represented visually. In mathematics various structures are used to encode data. A common structure used in combinatorics is the tree structure. A tree structure is a graph where the vertices are connected by one line, called a path or edge. Trees may not contain cycles, and may or may not be complete. It is possible to encode a word, since a word is constructed by symbols, and encode the data by using a tree.[1] This gives a visual representation of the object. Major contributions The first books on combinatorics on words that summarize the origins of the subject were written by a group of mathematicians that collectively went by the name of M. Lothaire. Their first book was published in 1983, when combinatorics on words became more widespread.[1] Patterns Patterns within words A main contributor to the development of combinatorics on words was Axel Thue (1863–1922); he researched repetition. Thue's main contribution was the proof of the existence of infinite square-free words. Square-free words do not have adjacent repeated factors.[1] To clarify, "dining" is not square-free since "in" is repeated consecutively, while "baggage" is square-free, its two "ag" factors not being adjacent. Thue proves his conjecture on the existence of infinite square-free words by using substitutions. A substitution is a way to take a symbol and replace it with a word. He uses this technique to describe his other contribution, the Thue–Morse sequence, or Thue–Morse word.[1] Thue wrote two papers on square-free words, the second of which was on the Thue–Morse word. Marston Morse is included in the name because he discovered the same result as Thue did, yet they worked independently. Thue also proved the existence of an overlap-free word. An overlap-free word is when, for two symbols x and y, the pattern xyxyx does not exist within the word. He continues in his second paper to prove a relationship between infinite overlap-free words and square-free words. He takes overlap-free words that are created using two different letters, and demonstrates how they can be transformed into square-free words of three letters using substitution.[1] As was previously described, words are studied by examining the sequences made by the symbols. Patterns are found, and they are able to be described mathematically. Patterns can be either avoidable patterns, or unavoidable. A significant contributor to the work of unavoidable patterns, or regularities, was Frank Ramsey in 1930. His important theorem states that for integers k, m≥2, there exists a least positive integer R(k,m) such that despite how a complete graph is colored with two colors, there will always exist a solid color subgraph of each color.[1] Other contributors to the study of unavoidable patterns include van der Waerden. His theorem states that if the positive integers are partitioned into k classes, then there exists a class c such that c contains an arithmetic progression of some unknown length. An arithmetic progression is a sequence of numbers in which the difference between adjacent numbers remains constant.[1] When examining unavoidable patterns sesquipowers are also studied. For some patterns x,y,z, a sesquipower is of the form x, xyx, xyxzxyx, .... This is another pattern such as square-free, or unavoidable patterns. Coudrain and Schützenberger mainly studied these sesquipowers for group theory applications. In addition, Zimin proved that sesquipowers are all unavoidable. Whether the entire pattern shows up, or only some piece of the sesquipower shows up repetitively, it is not possible to avoid it.[1] Patterns within alphabets Necklaces are constructed from words of circular sequences. They are most frequently used in music and astronomy. Flye Sainte-Marie in 1894 proved there are 22n−1−n binary de Bruijn necklaces of length 2n. A de Bruijn necklace contains factors made of words of length n over a certain number of letters. The words appear only once in the necklace.[1] In 1874, Baudot developed the code that would eventually take the place of Morse code by applying the theory of binary de Bruijn necklaces. The problem continued from Sainte-Marie to Martin in 1934, who began looking at algorithms to make words of the de Bruijn structure. It was then worked on by Posthumus in 1943.[1] Language hierarchy Main article: Chomsky hierarchy Possibly the most applied result in combinatorics on words is the Chomsky hierarchy, developed by Noam Chomsky. He studied formal language in the 1950s.[2] His way of looking at language simplified the subject. He disregards the actual meaning of the word, does not consider certain factors such as frequency and context, and applies patterns of short terms to all length terms. The basic idea of Chomsky's work is to divide language into four levels, or the language hierarchy. The four levels are: regular, context-free, context-sensitive, and computably enumerable or unrestricted.[2] Regular is the least complex while computably enumerable is the most complex. While his work grew out of combinatorics on words, it drastically affected other disciplines, especially computer science.[3] Word types Sturmian words Main article: Sturmian word Sturmian words, created by François Sturm, have roots in combinatorics on words. There exist several equivalent definitions of Sturmian words. For example, an infinite word is Sturmian if and only if it has n+1 distinct factors of length n, for every non-negative integer n.[1] Lyndon word Main article: Lyndon word A Lyndon word is a word over a given alphabet that is written in its simplest and most ordered form out of its respective conjugacy class. Lyndon words are important because for any given Lyndon word x, there exists Lyndon words y and z, with y<z, x=yz. Further, there exists a theorem by Chen, Fox, and Lyndon, that states any word has a unique factorization of Lyndon words, where the factorization words are non-increasing. Due to this property, Lyndon words are used to study algebra, specifically group theory. They form the basis for the idea of commutators.[1] Visual representation Cobham contributed work relating Eugène Prouhet's work with finite automata. A mathematical graph is made of edges and nodes. With finite automata, the edges are labeled with a letter in an alphabet. To use the graph, one starts at a node and travels along the edges to reach a final node. The path taken along the graph forms the word. It is a finite graph because there are a countable number of nodes and edges, and only one path connects two distinct nodes.[1] Gauss codes, created by Carl Friedrich Gauss in 1838, are developed from graphs. Specifically, a closed curve on a plane is needed. If the curve only crosses over itself a finite number of times, then one labels the intersections with a letter from the alphabet used. Traveling along the curve, the word is determined by recording each letter as an intersection is passed. Gauss noticed that the distance between when the same symbol shows up in a word is an even integer.[1] Group theory Walther Franz Anton von Dyck began the work of combinatorics on words in group theory by his published work in 1882 and 1883. He began by using words as group elements. Lagrange also contributed in 1771 with his work on permutation groups.[1] One aspect of combinatorics on words studied in group theory is reduced words. A group is constructed with words on some alphabet including generators and inverse elements, excluding factors that appear of the form aā or āa, for some a in the alphabet. Reduced words are formed when the factors aā, āa are used to cancel out elements until a unique word is reached.[1] Nielsen transformations were also developed. For a set of elements of a free group, a Nielsen transformation is achieved by three transformations; replacing an element with its inverse, replacing an element with the product of itself and another element, and eliminating any element equal to 1. By applying these transformations Nielsen reduced sets are formed. A reduced set means no element can be multiplied by other elements to cancel out completely. There are also connections with Nielsen transformations with Sturmian words.[1] Considered problems One problem considered in the study of combinatorics on words in group theory is the following: for two elements x,y of a semigroup, does x=y modulo the defining relations of x and y. Post and Markov studied this problem and determined it undecidable. Undecidable means the theory cannot be proved.[1] The Burnside question was proved using the existence of an infinite cube-free word. This question asks if a group is finite if the group has a definite number of generators and meets the criteria xn=1, for x in the group.[1] Many word problems are undecidable based on the Post correspondence problem. Any two homomorphisms $g,h$ with a common domain and a common codomain form an instance of the Post correspondence problem, which asks whether there exists a word $w$ in the domain such that $g(w)=h(w)$. Post proved that this problem is undecidable; consequently, any word problem that can be reduced to this basic problem is likewise undecidable.[1] Other applications Combinatorics on words have applications on equations. Makanin proved that it is possible to find a solution for a finite system of equations, when the equations are constructed from words.[1] See also • Fibonacci word • Kolakoski sequence • Levi's lemma • Partial word • Shift space • Word metric • Word problem (computability) • Word problem (mathematics) • Word problem for groups • Young–Fibonacci lattice References 1. Berstel, Jean; Dominique Perrin (April 2007). "The origins of combinatorics on words". European Journal of Combinatorics. 28 (3): 996–1022. doi:10.1016/j.ejc.2005.07.019. 2. Jäger, Gerhard; James Rogers (July 2012). "Formal language theory: refining the Chomsky hierarchy". Philosophical Transactions of the Royal Society B. 367 (1598): 1956–1970. doi:10.1098/rstb.2012.0077. PMC 3367686. PMID 22688632. 3. Morales-Bueno, Rafael; Baena-Garcia, Manuel; Carmona-Cejudo, Jose M.; Castillo, Gladys (2010). "Counting Word Avoiding Factors". Electronic Journal of Mathematics and Technology. 4 (3): 251. Further reading • The origins of combinatorics on words, Jean Berstel, Dominique Perrin, European Journal of Combinatorics 28 (2007) 996–1022 • Combinatorics on words – a tutorial, Jean Berstel and Juhani Karhumäki. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 79:178–228, 2003. • Combinatorics on Words: A New Challenging Topic, Juhani Karhumäki. • Choffrut, Christian; Karhumäki, Juhani (1997). "Combinatorics of words". In Rozenberg, Grzegorz; Salomaa, Arto (eds.). Handbook of formal languages. Vol. 1. Springer. CiteSeerX 10.1.1.54.3135. ISBN 978-3-540-60420-4. • Lothaire, M. (1983), Combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 17, Addison-Wesley Publishing Co., Reading, Mass., ISBN 978-0-201-13516-9, MR 0675953, Zbl 0514.20045 • Lothaire, M. (2002), Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, ISBN 978-0-521-81220-7, MR 1905123, Zbl 1001.68093 • "Infinite words: automata, semigroups, logic and games", Dominique Perrin, Jean Éric Pin, Academic Press, 2004, ISBN 978-0-12-532111-2. • Lothaire, M. (2005), Applied combinatorics on words, Encyclopedia of Mathematics and its Applications, vol. 105, Cambridge University Press, ISBN 978-0-521-84802-2, MR 2165687, Zbl 1133.68067 • "Algorithmic Combinatorics on Partial Words", Francine Blanchet-Sadri, CRC Press, 2008, ISBN 978-1-4200-6092-8. • Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009), Combinatorics on words. Christoffel words and repetitions in words, CRM Monograph Series, vol. 27, Providence, RI: American Mathematical Society, ISBN 978-0-8218-4480-9, Zbl 1161.68043 • "Combinatorics of Compositions and Words", Silvia Heubach, Toufik Mansour, CRC Press, 2009, ISBN 978-1-4200-7267-9. • "Word equations and related topics: 1st international workshop, IWWERT '90, Tübingen, Germany, October 1–3, 1990 : proceedings", Editor: Klaus Ulrich Schulz, Springer, 1992, ISBN 978-3-540-55124-9 • "Jewels of stringology: text algorithms", Maxime Crochemore, Wojciech Rytter, World Scientific, 2003, ISBN 978-981-02-4897-0 • Berthé, Valérie; Rigo, Michel, eds. (2010). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 135. Cambridge: Cambridge University Press. ISBN 978-0-521-51597-9. Zbl 1197.68006. • Berstel, Jean; Reutenauer, Christophe (2011). Noncommutative rational series with applications. Encyclopedia of Mathematics and Its Applications. Vol. 137. Cambridge: Cambridge University Press. ISBN 978-0-521-19022-0. Zbl 1250.68007. • "Patterns in Permutations and Words", Sergey Kitaev, Springer, 2011, ISBN 9783642173325 • "Distribution Modulo One and Diophantine Approximation", Yann Bugeaud, Cambridge University Press, 2012, ISBN 9780521111690 External links Wikimedia Commons has media related to Combinatorics on words. • Jean Berstel's page • Tero Harju's page • Guy Melançon's page
Word problem for groups In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A. Thus one can speak unambiguously of the decidability of the word problem for the finitely generated group G. The related but different uniform word problem for a class K of recursively presented groups is the algorithmic problem of deciding, given as input a presentation P for a group G in the class K and two words in the generators of G, whether the words represent the same element of G. Some authors require the class K to be definable by a recursively enumerable set of presentations. History Throughout the history of the subject, computations in groups have been carried out using various normal forms. These usually implicitly solve the word problem for the groups in question. In 1911 Max Dehn proposed that the word problem was an important area of study in its own right,[1] together with the conjugacy problem and the group isomorphism problem. In 1912 he gave an algorithm that solves both the word and conjugacy problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2.[2] Subsequent authors have greatly extended Dehn's algorithm and applied it to a wide range of group theoretic decision problems.[3][4][5] It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable.[6] It follows immediately that the uniform word problem is also undecidable. A different proof was obtained by William Boone in 1958.[7] The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra. As a result of its unsolvability, several other problems in combinatorial group theory have been shown to be unsolvable as well. It is important to realize that the word problem is in fact solvable for many groups G. For example, polycyclic groups have solvable word problems since the normal form of an arbitrary word in a polycyclic presentation is readily computable; other algorithms for groups may, in suitable circumstances, also solve the word problem, see the Todd–Coxeter algorithm[8] and the Knuth–Bendix completion algorithm.[9] On the other hand, the fact that a particular algorithm does not solve the word problem for a particular group does not show that the group has an unsolvable word problem. For instance Dehn's algorithm does not solve the word problem for the fundamental group of the torus. However this group is the direct product of two infinite cyclic groups and so has a solvable word problem. A more concrete description In more concrete terms, the uniform word problem can be expressed as a rewriting question, for literal strings.[10] For a presentation P of a group G, P will specify a certain number of generators x, y, z, ... for G. We need to introduce one letter for x and another (for convenience) for the group element represented by x−1. Call these letters (twice as many as the generators) the alphabet $\Sigma $ for our problem. Then each element in G is represented in some way by a product abc ... pqr of symbols from $\Sigma $, of some length, multiplied in G. The string of length 0 (null string) stands for the identity element e of G. The crux of the whole problem is to be able to recognise all the ways e can be represented, given some relations. The effect of the relations in G is to make various such strings represent the same element of G. In fact the relations provide a list of strings that can be either introduced where we want, or cancelled out whenever we see them, without changing the 'value', i.e. the group element that is the result of the multiplication. For a simple example, take the presentation {a \| a3}. Writing A for the inverse of a, we have possible strings combining any number of the symbols a and A. Whenever we see aaa, or aA or Aa we may strike these out. We should also remember to strike out AAA; this says that since the cube of a is the identity element of G, so is the cube of the inverse of a. Under these conditions the word problem becomes easy. First reduce strings to the empty string, a, aa, A or AA. Then note that we may also multiply by aaa, so we can convert A to aa and convert AA to a. The result is that the word problem, here for the cyclic group of order three, is solvable. This is not, however, the typical case. For the example, we have a canonical form available that reduces any string to one of length at most three, by decreasing the length monotonically. In general, it is not true that one can get a canonical form for the elements, by stepwise cancellation. One may have to use relations to expand a string many-fold, in order eventually to find a cancellation that brings the length right down. The upshot is, in the worst case, that the relation between strings that says they are equal in G is an Undecidable problem. Examples The following groups have a solvable word problem: • Automatic groups, including: • Finite groups • Negatively curved (aka. hyperbolic) groups • Euclidean groups • Coxeter groups • Braid groups • Geometrically finite groups • Finitely generated free groups • Finitely generated free abelian groups • Polycyclic groups • Finitely generated recursively absolutely presented groups,[11] including: • Finitely presented simple groups. • Finitely presented residually finite groups • One relator groups[12] (this is a theorem of Magnus), including: • Fundamental groups of closed orientable two-dimensional manifolds. • Combable groups • Autostackable groups Examples with unsolvable word problems are also known: • Given a recursively enumerable set A of positive integers that has insoluble membership problem, ⟨a,b,c,d \| anban = cndcn : n ∈ A⟩ is a finitely generated group with a recursively enumerable presentation whose word problem is insoluble[13] • Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem[14] • The number of relators in a finitely presented group with insoluble word problem may be as low as 14 [15] or even 12.[16][17] • An explicit example of a reasonable short presentation with insoluble word problem is given in Collins 1986:[18][19] ${\begin{array}{lllll}\langle &a,b,c,d,e,p,q,r,t,k&\|&&\\&p^{10}a=ap,&pacqr=rpcaq,&ra=ar,&\\&p^{10}b=bp,&p^{2}adq^{2}r=rp^{2}daq^{2},&rb=br,&\\&p^{10}c=cp,&p^{3}bcq^{3}r=rp^{3}cbq^{3},&rc=cr,&\\&p^{10}d=dp,&p^{4}bdq^{4}r=rp^{4}dbq^{4},&rd=dr,&\\&p^{10}e=ep,&p^{5}ceq^{5}r=rp^{5}ecaq^{5},&re=er,&\\&aq^{10}=qa,&p^{6}deq^{6}r=rp^{6}edbq^{6},&pt=tp,&\\&bq^{10}=qb,&p^{7}cdcq^{7}r=rp^{7}cdceq^{7},&qt=tq,&\\&cq^{10}=qc,&p^{8}ca^{3}q^{8}r=rp^{8}a^{3}q^{8},&&\\&dq^{10}=qd,&p^{9}da^{3}q^{9}r=rp^{9}a^{3}q^{9},&&\\&eq^{10}=qe,&a^{-3}ta^{3}k=ka^{-3}ta^{3}&&\rangle \end{array}}$ Partial solution of the word problem The word problem for a recursively presented group can be partially solved in the following sense: Given a recursive presentation P = ⟨X\|R⟩ for a group G, define: $S=\{\langle u,v\rangle :u{\text{ and }}v{\text{ are words in }}X{\text{ and }}u=v{\text{ in }}G\ \}$ then there is a partial recursive function fP such that: $f_{P}(\langle u,v\rangle )={\begin{cases}0&{\text{if}}\ \langle u,v\rangle \in S\\{\text{undefined/does not halt}}\ &{\text{if}}\ \langle u,v\rangle \notin S\end{cases}}$ More informally, there is an algorithm that halts if u=v, but does not do so otherwise. It follows that to solve the word problem for P it is sufficient to construct a recursive function g such that: $g(\langle u,v\rangle )={\begin{cases}0&{\text{if}}\ \langle u,v\rangle \notin S\\{\text{undefined/does not halt}}\ &{\text{if}}\ \langle u,v\rangle \in S\end{cases}}$ However u=v in G if and only if uv−1=1 in G. It follows that to solve the word problem for P it is sufficient to construct a recursive function h such that: $h(x)={\begin{cases}0&{\text{if}}\ x\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ x=1\ {\text{in}}\ G\end{cases}}$ Example The following will be proved as an example of the use of this technique: Theorem: A finitely presented residually finite group has solvable word problem. Proof: Suppose G = ⟨X\|R⟩ is a finitely presented, residually finite group. Let S be the group of all permutations of N, the natural numbers, that fixes all but finitely many numbers then: 1. S is locally finite and contains a copy of every finite group. 2. The word problem in S is solvable by calculating products of permutations. 3. There is a recursive enumeration of all mappings of the finite set X into S. 4. Since G is residually finite, if w is a word in the generators X of G then w ≠ 1 in G if and only of some mapping of X into S induces a homomorphism such that w ≠ 1 in S. Given these facts, algorithm defined by the following pseudocode: For every mapping of X into S If every relator in R is satisfied in S If w ≠ 1 in S return 0 End if End if End for defines a recursive function h such that: $h(x)={\begin{cases}0&{\text{if}}\ x\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ x=1\ {\text{in}}\ G\end{cases}}$ This shows that G has solvable word problem. Unsolvability of the uniform word problem The criterion given above, for the solvability of the word problem in a single group, can be extended by a straightforward argument. This gives the following criterion for the uniform solvability of the word problem for a class of finitely presented groups: To solve the uniform word problem for a class K of groups, it is sufficient to find a recursive function $f(P,w)$ that takes a finite presentation P for a group G and a word $w$ in the generators of G, such that whenever G ∈ K: $f(P,w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ G\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ G\end{cases}}$ Boone-Rogers Theorem: There is no uniform partial algorithm that solves the word problem in all finitely presented groups with solvable word problem. In other words, the uniform word problem for the class of all finitely presented groups with solvable word problem is unsolvable. This has some interesting consequences. For instance, the Higman embedding theorem can be used to construct a group containing an isomorphic copy of every finitely presented group with solvable word problem. It seems natural to ask whether this group can have solvable word problem. But it is a consequence of the Boone-Rogers result that: Corollary: There is no universal solvable word problem group. That is, if G is a finitely presented group that contains an isomorphic copy of every finitely presented group with solvable word problem, then G itself must have unsolvable word problem. Remark: Suppose G = ⟨X\|R⟩ is a finitely presented group with solvable word problem and H is a finite subset of G. Let H* = ⟨H⟩, be the group generated by H. Then the word problem in H* is solvable: given two words h, k in the generators H of H, write them as words in X and compare them using the solution to the word problem in G. It is easy to think that this demonstrates a uniform solution of the word problem for the class K (say) of finitely generated groups that can be embedded in G. If this were the case, the non-existence of a universal solvable word problem group would follow easily from Boone-Rogers. However, the solution just exhibited for the word problem for groups in K is not uniform. To see this, consider a group J = ⟨Y\|T⟩ ∈ K; in order to use the above argument to solve the word problem in J, it is first necessary to exhibit a mapping e: Y → G that extends to an embedding e: J → G. If there were a recursive function that mapped (finitely generated) presentations of groups in K to embeddings into G, then a uniform solution of the word problem in K could indeed be constructed. But there is no reason, in general, to suppose that such a recursive function exists. However, it turns out that, using a more sophisticated argument, the word problem in J can be solved without using an embedding e: J → G. Instead an enumeration of homomorphisms is used, and since such an enumeration can be constructed uniformly, it results in a uniform solution to the word problem in K. Proof that there is no universal solvable word problem group Suppose G were a universal solvable word problem group. Given a finite presentation P = ⟨X\|R⟩ of a group H, one can recursively enumerate all homomorphisms h: H → G by first enumerating all mappings h†: X → G. Not all of these mappings extend to homomorphisms, but, since h†(R) is finite, it is possible to distinguish between homomorphisms and non-homomorphisms, by using the solution to the word problem in G. "Weeding out" non-homomorphisms gives the required recursive enumeration: h1, h2, ..., hn, ... . If H has solvable word problem, then at least one of these homomorphisms must be an embedding. So given a word w in the generators of H: ${\text{If}}\ w\neq 1\ {\text{in}}\ H,\ h_{n}(w)\neq 1\ {\text{in}}\ G\ {\text{for some}}\ h_{n}$ ${\text{If}}\ w=1\ {\text{in}}\ H,\ h_{n}(w)=1\ {\text{in}}\ G\ {\text{for all}}\ h_{n}$ Consider the algorithm described by the pseudocode: Let n = 0 Let repeatable = TRUE while (repeatable) increase n by 1 if (solution to word problem in G reveals hn(w) ≠ 1 in G) Let repeatable = FALSE output 0. This describes a recursive function: $f(w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ H\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ H.\end{cases}}$ The function f clearly depends on the presentation P. Considering it to be a function of the two variables, a recursive function $f(P,w)$ has been constructed that takes a finite presentation P for a group H and a word w in the generators of a group G, such that whenever G has soluble word problem: $f(P,w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ H\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ H.\end{cases}}$ But this uniformly solves the word problem for the class of all finitely presented groups with solvable word problem, contradicting Boone-Rogers. This contradiction proves G cannot exist. Algebraic structure and the word problem There are a number of results that relate solvability of the word problem and algebraic structure. The most significant of these is the Boone-Higman theorem: A finitely presented group has solvable word problem if and only if it can be embedded in a simple group that can be embedded in a finitely presented group. It is widely believed that it should be possible to do the construction so that the simple group itself is finitely presented. If so one would expect it to be difficult to prove as the mapping from presentations to simple groups would have to be non-recursive. The following has been proved by Bernhard Neumann and Angus Macintyre: A finitely presented group has solvable word problem if and only if it can be embedded in every algebraically closed group What is remarkable about this is that the algebraically closed groups are so wild that none of them has a recursive presentation. The oldest result relating algebraic structure to solvability of the word problem is Kuznetsov's theorem: A recursively presented simple group S has solvable word problem. To prove this let ⟨X\|R⟩ be a recursive presentation for S. Choose a ∈ S such that a ≠ 1 in S. If w is a word on the generators X of S, then let: $S_{w}=\langle X\|R\cup \{w\}\rangle .$ There is a recursive function $f_{\langle X\|R\cup \{w\}\rangle }$ such that: $f_{\langle X\|R\cup \{w\}\rangle }(x)={\begin{cases}0&{\text{if}}\ x=1\ {\text{in}}\ S_{w}\\{\text{undefined/does not halt}}\ &{\text{if}}\ x\neq 1\ {\text{in}}\ S_{w}.\end{cases}}$ Write: $g(w,x)=f_{\langle X\|R\cup \{w\}\rangle }(x).$ Then because the construction of f was uniform, this is a recursive function of two variables. It follows that: $h(w)=g(w,a)$ is recursive. By construction: $h(w)={\begin{cases}0&{\text{if}}\ a=1\ {\text{in}}\ S_{w}\\{\text{undefined/does not halt}}\ &{\text{if}}\ a\neq 1\ {\text{in}}\ S_{w}.\end{cases}}$ Since S is a simple group, its only quotient groups are itself and the trivial group. Since a ≠ 1 in S, we see a = 1 in Sw if and only if Sw is trivial if and only if w ≠ 1 in S. Therefore: $h(w)={\begin{cases}0&{\text{if}}\ w\neq 1\ {\text{in}}\ S\\{\text{undefined/does not halt}}\ &{\text{if}}\ w=1\ {\text{in}}\ S.\end{cases}}$ The existence of such a function is sufficient to prove the word problem is solvable for S. This proof does not prove the existence of a uniform algorithm for solving the word problem for this class of groups. The non-uniformity resides in choosing a non-trivial element of the simple group. There is no reason to suppose that there is a recursive function that maps a presentation of a simple groups to a non-trivial element of the group. However, in the case of a finitely presented group we know that not all the generators can be trivial (Any individual generator could be, of course). Using this fact it is possible to modify the proof to show: The word problem is uniformly solvable for the class of finitely presented simple groups. See also • Combinatorics on words • SQ-universal group • Word problem (mathematics) • Reachability problem • Nested stack automata (have been used to solve the word problem for groups) Notes 1. Dehn 1911. 2. Dehn 1912. 3. Greendlinger, Martin (June 1959), "Dehn's algorithm for the word problem", Communications on Pure and Applied Mathematics, 13 (1): 67–83, doi:10.1002/cpa.3160130108. 4. Lyndon, Roger C. (September 1966), "On Dehn's algorithm", Mathematische Annalen, 166 (3): 208–228, doi:10.1007/BF01361168, hdl:2027.42/46211, S2CID 36469569. 5. Schupp, Paul E. (June 1968), "On Dehn's algorithm and the conjugacy problem", Mathematische Annalen, 178 (2): 119–130, doi:10.1007/BF01350654, S2CID 120429853. 6. Novikov, P. S. (1955), "On the algorithmic unsolvability of the word problem in group theory", Proceedings of the Steklov Institute of Mathematics (in Russian), 44: 1–143, Zbl 0068.01301 7. Boone, William W. (1958), "The word problem" (PDF), Proceedings of the National Academy of Sciences, 44 (10): 1061–1065, Bibcode:1958PNAS...44.1061B, doi:10.1073/pnas.44.10.1061, PMC 528693, PMID 16590307, Zbl 0086.24701 8. Todd, J.; Coxeter, H.S.M. (1936). "A practical method for enumerating cosets of a finite abstract group". Proceedings of the Edinburgh Mathematical Society. 5 (1): 26–34. doi:10.1017/S0013091500008221. 9. Knuth, D.; Bendix, P. (2014) [1970]. "Simple word problems in universal algebras". In Leech, J. (ed.). Computational Problems in Abstract Algebra: Proceedings of a Conference Held at Oxford Under the Auspices of the Science Research Council Atlas Computer Laboratory, 29th August to 2nd September 1967. Springer. pp. 263–297. ISBN 9781483159423. 10. Rotman 1994. 11. Simmons, H. (1973). "The word problem for absolute presentations". J. London Math. Soc. s2-6 (2): 275–280. doi:10.1112/jlms/s2-6.2.275. 12. Lyndon, Roger C.; Schupp, Paul E (2001). Combinatorial Group Theory. Springer. pp. 1–60. ISBN 9783540411581. 13. Collins & Zieschang 1990, p. 149. 14. Collins & Zieschang 1993, Cor. 7.2.6. sfn error: no target: CITEREFCollinsZieschang1993 (help) 15. Collins 1969. 16. Borisov 1969. 17. Collins 1972. 18. Collins 1986. 19. We use the corrected version from John Pedersen's A Catalogue of Algebraic Systems References • Boone, W.W.; Cannonito, F.B.; Lyndon, Roger C. (1973). Word problems : decision problems and the Burnside problem in group theory. Studies in logic and the foundations of mathematics. Vol. 71. North-Holland. ISBN 9780720422719. • Boone, W. W.; Higman, G. (1974). "An algebraic characterization of the solvability of the word problem". J. Austral. Math. Soc. 18: 41–53. doi:10.1017/s1446788700019108. • Boone, W. W.; Rogers Jr, H. (1966). "On a problem of J. H. C. Whitehead and a problem of Alonzo Church". Math. Scand. 19: 185–192. doi:10.7146/math.scand.a-10808. • Borisov, V. V. (1969), "Simple examples of groups with unsolvable word problem", Akademiya Nauk SSSR. Matematicheskie Zametki, 6: 521–532, ISSN 0025-567X, MR 0260851 • Collins, Donald J. (1969), "Word and conjugacy problems in groups with only a few defining relations", Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 15 (20–22): 305–324, doi:10.1002/malq.19690152001, MR 0263903 • Collins, Donald J. (1972), "On a group embedding theorem of V. V. Borisov", Bulletin of the London Mathematical Society, 4 (2): 145–147, doi:10.1112/blms/4.2.145, ISSN 0024-6093, MR 0314998 • Collins, Donald J. (1986), "A simple presentation of a group with unsolvable word problem", Illinois Journal of Mathematics, 30 (2): 230–234, doi:10.1215/ijm/1256044631, ISSN 0019-2082, MR 0840121 • Collins, Donald J.; Zieschang, H. (1990), Combinatorial group theory and fundamental groups, Springer-Verlag, p. 166, MR 1099152 • Dehn, Max (1911), "Über unendliche diskontinuierliche Gruppen", Mathematische Annalen, 71 (1): 116–144, doi:10.1007/BF01456932, ISSN 0025-5831, MR 1511645, S2CID 123478582 • Dehn, Max (1912), "Transformation der Kurven auf zweiseitigen Flächen", Mathematische Annalen, 72 (3): 413–421, doi:10.1007/BF01456725, ISSN 0025-5831, MR 1511705, S2CID 122988176 • Kuznetsov, A.V. (1958). "Algorithms as operations in algebraic systems". Izvestia Akad. Nauk SSSR Ser Mat. 13 (3): 81. • Miller, C.F. (1991). "Decision problems for groups — survey and reflections". Algorithms and Classification in Combinatorial Group Theory. Mathematical Sciences Research Institute Publications. Vol. 23. Springer. pp. 1–60. doi:10.1007/978-1-4613-9730-4_1. ISBN 978-1-4613-9730-4. • Nyberg-Brodda, Carl-Fredrik (2021), "The word problem for one-relation monoids: a survey", Semigroup Forum, 103 (2): 297–355, arXiv:2105.02853, doi:10.1007/s00233-021-10216-8 • Rotman, Joseph (1994), An introduction to the theory of groups, Springer-Verlag, ISBN 978-0-387-94285-8 • Stillwell, J. (1982). "The word problem and the isomorphism problem for groups". Bulletin of the AMS. 6: 33–56. doi:10.1090/s0273-0979-1982-14963-1.
Word problem (mathematics) In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable.[1] Background and motivation In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a solution to the word problem. For example, imagine that $x,y,z$ are symbols representing real numbers - then a relevant solution to the word problem would, given the input $(x\cdot y)/z\mathrel {\overset {?}{=}} (x/z)\cdot y$, produce the output EQUAL, and similarly produce NOT_EQUAL from $(x\cdot y)/z\mathrel {\overset {?}{=}} (x/x)\cdot y$. The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an equivalence class of expressions to a single encoding known as the normal form - the word problem is then solved by comparing these normal forms via syntactic equality.[1] For example one might decide that $x\cdot y\cdot z^{-1}$ is the normal form of $(x\cdot y)/z$, $(x/z)\cdot y$, and $(y/z)\cdot x$, and devise a transformation system to rewrite those expressions to that form, in the process proving that all equivalent expressions will be rewritten to the same normal form.[2] But not all solutions to the word problem use a normal form theorem - there are algebraic properties which indirectly imply the existence of an algorithm.[1] While the word problem asks whether two terms containing constants are equal, a proper extension of the word problem known as the unification problem asks whether two terms $t_{1},t_{2}$ containing variables have instances that are equal, or in other words whether the equation $t_{1}=t_{2}$ has any solutions. As a common example, $2+3\mathrel {\overset {?}{=}} 8+(-3)$ is a word problem in the integer group ℤ, while $2+x\mathrel {\overset {?}{=}} 8+(-x)$ is a unification problem in the same group; since the former terms happen to be equal in ℤ, the latter problem has the substitution $\{x\mapsto 3\}$ as a solution. History One of the most deeply studied cases of the word problem is in the theory of semigroups and groups. A timeline of papers relevant to the Novikov-Boone theorem is as follows:[3][4] • 1910 (1910): Axel Thue poses a general problem of term rewriting on tree-like structures. He states "A solution of this problem in the most general case may perhaps be connected with unsurmountable difficulties".[5][6] • 1911 (1911): Max Dehn poses the word problem for finitely presented groups.[7] • 1912 (1912): Dehn presents Dehn's algorithm, and proves it solves the word problem for the fundamental groups of closed orientable two-dimensional manifolds of genus greater than or equal to 2.[8] Subsequent authors have greatly extended it to a wide range of group-theoretic decision problems.[9][10][11] • 1914 (1914): Axel Thue poses the word problem for finitely presented semigroups.[12] • 1930 (1930) – 1938 (1938): The Church-Turing thesis emerges, defining formal notions of computability and undecidability.[13] • 1947 (1947): Emil Post and Andrey Markov Jr. independently construct finitely presented semigroups with unsolvable word problem.[14][15] Post's construction is built on Turing machines while Markov's uses Post's normal systems.[3] • 1950 (1950): Alan Turing shows the word problem for cancellation semigroups is unsolvable,[16] by furthering Post’s construction. The proof is difficult to follow but marks a turning point in the word problem for groups.[3]: 342 • 1955 (1955): Pyotr Novikov gives the first published proof that the word problem for groups is unsolvable, using Turing’s cancellation semigroup result.[17][3]: 354 The proof contains a "Principal Lemma" equivalent to Britton's Lemma.[3]: 355 • 1954 (1954) – 1957 (1957): William Boone independently shows the word problem for groups is unsolvable, using Post's semigroup construction.[18][19] • 1957 (1957) – 1958 (1958): John Britton gives another proof that the word problem for groups is unsolvable, based on Turing's cancellation semigroups result and some of Britton's earlier work.[20] An early version of Britton's Lemma appears.[3]: 355 • 1958 (1958) – 1959 (1959): Boone publishes a simplified version of his construction.[21][22] • 1961 (1961): Graham Higman characterises the subgroups of finitely presented groups with Higman's embedding theorem,[23] connecting recursion theory with group theory in an unexpected way and giving a very different proof of the unsolvability of the word problem.[3] • 1961 (1961) – 1963 (1963): Britton presents a greatly simplified version of Boone's 1959 proof that the word problem for groups is unsolvable.[24] It uses a group-theoretic approach, in particular Britton's Lemma. This proof has been used in a graduate course, although more modern and condensed proofs exist.[25] • 1977 (1977): Gennady Makanin proves that the existential theory of equations over free monoids is solvable.[26] The word problem for semi-Thue systems The accessibility problem for string rewriting systems (semi-Thue systems or semigroups) can be stated as follows: Given a semi-Thue system $T:=(\Sigma ,R)$ and two words (strings) $u,v\in \Sigma ^{}$, can $u$ be transformed into $v$ by applying rules from $R$? Note that the rewriting here is one-way. The word problem is the accessibility problem for symmetric rewrite relations, i.e. Thue systems.[27] The accessibility and word problems are undecidable, i.e. there is no general algorithm for solving this problem.[28] This even holds if we limit the systems to have finite presentations, i.e. a finite set of symbols and a finite set of relations on those symbols.[27] Even the word problem restricted to ground terms is not decidable for certain finitely presented semigroups.[29][30] The word problem for groups Main article: Word problem for groups Given a presentation $\langle S\mid {\mathcal {R}}\rangle $ for a group G, the word problem is the algorithmic problem of deciding, given as input two words in S, whether they represent the same element of G. The word problem is one of three algorithmic problems for groups proposed by Max Dehn in 1911. It was shown by Pyotr Novikov in 1955 that there exists a finitely presented group G such that the word problem for G is undecidable.[31] The word problem in combinatorial calculus and lambda calculus Main article: Combinatory logic § Undecidability of combinatorial calculus One of the earliest proofs that a word problem is undecidable was for combinatory logic: when are two strings of combinators equivalent? Because combinators encode all possible Turing machines, and the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable. Alonzo Church observed this in 1936.[32] Likewise, one has essentially the same problem in (untyped) lambda calculus: given two distinct lambda expressions, there is no algorithm which can discern whether they are equivalent or not; equivalence is undecidable. For several typed variants of the lambda calculus, equivalence is decidable by comparison of normal forms. The word problem for abstract rewriting systems The word problem for an abstract rewriting system (ARS) is quite succinct: given objects x and y are they equivalent under ${\stackrel {}{\leftrightarrow }}$?[29] The word problem for an ARS is undecidable in general. However, there is a computable solution for the word problem in the specific case where every object reduces to a unique normal form in a finite number of steps (i.e. the system is convergent): two objects are equivalent under ${\stackrel {*}{\leftrightarrow }}$ if and only if they reduce to the same normal form.[33] The Knuth-Bendix completion algorithm can be used to transform a set of equations into a convergent term rewriting system. The word problem in universal algebra In universal algebra one studies algebraic structures consisting of a generating set A, a collection of operations on A of finite arity, and a finite set of identities that these operations must satisfy. The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras.[1] The word problem on free Heyting algebras is difficult.[34] The only known results are that the free Heyting algebra on one generator is infinite, and that the free complete Heyting algebra on one generator exists (and has one more element than the free Heyting algebra). The word problem for free lattices Example computation of x∧z ~ x∧z∧(x∨y) x∧z∧(x∨y)≤~x∧z by 5. since x∧z≤~x∧z by 1. since x∧z=x∧z x∧z≤~x∧z∧(x∨y) by 7. since x∧z≤~x∧z and x∧z≤~x∨y by 1. since x∧z=x∧z by 6. since x∧z≤~x by 5. since x≤~x by 1. since x=x The word problem on free lattices and more generally free bounded lattices has a decidable solution. Bounded lattices are algebraic structures with the two binary operations ∨ and ∧ and the two constants (nullary operations) 0 and 1. The set of all well-formed expressions that can be formulated using these operations on elements from a given set of generators X will be called W(X). This set of words contains many expressions that turn out to denote equal values in every lattice. For example, if a is some element of X, then a ∨ 1 = 1 and a ∧ 1 = a. The word problem for free bounded lattices is the problem of determining which of these elements of W(X) denote the same element in the free bounded lattice FX, and hence in every bounded lattice. The word problem may be resolved as follows. A relation ≤~ on W(X) may be defined inductively by setting w ≤~ v if and only if one of the following holds: 1. w = v (this can be restricted to the case where w and v are elements of X), 2. w = 0, 3. v = 1, 4. w = w1 ∨ w2 and both w1 ≤~ v and w2 ≤~ v hold, 5. w = w1 ∧ w2 and either w1 ≤~ v or w2 ≤~ v holds, 6. v = v1 ∨ v2 and either w ≤~ v1 or w ≤~ v2 holds, 7. v = v1 ∧ v2 and both w ≤~ v1 and w ≤~ v2 hold. This defines a preorder ≤~ on W(X), so an equivalence relation can be defined by w ~ v when w ≤~ v and v ≤~ w. One may then show that the partially ordered quotient set W(X)/~ is the free bounded lattice FX.[35][36] The equivalence classes of W(X)/~ are the sets of all words w and v with w ≤~ v and v ≤~ w. Two well-formed words v and w in W(X) denote the same value in every bounded lattice if and only if w ≤~ v and v ≤~ w; the latter conditions can be effectively decided using the above inductive definition. The table shows an example computation to show that the words x∧z and x∧z∧(x∨y) denote the same value in every bounded lattice. The case of lattices that are not bounded is treated similarly, omitting rules 2 and 3 in the above construction of ≤~. Example: A term rewriting system to decide the word problem in the free group Bläsius and Bürckert [37] demonstrate the Knuth–Bendix algorithm on an axiom set for groups. The algorithm yields a confluent and noetherian term rewrite system that transforms every term into a unique normal form.[38] The rewrite rules are numbered incontiguous since some rules became redundant and were deleted during the algorithm run. The equality of two terms follows from the axioms if and only if both terms are transformed into literally the same normal form term. For example, the terms $((a^{-1}\cdot a)\cdot (b\cdot b^{-1}))^{-1}\mathrel {\overset {R2}{\rightsquigarrow }} (1\cdot (b\cdot b^{-1}))^{-1}\mathrel {\overset {R13}{\rightsquigarrow }} (1\cdot 1)^{-1}\mathrel {\overset {R1}{\rightsquigarrow }} 1^{-1}\mathrel {\overset {R8}{\rightsquigarrow }} 1$, and $b\cdot ((a\cdot b)^{-1}\cdot a)\mathrel {\overset {R17}{\rightsquigarrow }} b\cdot ((b^{-1}\cdot a^{-1})\cdot a)\mathrel {\overset {R3}{\rightsquigarrow }} b\cdot (b^{-1}\cdot (a^{-1}\cdot a))\mathrel {\overset {R2}{\rightsquigarrow }} b\cdot (b^{-1}\cdot 1)\mathrel {\overset {R11}{\rightsquigarrow }} b\cdot b^{-1}\mathrel {\overset {R13}{\rightsquigarrow }} 1$ share the same normal form, viz. $1$; therefore both terms are equal in every group. As another example, the term $1\cdot (a\cdot b)$ and $b\cdot (1\cdot a)$ has the normal form $a\cdot b$ and $b\cdot a$, respectively. Since the normal forms are literally different, the original terms cannot be equal in every group. In fact, they are usually different in non-abelian groups. Group axioms used in Knuth–Bendix completion A1$1\cdot x$$=x$ A2$x^{-1}\cdot x$$=1$ A3 $(x\cdot y)\cdot z$$=x\cdot (y\cdot z)$ Term rewrite system obtained from Knuth–Bendix completion R1$1\cdot x$$\rightsquigarrow x$ R2$x^{-1}\cdot x$$\rightsquigarrow 1$ R3$(x\cdot y)\cdot z$$\rightsquigarrow x\cdot (y\cdot z)$ R4$x^{-1}\cdot (x\cdot y)$$\rightsquigarrow y$ R8$1^{-1}$$\rightsquigarrow 1$ R11$x\cdot 1$$\rightsquigarrow x$ R12$(x^{-1})^{-1}$$\rightsquigarrow x$ R13$x\cdot x^{-1}$$\rightsquigarrow 1$ R14$x\cdot (x^{-1}\cdot y)$$\rightsquigarrow y$ R17 $(x\cdot y)^{-1}$$\rightsquigarrow y^{-1}\cdot x^{-1}$ See also • Conjugacy problem • Group isomorphism problem References 1. Evans, Trevor (1978). "Word problems". Bulletin of the American Mathematical Society. 84 (5): 790. doi:10.1090/S0002-9904-1978-14516-9. 2. Cohen, Joel S. (2002). Computer algebra and symbolic computation: elementary algorithms. Natick, Mass.: A K Peters. pp. 90–92. ISBN 1568811586. 3. Miller, Charles F. (2014). Downey, Rod (ed.). "Turing machines to word problems" (PDF). Turing's Legacy: 330. doi:10.1017/CBO9781107338579.010. hdl:11343/51723. ISBN 9781107338579. Retrieved 6 December 2021. 4. Stillwell, John (1982). "The word problem and the isomorphism problem for groups". Bulletin of the American Mathematical Society. 6 (1): 33–56. doi:10.1090/S0273-0979-1982-14963-1. 5. Müller-Stach, Stefan (12 September 2021). "Max Dehn, Axel Thue, and the Undecidable". p. 13. arXiv:1703.09750 [math.HO]. 6. Steinby, Magnus; Thomas, Wolfgang (2000). "Trees and term rewriting in 1910: on a paper by Axel Thue". Bulletin of the European Association for Theoretical Computer Science. 72: 256–269. CiteSeerX 10.1.1.32.8993. MR 1798015. 7. Dehn, Max (1911). "Über unendliche diskontinuierliche Gruppen". Mathematische Annalen. 71 (1): 116–144. doi:10.1007/BF01456932. ISSN 0025-5831. MR 1511645. S2CID 123478582. 8. Dehn, Max (1912). "Transformation der Kurven auf zweiseitigen Flächen". Mathematische Annalen. 72 (3): 413–421. doi:10.1007/BF01456725. ISSN 0025-5831. MR 1511705. S2CID 122988176. 9. Greendlinger, Martin (June 1959). "Dehn's algorithm for the word problem". Communications on Pure and Applied Mathematics. 13 (1): 67–83. doi:10.1002/cpa.3160130108. 10. Lyndon, Roger C. (September 1966). "On Dehn's algorithm". Mathematische Annalen. 166 (3): 208–228. doi:10.1007/BF01361168. hdl:2027.42/46211. S2CID 36469569. 11. Schupp, Paul E. (June 1968). "On Dehn's algorithm and the conjugacy problem". Mathematische Annalen. 178 (2): 119–130. doi:10.1007/BF01350654. S2CID 120429853. 12. Power, James F. (27 August 2013). "Thue's 1914 paper: a translation". arXiv:1308.5858 [cs.FL]. 13. See History of the Church–Turing thesis. The dates are based on On Formally Undecidable Propositions of Principia Mathematica and Related Systems and Systems of Logic Based on Ordinals. 14. Post, Emil L. (March 1947). "Recursive Unsolvability of a problem of Thue" (PDF). Journal of Symbolic Logic. 12 (1): 1–11. doi:10.2307/2267170. JSTOR 2267170. S2CID 30320278. Retrieved 6 December 2021. 15. Mostowski, Andrzej (September 1951). "A. Markov. Névožmoinost' nékotoryh algoritmov v téorii associativnyh sistém (Impossibility of certain algorithms in the theory of associative systems). Doklady Akadémii Nauk SSSR, vol. 77 (1951), pp. 19–20". Journal of Symbolic Logic. 16 (3): 215. doi:10.2307/2266407. JSTOR 2266407. 16. Turing, A. M. (September 1950). "The Word Problem in Semi-Groups With Cancellation". The Annals of Mathematics. 52 (2): 491–505. doi:10.2307/1969481. JSTOR 1969481. 17. Novikov, P. S. (1955). "On the algorithmic unsolvability of the word problem in group theory". Proceedings of the Steklov Institute of Mathematics (in Russian). 44: 1–143. Zbl 0068.01301. 18. Boone, William W. (1954). "Certain Simple, Unsolvable Problems of Group Theory. I". Indagationes Mathematicae (Proceedings). 57: 231–237. doi:10.1016/S1385-7258(54)50033-8. 19. Boone, William W. (1957). "Certain Simple, Unsolvable Problems of Group Theory. VI". Indagationes Mathematicae (Proceedings). 60: 227–232. doi:10.1016/S1385-7258(57)50030-9. 20. Britton, J. L. (October 1958). "The Word Problem for Groups". Proceedings of the London Mathematical Society. s3-8 (4): 493–506. doi:10.1112/plms/s3-8.4.493. 21. Boone, William W. (1958). "The word problem" (PDF). Proceedings of the National Academy of Sciences. 44 (10): 1061–1065. Bibcode:1958PNAS...44.1061B. doi:10.1073/pnas.44.10.1061. PMC 528693. PMID 16590307. Zbl 0086.24701. 22. Boone, William W. (September 1959). "The Word Problem". The Annals of Mathematics. 70 (2): 207–265. doi:10.2307/1970103. JSTOR 1970103. 23. Higman, G. (8 August 1961). "Subgroups of finitely presented groups". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 262 (1311): 455–475. Bibcode:1961RSPSA.262..455H. doi:10.1098/rspa.1961.0132. S2CID 120100270. 24. Britton, John L. (January 1963). "The Word Problem". The Annals of Mathematics. 77 (1): 16–32. doi:10.2307/1970200. JSTOR 1970200. 25. Simpson, Stephen G. (18 May 2005). "A Slick Proof of the Unsolvability of the Word Problem for Finitely Presented Groups" (PDF). Retrieved 6 December 2021. 26. "Subgroups of finitely presented groups". Mathematics of the USSR-Sbornik. 103 (145): 147–236. 13 February 1977. doi:10.1070/SM1977v032n02ABEH002376. 27. Matiyasevich, Yuri; Sénizergues, Géraud (January 2005). "Decision problems for semi-Thue systems with a few rules". Theoretical Computer Science. 330 (1): 145–169. doi:10.1016/j.tcs.2004.09.016. 28. Davis, Martin (1978). "What is a Computation?" (PDF). Mathematics Today Twelve Informal Essays: 257–259. doi:10.1007/978-1-4613-9435-8_10. ISBN 978-1-4613-9437-2. Retrieved 5 December 2021. 29. Baader, Franz; Nipkow, Tobias (5 August 1999). Term Rewriting and All That. Cambridge University Press. pp. 59–60. ISBN 978-0-521-77920-3. • Matiyasevich, Yu. V. (1967). "Простые примеры неразрешимых ассоциативных исчислений" [Simple examples of undecidable associative calculi]. Doklady Akademii Nauk SSSR (in Russian). 173 (6): 1264–1266. ISSN 0869-5652. • Matiyasevich, Yu. V. (1967). "Simple examples of undecidable associative calculi". Soviet Mathematics. 8 (2): 555–557. ISSN 0197-6788. 30. Novikov, P. S. (1955). "On the algorithmic unsolvability of the word problem in group theory". Trudy Mat. Inst. Steklov (in Russian). 44: 1–143. 31. Statman, Rick (2000). "On the Word Problem for Combinators". Rewriting Techniques and Applications. Lecture Notes in Computer Science. 1833: 203–213. doi:10.1007/10721975_14. ISBN 978-3-540-67778-9. 32. Beke, Tibor (May 2011). "Categorification, term rewriting and the Knuth–Bendix procedure". Journal of Pure and Applied Algebra. 215 (5): 730. doi:10.1016/j.jpaa.2010.06.019. 33. Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, Cambridge, ISBN 0-521-23893-5. (See chapter 1, paragraph 4.11) 34. Whitman, Philip M. (January 1941). "Free Lattices". The Annals of Mathematics. 42 (1): 325–329. doi:10.2307/1969001. JSTOR 1969001. 35. Whitman, Philip M. (1942). "Free Lattices II". Annals of Mathematics. 43 (1): 104–115. doi:10.2307/1968883. JSTOR 1968883. 36. K. H. Bläsius and H.-J. Bürckert, ed. (1992). Deduktionsssysteme. Oldenbourg. p. 291.; here: p.126, 134 37. Apply rules in any order to a term, as long as possible; the result doesn't depend on the order; it is the term's normal form. Authority control International • FAST National • Germany • 2 • Israel • United States
Ambigram An ambigram is a calligraphic or typographic design with multiple interpretations as written words.[2] Alternative meanings are often yielded when the design is transformed or the observer moves, but they can also result from a shift in mental perspective.[3] The term was coined by Douglas Hofstadter in 1983–1984.[3][4] Most often, ambigrams appear as visually symmetrical words. When flipped, they remain unchanged, or they mutate to reveal another meaning. "Half-turn" ambigrams undergo a point reflection (180-degree rotational symmetry) and can be read upside down, while mirror ambigrams have axial symmetry and can be read through a reflective surface like a mirror. Many other types of ambigrams exist.[5] Ambigrams can be constructed in various languages and alphabets, and the notion often extends to numbers and other symbols. It is a recent interdisciplinary concept, combining art, literature, mathematics, cognition, and optical illusions. Drawing symmetrical words constitutes also a recreational activity for amateurs. Numerous ambigram logos are famous, and ambigram tattoos have become increasingly popular. There are methods to design an ambigram, a field in which some artists have become specialists. Etymology The word ambigram was coined in 1983 by Douglas Hofstadter, an American scholar of cognitive science best known as the Pulitzer Prize-winning author of the book Gödel, Escher, Bach.[6][2][4] Hofstadter describes ambigrams as "calligraphic designs that manage to squeeze in two different readings."[7] "The essence is imbuing a single written form with ambiguity".[8] An ambigram is a visual pun of a special kind: a calligraphic design having two or more (clear) interpretations as written words. One can voluntarily jump back and forth between the rival readings usually by shifting one's physical point of view (moving the design in some way) but sometimes by simply altering one's perceptual bias towards a design (clicking an internal mental switch, so to speak). Sometimes the readings will say identical things, sometimes they will say different things.[2] — Douglas Hofstadter Hofstadter attributed the origin of the word ambigram to conversations among a small group of friends during 1983–1984.[4] Prior to Hofstadter's terminology, other names were used to refer to ambigrams. Among them, the expressions "vertical palindromes" by Dmitri Borgmann[9] (1965) and Georges Perec,[10][11] "designatures" (1979),[12] "inversions" (1980) by Scott Kim,[13][14] or simply "upside-down words" by John Langdon and Robert Petrick.[14] Ambigram was added to the Oxford English Dictionary in March 2011,[5][15] and to the Merriam-Webster dictionary in September 2020.[3][16] Scrabble included the word in its database in November 2022.[17][18][19] History Many ambigrams can be described as graphic palindromes. The first Sator square palindrome was found in the ruins of Pompeii, meaning it was created before the Eruption of Mount Vesuvius in 79 AD. A sator square using the mirror writing for the representation of the letters S and N was carved in a stone wall in Oppède (France) between the Roman Empire and the Middle Ages,[21] thus producing a work made up of 25 letters and 8 different characters, 3 naturally symmetrical (A, T, O), 3 others decipherable from left to right (R, P, E), and 2 others from right to left (S, N). This engraving is therefore readable in four directions.[22] Although the term is recent, the existence of mirror ambigrams has been attested since at least the first millennium. They are generally palindromes stylized to be visually symmetrical. In ancient Greek, the phrase "ΝΙΨΟΝ ΑΝΟΜΗΜΑΤΑ ΜΗ ΜΟΝΑΝ ΟΨΙΝ" (wash the sins, not only the face), is a palindrome found in several locations, including the site of the church Hagia Sophia in Turkey.[20][23] It is sometimes turned into a mirror ambigram when written in capital letters with the removal of spaces, and the stylization of the letter Ν (Ν). A boustrophedon is a type of bi-directional text, mostly seen in ancient manuscripts and other inscriptions. Every other line of writing is flipped or reversed, with reversed letters. Rather than going left-to-right as in modern European languages, or right-to-left as in Arabic and Hebrew, alternate lines in boustrophedon must be read in opposite directions. Also, the individual characters are reversed, or mirrored. This two-way writing system reveals that modern ambigrams can have quite ancient origins, with an intuitive component in some minds. Mirror writing in Islamic calligraphy flourished during the early modern period, but its origins may stretch as far back as pre-Islamic mirror-image rock inscriptions in the Hejaz.[24] The earliest known non-natural rotational ambigram dates to 1893 by artist Peter Newell.[25] Although better known for his children's books and illustrations for Mark Twain and Lewis Carroll, he published two books of reversible illustrations, in which the picture turns into a different image entirely when flipped upside down. The last page in his book Topsys & Turvys contains the phrase The end, which, when inverted, reads Puzzle. In Topsys & Turvys Number 2 (1902), Newell ended with a variation on the ambigram in which The end changes into Puzzle 2. In March 1904 the Dutch-American comic artist Gustave Verbeek used ambigrams in three consecutive strips of The UpsideDowns of old man Muffaroo and little lady Lovekins.[26] His comics were ambiguous images, made in such a way that one could read the six-panel comic, flip the book and keep reading. From June to September 1908, the British monthly The Strand Magazine published a series of ambigrams by different people in its "Curiosities" column.[27] Of particular interest is the fact that all four of the people submitting ambigrams believed them to be a rare property of particular words. Mitchell T. Lavin, whose "chump" was published in June, wrote, "I think it is in the only word in the English language which has this peculiarity," while Clarence Williams wrote, about his "Bet" ambigram, "Possibly B is the only letter of the alphabet that will produce such an interesting anomaly."[27][28] Characteristics Natural ambigrams In the Latin alphabet, many letters are symmetrical glyphs. The capital letters B, C, D, E, H, I, K, O, and X have a horizontal symmetry axis. This means that all words that can be written using only these letters are natural lake reflection ambigrams. For example, BOOK, CHOICE, or DECIDE. The lowercase letters l, o, s, x and z are rotationally symmetrical, while pairs such as b/q, d/p, m/w, n/u, and in some typefaces h/y and a/e, are rotations of each other. Thus, the words "sos", "pod", "suns", "yeah", "swims", "dollop", or "passed" form natural rotational ambigrams. More generally, a "natural ambigram" is a word that possesses one or more symmetries when written in its natural state, requiring no typographic styling. The words "bud", "bid", or "mom", form natural mirror ambigrams when reflected over a vertical axis, as does "ليبيا", the name of the country Libya in Arabic. The words "HIM", "TOY, "TOOTH" or "MAXIMUM", in all capitals, form natural mirror ambigrams when their letters are stacked vertically and reflected over a vertical axis. The uppercase word "OHIO" can flip a quarter to produce a 90° rotational ambigram when written in serif style (with large "feet" above and below the "I"). Like all strobogrammatic numbers, 69 is a natural rotational ambigram. Patterns in nature are visible regularities of form found in the natural world.[29] Similarly, patterns in ambigrams are regularities found in graphemes. As a consequence to this "natural" property, some shapes appear more or less appropriate to handle for the designer. Ambigram candidates can become "almost natural", when all the letters except maybe one or two are symmetrically cooperative, for example the word "awesome" possesses 5 compatible letters (the central s that flips around itself, and the couples a/e and w/m). Single words or several words A symmetrical ambigram can be called "homogram" (contraction of "homo-ambigram") when it remains unchanged after reflection, and "heterogram" when it transforms.[30][31] In the most common type of ambigram, the two interpretations arise when the image is rotated 180 degrees with respect to each other (in other words, a second reading is obtained from the first by simply rotating the sheet). Single word ambigrams Douglas Hofstadter coined the word "homogram" to define an ambigram with identical letters.[30][31] In this case, the first half of the word turns into the last half.[14] • Ambigram "Wikipedia", drawn by French artist Jean-Claude Pertuzé, 180° rotational symmetry. • "Candy", 180° symmetrical ambigram. • "Cloud", vertical axis mirror ambigram with a cloud occupying negative space in the letter O. • "Doug", hypocorism for Douglas Hofstadter, the "father" of the ambigram concept. Several words A symmetrical ambigram is called "heterogram"[30][31] (contraction of "hetero-ambigram") when it gives another word. Visually, a heterogram ambigram is symmetrical only when both versions of the pairing are shown together. The aesthetical appearance is more difficult to design, when a changing ambigram aims to be revealed in one way only, alternatively or separately, because symmetry generally enhances elegance. Technically, there are twice more combinations of letters involved in a hetero-ambigram than in a homo-ambigram. For example, the 180° rotational ambigram "yeah" contains only two pairs of letters: y/h and e/a, whereas the heterogram "yeah / good" contains four : y/d, e/o, a/o, and h/g. A single word ambigram cannot be hetero-, but a multiple words ambigram can be homo- type if the letters overlapse, like in "upsidedown" written attached, for example. The ambigram saying "upsidedown" one way and "upsidedown" again the other way, means it is a two words homogram. But the ambigram saying "upside" one way and "down" after rotation, means it is a two words heterogram. There is no limitation to the number of words potentially associable, and full ambigram sentences have even been published.[10][14] • "Ambigram / Wikipedia", hetero- type.[32] • "True flag", self-referential flag, horizontal axis mirror hetero- type. • Two words ambigram "Stay Here". • Two words ambigram "Real / Fake" showing alternatively one version of the pair. Types Ambigrams are exercises in graphic design that play with optical illusions, symmetry and visual perception. Some ambigrams feature a relationship between their form and their content. Ambigrams usually fall into one of several categories. 180° rotational ambigrams "Half-turn" ambigrams or point reflection ambigrams, commonly called "upside-down words", are 180° rotational symmetrical calligraphies.[6] We can read them right side up or upside down, or both. Rotation ambigrams are the most common type of ambigrams for good reason. When a word is turned upside down, the top halves of the letters turn into the bottom halves. And because our eyes pay attention primarily to the top halves of letters when we read, that means that you can essentially chop off the top half of a word, turn it upside down, and glue it to itself to make an ambigram. [...][14] — Scott Kim • Rotating ambigram "Say Yes", half-turn type with 8 occurrences of the same pattern. The phrase itself is a phonetic palindrome. • Point reflection ambigram merci. • "Home / Away", 180° rotational hetero-ambigram. • "Lift", half-turn ambigram logo. Mirror ambigrams A mirror ambigram, or reflection ambigram, is a design that can be read when reflected in a mirror vertically, horizontally, or at 45 degrees,[14] giving either the same word or another word or phrase. Vertical axis reflection ambigrams When the reflecting surface is vertical (like a mirror for example), the calligraphic design is a vertical axis mirror ambigram. The "museum" ambigram is almost natural with mirror symmetry, because the first two letters are easily exchanged with the last two, and the lowercase letter e can be transformed into s by a fairly obvious typographical acrobatics.[34] Vertical axis mirror ambigrams find clever applications in mirror writing (or specular writing), that is formed by writing in the direction that is the reverse of the natural way for a given language, such that the result is the mirror image of normal writing: it appears normal when it is reflected in a mirror. For example, the word "ambulance" could be read frontward and backward in a vertical axis reflective ambigram. Following this idea, the French artist Patrice Hamel created a mirror ambigram saying "entrée" (entrance, in French) one way, and "sortie" (exit) the other way, displayed in the giant glass façade of the Gare du Nord in Paris, so that the travelers coming in read entrance, and those leaving read way out.[35] Horizontal axis reflection ambigrams When the reflecting surface is horizontal (like a mirroring lake for example), the calligraphic design is a horizontal axis mirror ambigram. The book Ambigrams Revealed features several creations of this type, like the word "Failure" mirroring in the water of a pond to give "Success", or "Love" changing into "Lust".[14] Figure-ground ambigrams In a figure / ground ambigram, letters fit together so the negative space around and between one word spells another word.[14] In Gestalt psychology, figure–ground perception is known as identifying a figure from the background. For example, black words on a printed paper are seen as the "figure", and the white sheet as the "background". In ambigrams, the typographic space of the background is used as negative space to form new letters and new words. For example, inside a capital H, one can easily insert a lowercase i. The oil painting You & Me (US) by John Langdon (1996) belongs to this category. The word "me" fills the space between the letters of "you".[36] Ambigram tessellations With Escher-like tessellations associated to word patterns, ambigrams can be oriented in three, four, and up to six directions via rotational symmetries of 120°, 90° and 60° respectively,[37] such as those created by French artist Alain Nicolas.[38] Some words can also transform in the negative space, but the multiplication of constraints often has the effect of reducing either the readability or the complexity of the designed words. Ambigram tessellations are sorts of word puzzles, in which geometry set the rules.[38] • Tessellation build with the natural ambigram "Yeah". • 3-directional ambigram "Serie" (series, in French), tessellation using a 120° rotational symmetry. Created from a hexagon. Media related to Ambigram tessellations at Wikimedia Commons. Chain ambigrams A chain ambigram is a design where a word (or sometimes words) are interlinked, forming a repeating chain.[14] Letters are usually overlapped: a word will start partway through another word. Sometimes chain ambigrams are presented in the form of a circle. For example, the chain "...sunsunsunsun..." can flip upside down, but not the word "sun" alone, written horizontally. A chain ambigram can be constituted of one to several elements. A single element ambigram chain is like a snake eating its own tail. A two-elements ambigram chain is like a snake eating the neighbor's tail with the neighbor eating the first snake's, and so on. Scott Kim's "Infinity" works, and that of John Langdon "Chain reaction", are also self-referential, since the first is infinite in the literal sense of the word, and the second, both reversible at 180° and interfering around the letter O, evokes a chain reaction.[14] Spinonyms A spinonym is a type of ambigram in which a word is written using the same glyph repeated in different orientations.[14] WEB is an example of a word that can easily be made into a spinonym. • MBE (Motor Bike Expo) spinonym logo. The same glyph is repeated in three different orientations. • Spinonym "neun 9" (German for nine), five times the same glyph repeated in different orientations. • "Happy new year" spinonym, the same glyph in different orientations shapes the twelve letters of the sentence. Perceptual shift ambigrams Perceptual shift ambigrams, also called "oscillation" ambigrams, are designs with no symmetry but can be read as two different words depending on how the curves of the letters are interpreted.[14] These ambigrams work on the principle of rabbit-duck-style ambiguous images. For example Douglas Hofstadter expresses the dual nature of light as revealed by physics with his perceptual shift ambigram Wave / Particle. 90° rotational ambigrams "Quarter-turn" ambigrams or 90° rotational ambigrams turn clockwise or counterclockwise to express different meanings.[2] For example, the letter U can turn into a C and reciprocally, or the letters M or W into an E.[14] Totem ambigrams The Alabama A&M University has a totem mirror ambigram logo. Words crossing or totem ambigram "Hot dog", vertical axis reflection symmetry. A totem ambigram is an ambigram whose letters are stacked like a totem, most often offering a vertical axis mirror symmetry. This type helps when several letters fit together, but hardly the whole word. For example, in the Maria monogram, the letters M, A and I are individually symmetrical, and the pairing R/A is almost naturally mirroring. When adequately stacked, the 5 letters produce a nice totem ambigram, whereas the whole name "Maria" would not offer the same cooperativeness. The ambigrammist artist John Langdon designed several totemic assemblages, such as the word "METRO" composed of the symmetrical letter M, then section ETR, and below O; or the sentence "THANK YOU", vertical assembly of T, H, A, then of the symmetric NK couple, then finally Y, O, U.[39] Fractal ambigrams In mathematics, a fractal is a geometrical shape that exhibits invariance under scaling. A piece of the whole, if enlarged, has the same geometrical features as the entire object itself. A fractal ambigram is a sort of space-filling ambigrams where the tiled word branches from itself and then shrinks in a self-similar manner, forming a fractal.[40] In general, only a few letters are constrainted in a fractal ambigram. The other letters don't need to look like any other, and thus can be shaped freely. 3-dimensional ambigrams A 3D ambigram is a design where an object is presented that will appear to read several letters or words when viewed from different angles. Such designs can be generated using constructive solid geometry, a technique used in solid modeling, and then physically constructed with the rapid prototyping method. 3-dimensional ambigram sculptures can also be achieved in plastic arts. They are volume ambigrams. The original 1979 edition of Hofstadter's Gödel, Escher, Bach featured two 3-D ambigrams on the cover.[41] Complex ambigrams Complex ambigrams are ambigrams involving more than one symmetry, or satisfying the criteria for several types. For example, a complex ambigram can be both rotational and mirror with a 4-fold dihedral symmetry. Or a spinonym that reads upside down is also a complex ambigram. • The logo Oxo has a 4-fold dihedral symmetry (mirror and 180° rotational ambigram). • The famous DJ Étienne de Crécy has a complex ambigram logo "EDC", mirroring through a horizontal axis, and figure-ground type with a power plug pictogram inserted in the negative space. • 4-fold dihedral symmetrical ambigram (mirror and rotational) "Dig hole, Die". Symbols Other languages Ambigram 곰 / 문 (Bear / Door, in Korean), 180° rotational symmetry. The word "বাংলা" (Bangla or Bengali, in Bengali), half-turn ambigram. Ambigrams exist in many languages. With the Latin alphabet, they generally mix lowercase and uppercase letters. But words can also be symmetrical in other alphabets, like Arabic, Bengali, Cyrillic, Greek, and even in Chinese characters and Japanese kanji. In Korean, 곰 (bear) and 문 (door), 공 (ball) and 운 (luck), or 물 (water) and 롬 (ROM) form a natural rotational ambigram. Some syllables like 응 (yes), 표 (ticket/signage) or 를 (object particle), and words like "허리피라우" (straighten your back) also make full ambigrams. The han character meaning "hundred" is written 百, that makes a natural 90° rotational ambigram when the glyph makes a quarter turn counterclockwise, one sees "100".[42] Media related to Ambigrams by language at Wikimedia Commons. Numbers Although not totally symmetrical, the Sochi 2014 (Olympic games) official logo offers mirror and rotational symmetries, linking the numbers to the letters like an ambigram. Rio 2016 (Olympic games), half-turn rotational ambigram logo containing letters and digits. An ambigram of numbers, or numeral ambigram, contains numerical digits, like 1, 2, 3...[14] In mathematics, a palindromic number (also known as a numeral palindrome) is a number that remains the same when its digits are reversed through a vertical axis (but not necessarily visually). The palindromic numbers containing only 1, 8, and 0, constitute natural numeric ambigrams (visually symmetrical through a mirror). Also, because the glyph 2 is graphically the mirror image of 5, it means numbers like 205 or 85128 are natural numeral mirror ambigrams. Though not palindromic in the mathematical sense, they read frontward and backward like real ambigrams. A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. The numeral looks the same right-side up and upside down (e.g., 69, 96, 1001).[43][44][45] Some dates are natural numeral ambigrams.[46] In March 1961, artist Norman Mingo created an upside-down cover for Mad magazine featuring an ambigram of the current year. The title says "No matter how you look at it... it's gonna be a Mad year. 1961, the first upside-down year since 1881."[47] Tuesday, 22 February 2022, was a palindrome and ambigram date called "Twosday" because it contained reversible 2 (two).[48][49][50] Ambigrams of numbers receive most attention in the realm of recreational mathematics.[2][51] Ambigrams with numbers sometimes combine letters and numerical digits. Because the number 5 is approximately shaped like the letter S, the number 6 like a lowercase b, the number 9 like the letter g, it is possible to play on these similarities to design ambigrams. A good example is the Sochi 2014 (Olympic games) logo where the four glyphs contained in 2014 are exact symmetries of the four letters S, o, i and h, individually.[52] Other symbols As alphabet letters are glyphs used in the writing systems to express the languages visually, other symbols are also used in the world to code other fields, like the prosigns in the Morse code or the musical notes in music. Similarly to the ambigrams of letters, the ambigrams with other symbols are generally visually symmetrical, either point reflective or reflective through an axis. The international Morse code distress signal SOS ▄ ▄ ▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄ ▄ ▄ is a natural ambigram constituted of dots and dashes. It flips upside down or through a mirror. In morse code, the letter P coded ▄ ▄▄▄ ▄▄▄ ▄ and the letter R coded ▄ ▄▄▄ ▄ are individually symmetrical, like many other letters and numbers. Also, the letter G coded ▄▄▄ ▄▄▄ ▄ is the exact reverse of the letter W coded ▄ ▄▄▄ ▄▄▄ . Thus, the combination ▄▄▄ ▄▄▄ ▄ / ▄ ▄▄▄ ▄▄▄ coding the pairing G/W constitutes a natural ambigram. Consequently, meaningful natural ambigrams written in morse code certainly exist, like for example the words "wog" ▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄ , "Dou" ▄▄▄ ▄ ▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄ ▄ ▄▄▄ or "mom" ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ ▄▄▄ .[6][53][54] In music, the interlude from Alban Berg's opera Lulu is a palindrome, thus the score made up of musical notes is almost symmetrical through a vertical axis.[55] In biology, researchers study the ambigrammatic property of narnaviruses by using visual representations of the symmetrical sequences.[29][1][56] Fields Calligraphy and typography Calligraphic color-reversal ambigram Soul of Laos, published in the book Ambigrams Revealed.[14] Calligraphic design Danke (thanks, in German) and half-turn ambigram. Instead of simply writing them, ambigram lettering covers the art of drawing letters. In ambigram calligraphy, each letter acts as an illustration, each letter is created with attention to detail and has a unique role within a composition. Lettering ambigrams do not translate into combinations of alphabet letters that can be used like a typeface, since they are created with a specific candidate in mind. The calligrapher, graffiti writer and graphic designer Niels Shoe Meulman created several rotational ambigrams like the number "fifty",[14] the names "Shoe / Patta",[14] and the opposition "Love / Fear".[57] The cover of the 7th volume of the typography book Typism is an ambigram drawn by Nikita Prokhorov.[58] The American type designer Mark Simonson designed poetic and humorous ambigrams, such as the words "Revelation", "Typophile", and the symbiosis "Drink / Drunk".[14] The last one makes a visual pun when printed on a shot glass, sold commercially.[59] Logos The rotational logo New Man created by Raymond Loewy in 1968 is a natural ambigram. The online two-sided marketplace for residential cleaning Handy has a 180° rotational ambigram logo. Sun (Microsystems) logo designed by Vaughan Pratt in 1982, chain ambigram, spinonym, 90° and 180° rotational symmetries. Nissin (Foods) ambigram visual identity (half-turn). Since they are visually striking, and sometimes surprising, ambigram words find large application in corporate logos and wordmarks, setting the visual identity of many organizations, trademarks and brands.[60] In 1968[61] or 1969, Raymond Loewy designed the rotational New Man ambigram logo.[62][63][64] The mirror ambigram DeLorean Motor Company logo, designed by Phil Gibbon, was first used in 1975.[65][66][67] Robert Petrick designed the invertible Angel logo[14] in 1976. The logo Sun (Microsystems) designed by professor Vaughan Pratt[68] in 1982 fulfills the criteria of several types: chain ambigram, spinonym, 90° and 180° rotational symmetries. The Swedish pop group ABBA owns a mirror ambigram logo stylized AᗺBA with a reversed B, designed by Rune Söderqvist[69] in 1976.[70] The Ventura logo of the Visitors & Convention Bureau's board, in California, cost US$25,000 and was created in 2014 by the DuPuis group. It uses a 180° rotational symmetry.[71][72] Other famous ambigram logos include: the insurance company Aviva;[73] the acronym CRD (Capital Regional District) in the Canadian province of British Columbia;[74] the American multinational corporation DXC Technology; the two-sided marketplace for residential cleaning Handy;[75][76] the brand name of French premium high-speed train services InOui;[77] the French company specializing in ticketing and passenger information systems IXXI; the century-old brand Maoam of the confectionery manufacturer Haribo;[78] the American industrial rock band NIͶ; the Japanese food company Nissin; the biotechnology company Noxxon Pharma, founded in 1997; the online travel agency Opodo in 2001;[79] the brand of food products OXO[80] born in 1899; the video game Pod; the American developer and manufacturer of audio products Sonos;[81] the American professional basketball team Phoenix Suns;[82][83] the German manufacturer of adhesive products UHU; the quadruple symmetrical logo UA from the American clothing brand Under Armour ; the Canadian corporation mandated to operate intercity passenger rail service VIA in 1978;[84] the American international broadcaster VOA, born in 1942; and the Malaysian mobile virtual network operator XOX. The student edition of the Tesco Clubcard used 180° rotational symmetry.[85] Visual communication Because they are visual puns,[2] ambigrams generally attract attention, and thus can be used in visual communication to broadcast a marketing or political message. In France, a mirror ambigram "Penelope / benevole" legible through a horizontal axis became a meme on the web after its diffusion on Wikimedia Commons.[86] Penelope Fillon, wife of French politician and former Prime Minister of France François Fillon, is suspected of having received wages for a fictitious job. Ironically, her name through the mirror becomes benevole (voluntary in French), suggesting dedication for a free service. Shared tens of thousands of times on the social networks, this humorous ambigram made the buzz via several French,[87] Belgian[88][89] and Swiss[86] medias. Ambigrams are regularly used by communication agencies such as Publicis to engage the reader or the consumer through two-way messages.[90] Thus, in 2021, male first names transformed into female first names are included in a Swiss advertising campaign aimed at raising awareness about gender equality. An intriguing catchphrase typography upside down invites the reader to rotate the magazine, in which the first names "Michael" or "Peter" are transformed into "Nathalie" or "Alice".[91][92] In 2015 iSmart's logo on one of its travel chargers went viral because the brand's name turned out to be a natural ambigram that read "+Jews!" upside down. The company noted that "...we learned a powerful lesson of what not to do when creating a logo." [93] Cinema posters sometimes seduce observers with ambigram titles, such as that of Tenet by Christopher Nolan, by central symmetry.[22] or Anna by Luc Besson around a vertical axis,[94][95] • Ambigram meme "Penelope / benevole" with a political message. • Half-turn traffic sign using a directional arrow symbol to display alternatively "Station / Toilets". • Visual pun "Avoid the plane" to attract attention towards the environmental impact of aviation. • A practical application of mirror ambigrams in a banner reading "Idaplatz fest" front and back (Zurich, 2008). Comics Ambigrams in comics The Upside Downs of Little Lady Lovekins and Old Man Muffaroo by Gustave Verbeek containing ambigram sentences in 1904. Another frame. The American artist and writer Peter Newell published a rotational ambigram in 1893 saying "Puzzle / The end" in the book containing reversible illustrations Topsys & Turvys.[25] In March 1904 the Dutch-American comic artist Gustave Verbeek used ambigrams in three consecutive strips of The UpsideDowns of old man Muffaroo and little lady Lovekins.[26] His comics were ambiguous images, made in such a way that one could read the six-panel comic, flip the book and keep reading. In The Wonderful Cure of the Waterfall (13 March 1904) an Indian medicine man says 'Big waters would make her very sound', while when flipped the medicine man turns into an Indian woman who says 'punos dery, eay apew poom, serlem big'. Which is explained as, 'poor deary' several foreign words that meant that she would call the 'Serlem Big'. The next comic called At the House of the Writing Pig (20 March 1904), where two ambigram word balloons are featured. The first features an angry pig trying to make the main protagonist leave by showing a sign that says; 'big boy go away, dis am home of mr h hog', up side down it reads 'Boy yew go away. We sip. Home of hog pig.' The protagonist asks the pig if it wants a big bun, upon which it replies 'Why big buns? Am mad u!', which flips into 'In pew we sang big hym'. Finally in The Bad Snake and the Good Wizard (1904 Mar 27) there are two more ambigrams. The first turns 'How do you do' into the name of a wizard called 'Opnohop Moy', the second features a squirrel telling the protagonist 'Yes further on' only to inform it that there are 'No serpents here' on his way back. In a 2012 Swedish remake of the book,[96] the artist Marcus Ivarsson redraws The Bad Snake and the Good Wizard in his own style. He removes the squirrel, but keeps the other ambigram. 'How do you do' is replaced by 'Nejnej' (Swedish for no) and the wizard is now called 'Laulau'. Media related to Ambigrams by Gustave Verbeek at Wikimedia Commons. Oubapo, workshop of potential comic book art, is a comics movement which believes in the use of formal constraints to push the boundaries of the medium. Étienne Lécroart, cartoonist, is a founder and key member of Oubapo association, and has composed cartoons that could be read either horizontally, vertically, or in diagonal, and vice versa, sometimes including appropriate ambigrams.[97] Drawings and paintings Ambigram "¡OHO!" published by Rex Whistler in 1946. Ambigram "¡OHO!" with reversible faces by Rex Whistler created before 1944. A young woman transforms into a grandmother. The British painter, designer and illustrator Rex Whistler, published in 1946 a rotational ambigram "¡OHO!" for the cover of a book gathering reversible drawings.[98] The artist John Langdon, specialist of ambigrams,[60] designed many color paintings featuring ambigrams of all kinds, figure-ground, rotational, mirror or totem. Among other influences, he particularly admires M. C. Escher's drawings.[14] The Canadian artist Kelly Klages painted several acrylics on canvas with ambigram words and sentences referring to famous writers' novels written by William Shakespeare or Agatha Christie, such as Third Girl, The Tempest, After the Funeral, The Hollow, Reformation, Sherlock Holmes, and Elephants Can Remember.[99] Sculptures Mia Florentine Weiss "Love Hate" sculpture in Munich, Germany, in 2020. "Now / Won" installation in front of the Reichstag building, Berlin, Germany, 2017. The German conceptual artist Mia Florentine Weiss built a sculptural ambigram Love Hate,[100] that has traveled Europe as a symbol of peace and change of perspective.[101] Depending on which side the viewer looks at it, the sculpture says "Love" or "Hate". A similar concept was installed in front of the Reichstag building in Berlin with the words "Now / Won". Both sculptures are mirror type ambigrams, symmetrical around a vertical axis.[102] The Swiss sculptor Markus Raetz made several three-dimensional ambigram works, featuring words generally with related meanings, such as YES-NO (2003),[103] ME-WE (2004, 2010),[104] OUI-NON (2000–2002) in French,[105][106] SI–NO (1996)[107] and TODO-NADA (1998) in Spanish[108][109] These are anamorphic works, which change in appearance depending on the angle of view of the observer. The OUI–NON ambigram is installed on the Place du Rhône, in Geneva, Switzerland, at the top of a metal pole. Physically, the letters have the appearance of iron twists. With the perspective, this work demonstrates that reality can be ambiguous.[106] Some ambigram sculptures by the French conjurer Francis Tabary are reversible by a half-turn rotation, and can therefore be exhibited on a support in two different ways.[110][111] Palindromes Ambigrams are sorts of visual palindromes.[112] Some words turn upside down, others are symmetrical through a mirror. Natural ambigram palindromes exist, like the words "wow", "malayalam"[113] (Dravidian language), or the biotechnology company Noxxon that possesses a palindromic name associated to a rotational ambigram logo. But some words are natural ambigrams, though not palindromes in the literary acception, like "bud" for example, because b and d are different letters. As a result, some words and sentences are good candidates for ambigrammists, but not for palindromists, and reciprocally, since the constraint slightly differ. Authors of ambigrams also benefit from a certain flexibility by playing on the typeface and graphical adjustments to influence the reading of their visual palindromes. Oulipo, workshop of potential literature, seeks to create works using constrained writing techniques. Georges Perec, French novelist and member of the Oulipo group, designed a rotational ambigram, that he called "vertical palindrome".[10] Sibylline, the sentence "Andin Basnoda a une épouse qui pue" in French means "Andin Basnoda has a smelly wife". Perec did not care about punctuation spaces, but his creation flips easily with a classical font like Arial. Visual palindromes sometimes perfectly illustrate literary contents. The American author Dan Brown incorporated John Langdon's designs into the plot of his bestseller Angels & Demons, and his fictional character Robert Langdon's surname was a homage to the ambigram artist.[114] The fantasy novel Abarat, written and illustrated by Clive Barker, features an ambigram of the title on its cover.[115] Calligrams Reflective calligram hat in Alevism forming a human face with Arabic letters. Oslo Climbing Club official logo[116] "OK" (acronym for Oslo Klatreklubb) 90° rotational ambigram showing a human silhouette vertically. A calligram is text arranged in such a way that it forms a thematically related image. It can be a poem, a phrase, a portion of scripture, or a single word. The visual arrangement can rely on certain use of the typeface, calligraphy or handwriting. The image created by the words illustrates the text by expressing visually what it says, or something closely associated. In Islamic calligraphy, symmetrical calligrams appear in ancient and modern periods, forming mirror ambigrams in Arabic language.[24] The word "OK" turned 90° counterclockwise evokes a human icon, with the letter O forming the head and the letter K the arms and the legs. The Norwegian Climbing Club Oslo Klatreklubb (acronym "OK") borrowed the concept of this natural calligram for their official logo.[116] Semantics As described by Douglas Hofstadter, ambigrams are visual puns having two or more (clear) interpretations as written words.[2] Multilingual ambigrams can be read one way in a language, and another way in a different language or alphabet.[14] Multi-lingual ambigrams can occur in all of the various types of ambigrams, with multi-lingual perceptual shift ambigrams being particularly striking. Like certain anagrams with providential meanings such as "Listen / Silent" or "The eyes / They see", ambigrams also sometimes take on a timely sense, for example "up" becomes the abbreviation "dn", very naturally by rotation of 180°.[117] But on the other hand, it happens that the luck of the letters makes things bad. This is the case with the weird anagram "Santa / Satan", as it is with a rotational ambigram that has gone viral because of the paradoxical and unintentional message it expresses. Spotted in 2015 on a metal medal marketed without bad intention, the text "hope" displays upside down with a fairly obvious reading "Adolf", first name of the Nazi leader situated at the antipodes of optimism. This coincidence photographed by an Internet user was relayed by several media and constitutes an ambiguous image.[118][119] Mathematics Recreational mathematics is carried out for entertainment rather than as a strictly research and application-based professional activity.[51] An ambigram magic square exists, with the sums of the numbers in each row, each column, and both main diagonals the same right side up and upside down (180° rotational design). Numeral ambigrams also associate with alphabet letters. A "dissection" ambigram of "squaring the circle" was achieved in a puzzle where each piece of the word "circle" fits inside a perfect square.[2] Burkard Polster, professor of mathematics in Melbourne[120] conducted researches on ambigrams and published several books dealing with the topic, including Eye Twisters, Ambigrams & Other Visual Puzzles to Amaze and Entertain.[121] In the abstract Mathemagical Ambigrams, Polster performs several ambigrams closely related to his realm, like the words "algebra", "geometry", "math", "maths", or "mathematics".[2] Message written with the digits "07734" upside down. Calculator spelling is an unintended characteristic of the seven-segment display traditionally used by calculators, in which, when read upside-down, the digits resemble letters of the Latin alphabet. Also, palindromic numbers and strobogrammatic numbers sometimes attract attention of mathematician ambigrammists.[44][43] Ambigram tessellations and 3D ambigrams are two types particularly fun for the mathematician in geometry. Word patterns in tessellations can start from 35 different fundamental polygons, such as the rhombus, the isosceles right triangle, or the parallelogram.[37] Word puzzles are used as a source of entertainment, but can additionally serve an educational purpose. The American puzzle designer Scott Kim published several ambigrams in Scientific American in Martin Gardner's "Mathematical Games" column, among them long sentences like "Martin Gardner's celebration of mind" turning into "Physics, patterns and prestidigitation".[14] Duality and analogy In the word "ambigram", the root ambi- means "both" and is a popular prefix in a world of dualities, such as day/night, left/right, birth/death, good/evil.[14] In Wordplay: The Philosophy, Art, and Science of Ambigrams,[60] John Langdon mentions the yin and yang symbol as one of his major influences to create upside down words. Ambigrams are mentioned in Metamagical Themas, an eclectic collection of articles that Douglas Hofstadter wrote for the popular science magazine Scientific American during the early 1980s.[8] Seeking the balance point of analogies is an aesthetic exercise closely related to the aesthetically pleasing activity of doing ambigrams, where shapes must be concocted that are poised exactly at the midpoint between two interpretations. But seeking the balance point is far more than just aesthetic play; it probes the very core of how people perceive abstractions, and it does so without their even knowing it. It is a crucial aspect of Copycat research.[8] — Douglas Hofstadter Cognition and psychology Legibility is an important aspect in successful ambigrams. It concerns the ease with which a reader decodes symbols. If the message is lost or difficult to perceive, an ambigram does not work.[7] Readability is related to perception, or how our brain interprets the forms we see through our eyes.[122] Symmetry in ambigrams generally improves the visual appearance of the calligraphic words.[14] Hermann Rorschach, inventor of the Rorschach Test notices that asymmetric figures are rejected by many subjects. Symmetry supplies part of the necessary artistic composition.[123] For many amateurs, designing ambigrams represents a recreational activity, where serendipity can play a fertile role, when the author makes an unplanned fortunate discovery.[2][27] Magic Ambigram "Magic / Dream", with a handheld pattern giving a reversed shadow. "incredible!" Magical ambigram.[124] In magic, ambigrams work like visual illusions, revealing an unexpected new message from a particular written word.[125] In the first series of the British show Trick or Treat, the show's host and creator Derren Brown uses cards with rotational ambigrams.[126][127] These cards can read either 'Trick' or 'Treat'. Ambiguous images, of which ambigrams are a part, cause ambiguity in different ways. For example, by rotational symmetry, as in the Illusion of The Cook by Giuseppe Arcimboldo (1570);[128] sometimes by a figure-ground ambivalence as in Rubin vase; by perceptual shift as in the rabbit–duck illusion, or through pareidolias; or again, by the representation of impossible objects, such as Necker cube or Penrose triangle. For all these types of images, certain ambigrams exist, and can be combined with visuals of the same type. John Langdon designed a figure-ground ambigram "optical illusion" with the two words "optical" and "illusion", one forming the figure and the other the background. "Optical" is easier to see initially but "illusion" emerges with longer observation.[129] Tattooing Mirror ambigram tattoos on wrists "Love / Eros". Handmade ambigram in tattoo "New York / Rich Man", right side up and upside down. 180° rotational ambigram tattoo "No religion". Ambigram tattoo Texas / Sexy, 180° rotational symmetry. One of the most dynamic sectors that harbors ambigrams is tattooing. Because they possess two ways of reading, ambigram tattoos inked on the skin benefit from a "mind-blowing" effect. On the arm, sleeve tattoos flip upside-down, on the back or jointly on two wrists they are more striking with a mirror symmetry. A large range of scripts and fonts is available. Experienced ambigram artists can create an optical illusion with a complex visual design.[130] In 2015, an ambigram tattoo went viral following an advertising campaign developed by the Publicis group two years earlier. The Samaritans of Singapore organization, active in suicide prevention, has a 180° reversible "SOS" ambigram logo, acronym of its name and homonym of the famous SOS distress signal. In 2013, this center orders advertisements that could be inserted in magazines to make readers aware of the problem of depression among young people, and the communication agency notices the symmetrical aspect of the logo. As a result, it begins to produce several ambigrammatic visuals, staged in photographic contexts, where sentences such as "I'm fine", "I feel fantastic" or "Life is great" turn into "Save me", "I'm falling apart", and "I hate myself". Readers noticing this logo placed at the upper left corner of the page with an upside-down typographical catchphrase rotate the newspaper and visualize the double calligraphed messages, which call out with the SOS.[90][131] These ads are so influential that Bekah Miles, an American student herself coming out of a severe depression, chooses to use the "I'm fine / Save me" ambigram to get a tattoo on her thigh. Posted on Facebook, the two-sided photography immediately appeals to many young people, impressed or sensitive to this difficulty.[132][133] To educate its students, George Fox University in the United States then relays the optical illusion in its official journal, through a video totaling more than three million views[134] and the information is also reproduced in several local media and international organizations, thus helping to popularize this famous two-way tattoo.[135][136] Less fortunate, another teenage girl, aged 16, committed suicide, with her also this ambigram found on a note in her room, "I'm fine / Save me", reversible calligraphy today printed on badges and bracelets, for educational purposes.[137] Clothing and fashion Adidas marketed a line of sneakers called "Bounce", with an ambigram typography printed inside the shoe. Several clothing brands, such as Helly Hansen (HH), Under Armour (UA), or New Man, raise an ambigram logo as their visual identity.[64] Mirror ambigrams are also sometimes placed on T-shirts, towels and hats, while socks are more adapted to rotational ambigrams. The conceptual artist Mia Florentine Weiss marketed T-shirts and other products with her mirror ambigram Love Hate.[138][101] Likewise, the city of Ventura in California sells sweatshirts, caps, jackets, and other fashion accessories printed with its rotational ambigram logo.[139] • Rotational and reflective ambigram "Ideal", printed on a T-shirt. • "Zen Yes" embroidered on a blue T-shirt with a meditation symmetrical pictogram. • Helly Hansen, Norwegian manufacturer and retailer of clothing and sports equipment, has an ambigram logo. Accessories The CD cover of the thirteenth studio album Funeral by American rapper Lil Wayne features a 180° rotational ambigram reading "Funeral / Lil Wayne".[140] The special edition paper sleeve (CD with DVD) of the solo album Chaos and Creation in the Backyard by Paul McCartney features an ambigram of the singer's name.[141] The Grateful Dead have used ambigrams several times, including on their albums Aoxomoxoa and American Beauty. Although the words spelled by most ambigrams are relatively short in length, one DVD cover for The Princess Bride movie creates a rotational ambigram out of two words "Princess Bride," whether viewed right side up or upside down. [142] The cover of the studio album Create/Destroy/Create by rock band Goodnight, Sunrise is an ambigram composition constituted of two invariant words, "create" and "destroy", designed by Polish artist Daniel Dostal.[143] The reversible shot glass containing a changing message "Drink / Drunk", created by the typographer Mark Simonson was manufactured and sold in the market.[59] The concept of reversible sign that some merchants use through their windows to indicate that the store is sometimes "open", sometimes "closed", was inaugurated at the beginning of the 2000s, by a rotational ambigram "Open / Closed" developed by David Holst.[34] Creating ambigrams Different ambigram artists, sometimes called ambigrammists,[8][14] may create distinctive ambigrams from the same words, differing in both style and form. Handmade designs There are no universal guidelines for creating ambigrams, and different ways of approaching problems coexist. A number of books suggest methods for creation, including WordPlay,[60] Eye Twisters,[121] and Ambigrams Revealed,[14] in English. Ambigram generators Computerized methods to automatically create ambigrams have been developed. Most of them function on the simplified principle of mapping a single letter to another single letter. Because of this weakness, most of them can only map a word to itself or to another word that is the same length and do not combine letters. Thus, the generated ambigrams are in general of poor quality when compared to hand made ambigrams. More sophisticated techniques employ databases of thousands of curves to create complex ambigrams. Some ambigram generators are free, while some others require payment. Artists John Langdon and Scott Kim each believed that they had invented ambigrams in the 1970s.[144] Douglas Hofstadter Douglas Hofstadter coined the term.[2] To explain visually the numerous types of possible ambigrams, Hofstadter created many pieces with different constraints and symmetries.[145] Hofstadter has had several exhibitions of his artwork in various university galleries. According to Scott Kim, Hofstadter once created a series of 50 ambigrams on the name of all the states in the US.[14] In 1987 a book of 200 of his ambigrams, together with a long dialogue with his alter ego Egbert G. Gebstadter on ambigrams and creativity, was published in Italy.[4] John Langdon John Langdon is a self-taught artist, graphic designer and painter, who started designing ambigrams in the late 1960s and early 70s. Lettering specialist, Langdon is a professor of typography and corporate identity at Drexel University in Philadelphia.[146] John Langdon produced a mirror image logo "Starship" in 1972[147] or 1975, that was sold to the rock band Jefferson Starship. Langdon's ambigram book Wordplay was published in 1992. It contains about 60 ambigrams. Each design is accompanied by a brief essay that explores the word's definition, its etymology, its relationship to philosophy and science, and its use in everyday life.[60] Ambigrams became more popular as a result of Dan Brown incorporating John Langdon's designs into the plot of his bestseller, Angels & Demons, and the DVD release of the Angels & Demons movie contains a bonus chapter called "This is an Ambigram". Langdon also produced the ambigram that was used for some versions of the book's cover.[144] Brown used the name Robert Langdon for the hero in his novels as an homage to John Langdon.[114][148] Blacksmith Records, the music management company and record label, possesses a rotational ambigram logo[149] designed by John Langdon.[150] Scott Kim Scott Kim is one of the best-known masters of the art of ambigrams.[63] He is an American puzzle designer and artist who published in 1981 a book called Inversions with ambigrams of many types.[13][148] Other artists Nikita Prokhorov is a graphic artist, typographer and professional ambigrammist. His book Ambigrams Revealed showcases ambigram designs of all types, from all around the world.[14][151] Born in 1946, Alain Nicolas is a specialist of figurative and ambigram tessellations. 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Further reading • Hofstadter, Douglas R., "Metafont, Metamathematics, and Metaphysics: Comments on Donald Knuth's Article 'The Concept of a Meta-Font'" Scientific American (August 1982) (republished, with a postscript, as chapter 13 in the book Metamagical Themas ISBN 978-0-553-34683-1) • Hofstadter, Douglas R., Ambigrammi: Un microcosmo ideale per lo studio della creativita (Ambigrams: An Ideal Microworld for the Study of Creativity), Hopefulmonster Editore Firenze (1987) (in Italian) ISBN 978-88-7757-006-2 • Langdon, John, Wordplay: Ambigrams and Reflections on the Art of Ambigrams, Harcourt Brace (1992, republished 2005) ISBN 978-0-15-198454-1 • Polster, Burkard, Eye Twisters: Ambigrams & Other Visual Puzzles to Amaze and Entertain, Constable (2008) ISBN 978-1-4027-5798-3 External links • Ambigrams at Curlie Optical illusions (list) Illusions • Afterimage • Ambigram • Ambiguous image • Ames room • Autostereogram • Barberpole • Bezold • Café wall • Checker shadow • Chubb • Cornsweet • Delboeuf • Ebbinghaus • Ehrenstein • Flash lag • Fraser spiral • Gravity hill • Grid • Hering • Impossible trident • Jastrow • Lilac chaser • Mach bands • McCollough • Müller-Lyer • Necker cube • Oppel-Kundt • Orbison • Penrose stairs • Penrose triangle • Peripheral drift • Poggendorff • Ponzo • Rubin vase • Sander • Schroeder stairs • Shepard tables • Spinning dancer • Ternus • Vertical–horizontal • White's • Wundt • Zöllner Popular culture • Op art • Trompe-l'œil • Spectropia (1864 book) • Ascending and Descending (1960 drawing) • Waterfall (1961 drawing) • The dress (2015 photograph) Related • Accidental viewpoint • Auditory illusions • Tactile illusions • Temporal illusion
Names of large numbers Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America. These naming procedures are based on taking the number n occurring in 103n+3 (short scale) or 106n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion. Names of numbers above a trillion are rarely used in practice; such large numbers have practical usage primarily in the scientific domain, where powers of ten are expressed as 10 with a numeric superscript. However, these somewhat rare names are considered acceptable for approximate statements. For example, the statement "There are approximately 7.1 octillion atoms in an adult human body" is understood to be in short scale of the table below (and is only accurate if referring to short scale rather than long scale). Indian English does not use millions, but has its own system of large numbers including lakhs and crores.[1] English also has many words, such as "zillion", used informally to mean large but unspecified amounts; see indefinite and fictitious numbers. Standard dictionary numbers x Name (SS/LS, LS) SS (103x+3) LS (106x, 106x+3) Authorities AHD4[2] CED[3] COD[4] OED2[5] OEDweb[6] RHD2[7] SOED3[8] W3[9] HM[10] 1Million 106106 ✓✓✓✓✓✓✓✓✓ Milliard 109 ✓✓✓✓✓✓ 2Billion 1091012 ✓✓✓✓✓✓✓✓✓ 3Trillion 10121018 ✓✓✓✓✓✓✓✓✓ 4Quadrillion 10151024 ✓✓✓✓✓✓✓✓✓ 5Quintillion 10181030 ✓✓ ✓✓✓✓✓✓ 6Sextillion 10211036 ✓✓ ✓✓✓✓✓✓ 7Septillion 10241042 ✓✓ ✓✓✓✓✓✓ 8Octillion 10271048 ✓✓ ✓✓✓✓✓✓ 9Nonillion 10301054 ✓✓ ✓✓✓✓✓✓ 10Decillion 10331060 ✓✓ ✓✓✓✓✓✓ 11Undecillion 10361066 ✓✓ ✓ ✓✓ 12Duodecillion 10391072 ✓✓ ✓ ✓✓ 13Tredecillion 10421078 ✓✓ ✓ ✓✓ 14Quattuordecillion 10451084 ✓✓ ✓ ✓✓ 15Quindecillion 10481090 ✓✓ ✓ ✓✓ 16Sexdecillion 10511096 ✓✓ ✓ ✓✓ 17Septendecillion 105410102 ✓✓ ✓ ✓✓ 18Octodecillion 105710108 ✓✓ ✓ ✓✓ 19Novemdecillion 106010114 ✓✓ ✓ ✓✓ 20Vigintillion 106310120 ✓✓ ✓✓✓✓✓✓ 100Centillion 1030310600 ✓✓ ✓✓✓ ✓ Usage: • Short scale: US, English Canada, modern British, Australia, and Eastern Europe • Long scale: French Canada, older British, Western & Central Europe Apart from million, the words in this list ending with -illion are all derived by adding prefixes (bi-, tri-, etc., derived from Latin) to the stem -illion.[11] Centillion[12] appears to be the highest name ending in -"illion" that is included in these dictionaries. Trigintillion, often cited as a word in discussions of names of large numbers, is not included in any of them, nor are any of the names that can easily be created by extending the naming pattern (unvigintillion, duovigintillion, duoquinquagintillion, etc.). Name Value Authorities AHD4CEDCODOED2OEDnewRHD2SOED3W3UM Googol 10100 ✓✓✓✓✓✓✓✓✓ Googolplex10googol (1010100) ✓✓✓✓✓✓✓✓✓ All of the dictionaries included googol and googolplex, generally crediting it to the Kasner and Newman book and to Kasner's nephew (see below). None include any higher names in the googol family (googolduplex, etc.). The Oxford English Dictionary comments that googol and googolplex are "not in formal mathematical use". Usage of names of large numbers Some names of large numbers, such as million, billion, and trillion, have real referents in human experience, and are encountered in many contexts. At times, the names of large numbers have been forced into common usage as a result of hyperinflation. The highest numerical value banknote ever printed was a note for 1 sextillion pengő (1021 or 1 milliard bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion (1014) Zimbabwean dollar note, which at the time of printing was worth about US$30.[13] Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of how large numbers are named. Even well-established names like sextillion are rarely used, since in the context of science, including astronomy, where such large numbers often occur, they are nearly always written using scientific notation. In this notation, powers of ten are expressed as 10 with a numeric superscript, e.g. "The X-ray emission of the radio galaxy is 1.3×1045 joules." When a number such as 1045 needs to be referred to in words, it is simply read out as "ten to the forty-fifth". This is easier to say and less ambiguous than "quattuordecillion", which means something different in the long scale and the short scale. When a number represents a quantity rather than a count, SI prefixes can be used—thus "femtosecond", not "one quadrillionth of a second"—although often powers of ten are used instead of some of the very high and very low prefixes. In some cases, specialized units are used, such as the astronomer's parsec and light year or the particle physicist's barn. Nevertheless, large numbers have an intellectual fascination and are of mathematical interest, and giving them names is one way people try to conceptualize and understand them. One of the earliest examples of this is The Sand Reckoner, in which Archimedes gave a system for naming large numbers. To do this, he called the numbers up to a myriad myriad (108) "first numbers" and called 108 itself the "unit of the second numbers". Multiples of this unit then became the second numbers, up to this unit taken a myriad myriad times, 108·108=1016. This became the "unit of the third numbers", whose multiples were the third numbers, and so on. Archimedes continued naming numbers in this way up to a myriad myriad times the unit of the 108-th numbers, i.e. $(10^{8})^{(10^{8})}=10^{8\cdot 10^{8}},$ and embedded this construction within another copy of itself to produce names for numbers up to $((10^{8})^{(10^{8})})^{(10^{8})}=10^{8\cdot 10^{16}}.$ Archimedes then estimated the number of grains of sand that would be required to fill the known universe, and found that it was no more than "one thousand myriad of the eighth numbers" (1063). Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that have no existence outside the imagination. One motivation for such a pursuit is that attributed to the inventor of the word googol, who was certain that any finite number "had to have a name". Another possible motivation is competition between students in computer programming courses, where a common exercise is that of writing a program to output numbers in the form of English words. Most names proposed for large numbers belong to systematic schemes which are extensible. Thus, many names for large numbers are simply the result of following a naming system to its logical conclusion—or extending it further. Origins of the "standard dictionary numbers" The words bymillion and trimillion were first recorded in 1475 in a manuscript of Jehan Adam. Subsequently, Nicolas Chuquet wrote a book Triparty en la science des nombres which was not published during Chuquet's lifetime. However, most of it was copied by Estienne de La Roche for a portion of his 1520 book, L'arismetique. Chuquet's book contains a passage in which he shows a large number marked off into groups of six digits, with the comment: Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers point tryllion Le quart quadrillion Le cinqe quyllion Le sixe sixlion Le sept.e septyllion Le huyte ottyllion Le neufe nonyllion et ainsi des ault's se plus oultre on vouloit preceder (Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go). Adam and Chuquet used the long scale of powers of a million; that is, Adam's bymillion (Chuquet's byllion) denoted 1012, and Adam's trimillion (Chuquet's tryllion) denoted 1018. The googol family The names googol and googolplex were invented by Edward Kasner's nephew Milton Sirotta and introduced in Kasner and Newman's 1940 book Mathematics and the Imagination[14] in the following passage: The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would happen if one tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it. Value Name Authority 10100GoogolKasner and Newman, dictionaries (see above) 10googol = 1010100GoogolplexKasner and Newman, dictionaries (see above) John Horton Conway and Richard K. Guy[15] have suggested that N-plex be used as a name for 10N. This gives rise to the name googolplexplex for 10googolplex = 101010100. Conway and Guy[15] have proposed that N-minex be used as a name for 10−N, giving rise to the name googolminex for the reciprocal of a googolplex, which is written as 10-(10100). None of these names are in wide use. The names googol and googolplex inspired the name of the Internet company Google and its corporate headquarters, the Googleplex, respectively. Extensions of the standard dictionary numbers This section illustrates several systems for naming large numbers, and shows how they can be extended past vigintillion. Traditional British usage assigned new names for each power of one million (the long scale): 1,000,000 = 1 million; 1,000,0002 = 1 billion; 1,000,0003 = 1 trillion; and so on. It was adapted from French usage, and is similar to the system that was documented or invented by Chuquet. Traditional American usage (which was also adapted from French usage but at a later date), Canadian, and modern British usage assign new names for each power of one thousand (the short scale.) Thus, a billion is 1000 × 10002 = 109; a trillion is 1000 × 10003 = 1012; and so forth. Due to its dominance in the financial world (and by the US dollar), this was adopted for official United Nations documents. Traditional French usage has varied; in 1948, France, which had originally popularized the short scale worldwide, reverted to the long scale. The term milliard is unambiguous and always means 109. It is seldom seen in American usage and rarely in British usage, but frequently in continental European usage. The term is sometimes attributed to French mathematician Jacques Peletier du Mans circa 1550 (for this reason, the long scale is also known as the Chuquet-Peletier system), but the Oxford English Dictionary states that the term derives from post-Classical Latin term milliartum, which became milliare and then milliart and finally our modern term. Concerning names ending in -illiard for numbers 106n+3, milliard is certainly in widespread use in languages other than English, but the degree of actual use of the larger terms is questionable. The terms "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish, and "миллиард," milliard (transliterated) in Russian, are standard usage when discussing financial topics. For additional details, see billion and long and short scale. The naming procedure for large numbers is based on taking the number n occurring in 103n+3 (short scale) or 106n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion. In this way, numbers up to 103·999+3 = 103000 (short scale) or 106·999 = 105994 (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers if n is 9 or smaller. For larger n (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy.[15] Today, sexdecillion and novemdecillion are standard dictionary numbers and, using the same reasoning as Conway and Guy did for the numbers up to nonillion, could probably be used to form acceptable prefixes. The Conway–Guy system for forming prefixes: Units Tens Hundreds 1 Un N Deci NX Centi 2 Duo MS Viginti N Ducenti 3 Tre () NS Triginta NS Trecenti 4 Quattuor NS Quadraginta NS Quadringenti 5 Quinqua NS Quinquaginta NS Quingenti 6 Se () N Sexaginta N Sescenti 7 Septe () N Septuaginta N Septingenti 8 Octo MX Octoginta MX Octingenti 9 Nove () Nonaginta Nongenti (*) ^ When preceding a component marked S or X, "tre" changes to "tres" and "se" to "ses" or "sex"; similarly, when preceding a component marked M or N, "septe" and "nove" change to "septem" and "novem" or "septen" and "noven". Since the system of using Latin prefixes will become ambiguous for numbers with exponents of a size which the Romans rarely counted to, like 106,000,258, Conway and Guy co-devised with Allan Wechsler the following set of consistent conventions that permit, in principle, the extension of this system indefinitely to provide English short-scale names for any integer whatsoever.[15] The name of a number 103n+3, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 103m+3, where m represents each group of comma-separated digits of n, with each but the last "-illion" trimmed to "-illi-", or, in the case of m = 0, either "-nilli-" or "-nillion".[15] For example, 103,000,012, the 1,000,003rd "-illion" number, equals one "millinillitrillion"; 1033,002,010,111, the 11,000,670,036th "-illion" number, equals one "undecillinilliseptuagintasescentillisestrigintillion"; and 1029,629,629,633, the 9,876,543,210th "-illion" number, equals one "nonilliseseptuagintaoctingentillitresquadragintaquingentillideciducentillion".[15] The following table shows number names generated by the system described by Conway and Guy for the short and long scales.[16] Base -illion (short scale) Base -illion (long scale) Value US, Canada and modern British (short scale) Traditional British (long scale) Traditional European (Peletier) (long scale) SI Symbol SI Prefix 1 1 106 Million Million Million M Mega- 2 1 109 Billion Thousand million Milliard G Giga- 3 2 1012 Trillion Billion Billion T Tera- 4 2 1015 Quadrillion Thousand billion Billiard P Peta- 5 3 1018 Quintillion Trillion Trillion E Exa- 6 3 1021 Sextillion Thousand trillion Trilliard Z Zetta- 7 4 1024 Septillion Quadrillion Quadrillion Y Yotta- 8 4 1027 Octillion Thousand quadrillion Quadrilliard R Ronna- 9 5 1030 Nonillion Quintillion Quintillion Q Quetta- 10 5 1033 Decillion Thousand quintillion Quintilliard 11 6 1036 Undecillion Sextillion Sextillion 12 6 1039 Duodecillion Thousand sextillion Sextilliard 13 7 1042 Tredecillion Septillion Septillion 14 7 1045 Quattuordecillion Thousand septillion Septilliard 15 8 1048 Quindecillion Octillion Octillion 16 8 1051 Sedecillion Thousand octillion Octilliard 17 9 1054 Septendecillion Nonillion Nonillion 18 9 1057 Octodecillion Thousand nonillion Nonilliard 19 10 1060 Novendecillion Decillion Decillion 20 10 1063 Vigintillion Thousand decillion Decilliard 21 11 1066 Unvigintillion Undecillion Undecillion 22 11 1069 Duovigintillion Thousand undecillion Undecilliard 23 12 1072 Tresvigintillion Duodecillion Duodecillion 24 12 1075 Quattuorvigintillion Thousand duodecillion Duodecilliard 25 13 1078 Quinvigintillion Tredecillion Tredecillion 26 13 1081 Sesvigintillion Thousand tredecillion Tredecilliard 27 14 1084 Septemvigintillion Quattuordecillion Quattuordecillion 28 14 1087 Octovigintillion Thousand quattuordecillion Quattuordecilliard 29 15 1090 Novemvigintillion Quindecillion Quindecillion 30 15 1093 Trigintillion Thousand quindecillion Quindecilliard 31 16 1096 Untrigintillion Sedecillion Sedecillion 32 16 1099 Duotrigintillion Thousand sedecillion Sedecilliard 33 17 10102 Trestrigintillion Septendecillion Septendecillion 34 17 10105 Quattuortrigintillion Thousand septendecillion Septendecilliard 35 18 10108 Quintrigintillion Octodecillion Octodecillion 36 18 10111 Sestrigintillion Thousand octodecillion Octodecilliard 37 19 10114 Septentrigintillion Novendecillion Novendecillion 38 19 10117 Octotrigintillion Thousand novendecillion Novendecilliard 39 20 10120 Noventrigintillion Vigintillion Vigintillion 40 20 10123 Quadragintillion Thousand vigintillion Vigintilliard 50 25 10153 Quinquagintillion Thousand quinvigintillion Quinvigintilliard 60 30 10183 Sexagintillion Thousand trigintillion Trigintilliard 70 35 10213 Septuagintillion Thousand quintrigintillion Quintrigintilliard 80 40 10243 Octogintillion Thousand quadragintillion Quadragintilliard 90 45 10273 Nonagintillion Thousand quinquadragintillion Quinquadragintilliard 100 50 10303 Centillion Thousand quinquagintillion Quinquagintilliard 101 51 10306 Uncentillion Unquinquagintillion Unquinquagintillion 110 55 10333 Decicentillion Thousand quinquinquagintillion Quinquinquagintilliard 111 56 10336 Undecicentillion Sesquinquagintillion Sesquinquagintillion 120 60 10363 Viginticentillion Thousand sexagintillion Sexagintilliard 121 61 10366 Unviginticentillion Unsexagintillion Unsexagintillion 130 65 10393 Trigintacentillion Thousand quinsexagintillion Quinsexagintilliard 140 70 10423 Quadragintacentillion Thousand septuagintillion Septuagintilliard 150 75 10453 Quinquagintacentillion Thousand quinseptuagintillion Quinseptuagintilliard 160 80 10483 Sexagintacentillion Thousand octogintillion Octogintilliard 170 85 10513 Septuagintacentillion Thousand quinoctogintillion Quinoctogintilliard 180 90 10543 Octogintacentillion Thousand nonagintillion Nonagintilliard 190 95 10573 Nonagintacentillion Thousand quinnonagintillion Quinnonagintilliard 200 100 10603 Ducentillion Thousand centillion Centilliard 300 150 10903 Trecentillion Thousand quinquagintacentillion Quinquagintacentilliard 400 200 101203 Quadringentillion Thousand ducentillion Ducentilliard 500 250 101503 Quingentillion Thousand quinquagintaducentillion Quinquagintaducentilliard 600 300 101803 Sescentillion Thousand trecentillion Trecentilliard 700 350 102103 Septingentillion Thousand quinquagintatrecentillion Quinquagintatrecentilliard 800 400 102403 Octingentillion Thousand quadringentillion Quadringentilliard 900 450 102703 Nongentillion Thousand quinquagintaquadringentillion Quinquagintaquadringentilliard 1000 500 103003 Millinillion [17] Thousand quingentillion Quingentilliard Value Name Equivalent US, Canadian and modern British (short scale) Traditional British (long scale) Traditional European (Peletier) (long scale) 10100 Googol Ten duotrigintillion Ten thousand sedecillion Ten sedecilliard 1010100 Googolplex [1] Ten trillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentillitrestrigintatrecentilliduotrigintatrecentillion [2] Ten thousand millisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillion [2] Ten millisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentillisesexagintasescentilliard ^[1] Googolplex's short scale name is derived from it equal to ten of the 3,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,332nd "-illion"s (This is the value of n when 10 × 10(3n + 3) = 1010100) ^[2] Googolplex's long scale name (both traditional British and traditional European) is derived from it being equal to ten thousand of the 1,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666,666th "-illion"s (This is the value of n when 10,000 × 106n = 1010100). Binary prefixes The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 10241 and 10248.[18] Power Value ISQ symbol ISQ prefix 1 10241 Ki Kibi- 2 10242 Mi Mebi- 3 10243 Gi Gibi- 4 10244 Ti Tebi- 5 10245 Pi Pebi- 6 10246 Ei Exbi- 7 10247 Zi Zebi- 8 10248 Yi Yobi- Other named large numbers used in mathematics, physics and chemistry • Avogadro number • Graham's number • Skewes's number • Steinhaus–Moser notation • TREE(3) • Rayo's number See also • -yllion – Mathematical notation • Asaṃkhyeya – Buddhist name for a large number • Chinese numerals – Words and characters used to denote numbers in Chinese • History of large numbers • Indefinite and fictitious numbers • Indian numbering system – Indic methods of naming large numbers • Knuth's up-arrow notation – Method of notation of very large integers • Law of large numbers – Averages of repeated trials converge to the expected value • List of numbers – Notable numbers • Long and short scale – Two meanings of "billion" and "trillion"Pages displaying short descriptions of redirect targets • Metric prefix – Order of magnitude indicator • Names of small numbers – Usage and derivation of names • Number names – Word or phrase which describes a numerical quantity • Number prefix – Prefix derived from numerals or other numbersPages displaying short descriptions of redirect targets • Orders of magnitude – Scale of numbers with a fixed ratioPages displaying short descriptions of redirect targets • Orders of magnitude (data) – Computer data measurements and scales. • Orders of magnitude (numbers) – Scale of numbers of interest arranged from small to large • Power of 10 – Ten raised to an integer power References 1. Bellos, Alex (2011). Alex's Adventures in Numberland. A&C Black. p. 114. ISBN 978-1-4088-0959-4. 2. The American Heritage Dictionary of the English Language (4th ed.). Houghton Mifflin. 2000. ISBN 0-395-82517-2. 3. "Collins English Dictionary". HarperCollins. 4. "Cambridge Dictionaries Online". Cambridge University Press. 5. The Oxford English Dictionary (2nd ed.). Clarendon Press. 1991. ISBN 0-19-861186-2. 6. "Oxford English Dictionary". Oxford University Press. 7. The Random House Dictionary of the English Language (2nd ed.). Random House. 1987. 8. Brown, Lesley; Little, William (1993). The New Shorter Oxford English Dictionary. Oxford University Press. ISBN 0198612710. 9. Webster, Noah (1981). Webster's Third New International Dictionary of the English Language, Unabridged. Merriam-Webster. ISBN 0877792011. 10. Rowlett, Russ. "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. Archived from the original on 1 March 2000. Retrieved 25 September 2022. 11. Emerson, Oliver Farrar (1894). The History of the English Language. Macmillan and Co. p. 316. 12. "Entry for centillion in dictionary.com". dictionary.com. Retrieved 25 September 2022. 13. "Zimbabwe rolls out Z$100tr note". BBC News. 16 January 2009. Retrieved 25 September 2022. 14. Kasner, Edward; Newman, James (1940). Mathematics and the Imagination. Simon and Schuster. ISBN 0-486-41703-4. 15. Conway, J. H.; Guy, R. K. (1998). The Book of Numbers. Springer Science & Business Media. pp. 15–16. ISBN 0-387-97993-X. 16. Fish. "Conway's illion converter". Retrieved 1 March 2023. 17. Stewart, Ian (2017). Infinity: A Very Short Introduction. Oxford University Press. p. 20. ISBN 978-0-19-875523-4. 18. "IEC 80000-13:2008". International Organization for Standardization. 15 April 2008. Retrieved 25 September 2022. Large numbers Examples in numerical order • Thousand • Ten thousand • Hundred thousand • Million • Ten million • Hundred million • Billion • Trillion • Quadrillion • Quintillion • Sextillion • Septillion • Octillion • Nonillion • Decillion • Eddington number • Googol • Shannon number • Googolplex • Skewes's number • Moser's number • Graham's number • TREE(3) • SSCG(3) • BH(3) • Rayo's number • Transfinite numbers Expression methods Notations • Scientific notation • Knuth's up-arrow notation • Conway chained arrow notation • Steinhaus–Moser notation Operators • Hyperoperation • Tetration • Pentation • Ackermann function • Grzegorczyk hierarchy • Fast-growing hierarchy Related articles (alphabetical order) • Busy beaver • Extended real number line • Indefinite and fictitious numbers • Infinitesimal • Largest known prime number • List of numbers • Long and short scales • Number systems • Number names • Orders of magnitude • Power of two • Power of three • Power of 10 • Sagan Unit • Names • History
Workshop on Geometric Methods in Physics The Workshop on Geometric Methods in Physics (WGMP) is a conference on mathematical physics focusing on geometric methods in physics . It is organized each year since 1982 in the village of Białowieża, Poland. It is organized by the Chair of Mathematical Physics of Faculty of Mathematics, University of Białystok. Its founder and main organizer is Anatol Odzijewicz.[1] Workshop on Geometric Methods in Physics StatusActive GenreMathematics conference FrequencyAnnual Location(s)Białowieża CountryPoland Years active1982–present Inaugurated1982 (1982) FounderAnatol Odzijewicz Most recent19 June – 25 June 2022 Next eventJune – July 2023 ActivityActive Organised byUniversity of Białystok Websitewgmp.uwb.edu.pl WGMP takes place in its home venue, in the heart of the Białowieża National Park. A number of social events, including campfire, an excursion to the Białowieża forest and a banquet, are usually organized during the week. Notable participants In the past, Workshops were attended by scientists including: Roy Glauber, Francesco Calogero, Ludvig Faddeev, Martin Kruskal, es:Bogdan Mielnik, Emma Previato, Stanisław Lech Woronowicz, Vladimir E. Zakharov, Dmitry Anosov, de:Gérard Emch, George Mackey, fr:Moshé Flato, Daniel Sternheimer, Tudor Ratiu, Simon Gindikin, Boris Fedosov, pl:Iwo Białynicki-Birula, Jędrzej Śniatycki, Askolʹd Perelomov, Alexander Belavin, Yvette Kosmann-Schwarzbach, pl:Krzysztof Maurin, Mikhail Shubin, Kirill Mackenzie.[2] Special sessions Many times special sessions were scheduled within the programme of the Workshop. In the year 2016 there was a session "Integrability and Geometry" financed by National Science Foundation.[3][4] In the year 2017 there was a session dedicated to the memory and scientific achievements of S. Twareque Ali, long time participant and co-organizer of the Workshop. In the year 2018 there was a session dedicated to scientific achievements of prof. Daniel Sternheimer on the occasion of his 80th birthday. In the previous years, there were sessions dedicated to other prominent mathematicians and physicists such as S.L. Woronowicz, G. Emch, B. Mielnik, F. Berezin.[5] School on Geometry and Physics Since 2012 the Workshop is accompanied by a School on Geometry and Physics, which is targeted at young researchers and graduate students. During the School several courses by leading experts in mathematical physics take place. Proceedings Starting at 1992, after the Workshop a volume of proceedings is published. In the recent years it was published in the series Trends in Mathematics by Birkhäuser.[6] In 2005 a commemorative tome Twenty Years of Bialowieza: A Mathematical Anthology. Aspects of Differential Geometric Methods in Physics was published by World Scientific.[7] References 1. Webpage of Faculty of Mathematics, University of Białystok 2. Voronov, Theodore; Ali, Syed Twareque; Goliński, Tomasz (March 2010). "The Białowieża Meetings on Geometric Methods in Physics: Thirty Years of Success and Inspiration" (PDF). European Mathematical Society Newsletter. No. 75. 3. NSF Award Abstract 4. Integrability and Geometry at WGMP 2016. Post-conference materials. 5. Berceanu, Stefan (August 2013). "Berezin la Białowieża XXX – o perspectivă personală" (PDF). Curierul de Fizica. No. 75. 6. List of proceedings volumes, WGMP webpage 7. Ali, Syed Twareque; Emch, Gerard G.; Odzijewicz, Anatol; Sclichenmaier, Martin; Woronowicz, Stanisław Lech, eds. (2005). Twenty Years of Bialowieza: A Mathematical Anthology. Aspects of Differential Geometric Methods in Physics. World Scientific Monograph Series in Mathematics. Vol. 8. World Scientific. doi:10.1142/5744. ISBN 978-981-256-146-6. Further reading • Voronov, Theodore; Ali, Syed Twareque; Goliński, Tomasz (March 2010). "The Białowieża Meetings on Geometric Methods in Physics: Thirty Years of Success and Inspiration" (PDF). European Mathematical Society Newsletter. No. 75. External links • Conference webpage • Workshop on Geometric Methods in Physics on Facebook
World Education Games The World Education Games[1] is a global online event for all schools and students around the world and is held semi annually during the month of October. It is the expanded format of what was once known as World Maths Day but it now includes World Literacy Day and World Science Day too. It is organized by the 3PLearning and sponsored by Microsoft,[2] UNICEF,[3] 3P Learning[2] and MACQUARIE.[4] The World Maths Day holds the 'Guinness World Record' for the Largest Online Maths Competition in 2010.[5][6] Its Global Ambassador is 'Scott Flansburg' aka the Human Calculator[7]. World Education Games Logo of World Education Games GenreInternational Event for Students around the Globe FrequencyBiennial Location(s)Worldwide Inaugurated2007 Most recentOctober 2015 Next eventMarch 2018 ParticipantsOpen to any student 4-18 years Attendance5,960,862 students from 240 Countries Patron(s)Microsoft UNICEF 3P Learning Macquarie Group Organised by3P Learning Websiteworldeducationgames.com Its inception with the expanded format was in 2012 when 5,960,862 students from 240 countries and territories around the world competed with each other. In 2013, it was held March 5–7. The World Education Games had taken place October 13 through 15, 2015,[8] where over 6 Million students joined worldwide from over 20,000 schools in 159 countries and raised over $100,000 which will help send 33,000 students to school.[9] History The World Education Games is a major free online educational competition-style event, hosted by the global e-learning provider 3P Learning (creators of subscription-based e-learning platforms designed primarily for schools - such as Mathletics, Spellodrome and IntoScience). The World Education Games had its origins purely as a mathematics-based event, then known as World Maths Day in 2007. The event was powered by 3P Learning's flagship online learning resource, Mathletics. In 2011, the event expanded to include a second subject (World Spelling Day, renamed World Literacy Day in 2013), followed a year later by a third subject (World Science Day) and at which point the event took on the fully encompassing World Education Games[10] name and branding. Since 2012 The World Education Games has been collaborating with UNICEF[11] in the framework of a program called "School in a Box"[12] that supports the development of education in regions that are affected by various disasters and poverty. Rules Participation in the games is open to all students from any country and is free. Registration is required and an access to the internet is a must. Students are matched according to their age and grade levels or abilities if such is requested by their teachers. Students play randomly against other students from all over the world. Students answer as many questions as possible during the allotted time for each game. Correct answers get points, wrong answers no points and three wrong answers end the game prematurely. Each student plays and scored for the points accumulated during the first 20 games only. Results are announced after counting all the points and after the organizers had communicated with the parents/ teachers of possible winners and ensured that they had participated in the Games under their own registered accounts. Multiple registrations and/ or playing under someone's else account is a violation punishable by annulling the results of everyone involved. Winners are the students who score the highest points in their grade level in each competition separately and in total. Schools are also awarded for receiving the highest point-average in each grade level as long as at least 10 students had participated from that school. Top 100 participants also get their achievements listed on the Hall of Fame.[13] World Education Games Ambassadors Students who win their regional lead-up events in their countries are hand-picked to become ambassadors.[14] No.AmbassadorCountry 1Alexander Y United Kingdom 2Alexandra B Australia 3Amy M United Kingdom 4Anna A Russia 5Creedon C Canada 6Ellie E United Kingdom 7Emmanuel M Mexico 8Fatima Y Pakistan 9Geoffrey M Canada 10 Gerania R United States 11Hui Qing L United Kingdom 12 Imaan C United Kingdom 13Kalliopi C[15] Australia 14 Luke W United States 15Maathangi A United Arab Emirates 16Meeral N Pakistan 17 Melina S United States 18Michael Murray United States 19Musaab H Canada 20 Steve Jobs United Kingdom 21 Peyton H Canada 22 Remi L Australia 23 Mariam K[16] Australia 24 Samuel O[17] Nigeria 25 Thomas United Kingdom 26 Tristan G Australia 27 Ursula H Australia 28 Vikayra G South Africa Prizes[18] Platinum Prizes • A glittering award ceremony to celebrate the winners will be held in November 2015 at the Sydney Opera House. • The student with the highest total World Education Games score (in each of the age categories) will be invited to attend the award ceremony to receive their medal. • Winning students will be flown to Sydney, Australia, along with one parent, to attend the ceremony. The trip includes flights, accommodation and a VIP Sydney tour. Trophies • Trophies will be awarded to the top scoring school in each of the three World Education Games events. • Trophies will be awarded across each of the ten year/grade categories. • Trophies will be specially engraved with the details of the winning schools. Medals • Medals will be awarded to the Top Scoring Students in each of the three World Education Games events. • Medals will be awarded across each of the ten year/grade categories. • Students finishing in first, second and third place in each of the age categories will receive a gold, silver or bronze medal. • Medals will be specially engraved with the details of the winning students. The winners of each group are awarded a 'minted gold medal' and the top ten in each group receive 'gold medals'. There are also various other prizes including trophies and certificates. A full list of winners including top ten in each category is available at the official website of World Education Games. The complete list of various prizes and cups over the years can be found in the official website of World Education Games Winners by countries Literacy Maths Science WEG 2015 Pending Pending Pending Pending 2014 No Games No Games No Games No Games 2013[19] Malaysia Turkey Malaysia Malaysia 2012 [20] United Kingdom Australia Malaysia United Kingdom Winners by names 4-7 yrs 8-10 yrs 11-13 yrs 14-18 yrs 2015 Pending Pending Pending Pending 2014 No Games No Games No Games No Games 2013[19] Sandali Rajapakse Salcombe Prep School, UK Vihangi Rajapakse Salcombe Prep School, UK Sachin Kumar Mital Canadian International School, Hong Kong & Shoaib hassan Beaconhouse School System, Mandi, Bahauddin, Pakistan Danial bin Muhammad Syafiq Cempaka Schools, CH, Malaysia 2012[20] Sandali Rajapakse Salcombe Prep School, UK Oliver Papillo Balwyn Primary School, Australia Sharan Maiya The Glasgow Academy, UK Malayandi P Cempaka Schools DH, Malaysia Winners of individual events World Literacy Day World Maths Day World Science Day 2015 11-13 yrs: Sydny Lum Shen Li 11-13 yrs: Sydny Lum Shen Li 11-13 yrs: Sydny Lum Shen Li 2014 No Games No Games No Games 2013[19] 4-7 yrs: Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Alastair Gibson, Hexham Middle School, UK 11-13 yrs: Sydny Lum Shen Li 14-18 yrs: Kianna Wan, Team Canada, Canada 4-7yrs : Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Rohith Niranjan, Global Indian International School, Japan 11-13 yrs: Ata Çağın Kolbaşı, Ata College, Izmir, Turkey 14-18 yrs: Husnain Ali Abid, FFC Grammar School, Pakistan 4-7 yrs: Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Chiacia Putri Effendy, Cahaya Harapan Sejahtera, Indonesia 11-13 yrs: Aryan Saju, The British Al Khubairat, UAE 14-18yrs: Danial Bin Muhammad Syafiq, Cempaka Schools CH, Malaysia 2012[20] 4-7 yrs: Sandali Rajapakse, Salcombe Prep School, UK 8-10 yrs: Dylan.C, Linn Primary School, UK 11-13 yrs: Edryna Syfinaz Z A, Cempaka Schools DH, Malaysia 14-18 yrs: Phoebe M, Sha Tin College, Hong Kong 4-7yrs: Yousuf Mohammad, Orbit International School, Saudi Arabia 8-10 yrs: Darshan.S, Indian Public School, India 11-13 yrs: Moosa FerozeTarrar, Beaconhouse School System, Pakistan 14-18 yrs: Kaya Genc, Southport College, Australia 4-7 yrs: Ashwati. N, Christ the Sower School, UK 8-10 yrs: Derek.L, Monterey Ridge Elementary School, USA 11-13 yrs: Sharan Maiya, The Glasgow Academy, UK 14-18yrs: Malayandi P, Cempaka Schools DH, Malaysia 2011 4-7 yrs: Vihangi Rajapakse, Salcombe Prep School, UK 8-10 yrs: Dylan.C, Linn Primary School, UK 11-13 yrs: George.W, Team United Kingdom, UK 14-18 yrs: Phoebe M, Sha Tin College, Hong Kong 4-7yrs: Eric Z, Team Australia, Australia 8-10 yrs: Mason F, Team New Zealand, New Zealand 11-13 yrs: Kaya Genc, The Southport School, Australia 14-18 yrs: David Andersen, Fraser Coast Anglican College, Australia No Games 2010 No Games 5-8yrs: Rohith Niranjan, Team Japan, Japan 9-13 yrs: Kaya Genc, The Southport School, Australia 14-18 yrs: David Andersen, Fraser Coast Anglican College, Australia No Games 2009 No Games 5-8yrs : N.S, The Sikh International School, Thailand 9-13 yrs: Kaya Genc, The Southport School, Australia 14-18 yrs: David Andersen, Fraser Coast Anglican College, Australia No Games 2008 No Games All Ages: Tatiana Devendranath, Haileybury College, Australia No Games 2007 No Games All Ages: Stefan L, Christian Alliance P.C. Lau Memorial International School, Hong Kong No Games See also • Mathletics • 3P Learning References 1. "World Education Games – home". 3P Learning. Retrieved 2015-09-25. 2. Games, The World Education. "The World Education Games Brings Together 8MM Students From 30,000 Schools in 200 Countries and Challenges Kids in Math, Literacy and Science". www.prnewswire.com. Retrieved 2015-09-24. 3. "3P Learning". 4. "3P Learning Pty Ltd (Organizations on EdSurge)". EdSurge. Retrieved 2015-09-24. 5. "The Worlds Largest Online Maths Competition". Archived from the original on March 7, 2012. 6. "Top Ten Facts About Maths". 7. "Fastest human calculator". Guinness World Records. Retrieved 2015-10-17. 8. "World Education Games to Return in 2015". 9. "The World Education Games Brings Together 8MM Students From 30,000 Schools in 200 Countries and Challenges Kids in Math, Literacy and Science". Reuters. 2015-09-21. Archived from the original on 2015-09-30. Retrieved 2015-09-29. 10. Games, The World Education. "The World Education Games Brings Together 8MM Students From 30,000 Schools in 200 Countries and Challenges Kids in Math, Literacy and Science". www.prnewswire.com. Retrieved 2015-09-25. 11. "Support UNICEF". support.unicef.org. Retrieved 2015-10-27. 12. "World Education Games". secured.unicef.org.au. Retrieved 2015-10-27. 13. "World Education Games 2015". worldeducationgames.com. Retrieved 2015-10-26. 14. "Hall of Fame Home". World Maths Day 2018. Retrieved 2018-06-03. 15. Media, Fairfax Regional. "Mandurah student crowned ambassador for games". Retrieved 2015-09-29. 16. "Teen wants to show positive side of Islam". 17. "African Ambassador makes his debut speech!". 3P Learning. Retrieved 2015-09-29. 18. "WEG Trophies and Medals". 3P Learning. Retrieved 2015-09-24. 19. "World Education Games 2013 Winners" (PDF). 20. "The 2012 World Education Games Results Fact Sheet" (PDF). External links • Official website • The World Education Games Blog • Всемирные Образовательные Игры (in Russian).
World Maths Day World Maths Day (World Math Day in American English) is an online international mathematics competition, powered by Mathletics (a learning platform from 3P Learning, the same organisation behind Reading Eggs and Mathseeds).[1] Smaller elements of the wider Mathletics program effectively power the World Maths Day event. World Maths Day GenreInternational Event Years active16 Inaugurated2007 Founder3P Learning Most recent2022 Next event8 March 2023 Attendance5,960,862 students from 240 Countries Organised by3P Learning Websitewww.worldmathsday.com The first World Maths Day started in 2007.[2] Despite these origins, the phrases "World Maths Day" and "World Math Day" are trademarks, and not to be confused with other competitions such as the International Mathematical Olympiad or days such as Pi Day. In 2010, World Maths Day created a Guinness World Record for the Largest Online Maths Competition.[3][4] The next World Maths Day will take place on the 8th of March 2023. Overview Open to all school-aged students (4 to 18 years old), World Maths Day involves participants playing 20 × 60-second games, with the platform heavily based on "Live Mathletics" found in Mathletics. The contests involve mental maths problems appropriate for each age group, which test the accuracy and speed of the students as they compete against other students across the globe. The simple but innovative idea of combining the aspects of multi-player online gaming with maths problems has contributed to its popularity around the world. There will be 10 Year group divisions for students to compete in from Kindergarten to Year 9 and above. An online Hall of Fame will track points throughout the competition with prizes to be awarded to the top students and schools.[5] The Champions Challenge is a new addition to the 2021 competition. Top Year/Grade 9 and above World Maths Day student come together to compete in a knockout tournament. As part of the challenge, students will have their event live streamed, bringing mathematics and Esports together.[6] History The inaugural World Maths Day was held on March 13, 2007. 287,000 students from 98 countries answered 38,904,275 questions. The student numbers and the participating countries have steadily increased in the following years. In 2009, 1.9 million students took part in World Maths Day. In 2011, World Maths Day sets a Guinness World Record for the Largest Online Maths Competition,[7] with almost 500 million maths questions answered during the event. In 2012, 3P Learning launched the World Education Games. Over 5.9 Million students from 240 Countries and Territories around the world registered to take part, with World Maths Day being the biggest attraction. In 2013, it was held between 5–7 March and the awards were presented at the Sydney Opera House to the Champions. In 2015, there were participants from 150 countries. US, UK and Australia all had over 1 million registrations. The 2019 World Maths Day event was combined with a social media competition, where students around the world were encouraged to dress up in a maths-themed outfit to celebrate maths. Entries included famous mathematicians, an aerial shot of students forming a pi symbol, and human calculators.[8] In 2022, World Maths Day celebrates 15 years in the making. Awards A number of awards are offered to the students who take part and for those who do well in the event. Additionally the champions and the top ten students in the world are awarded gold medals every year. There are also a number of national lead-up events in different regions around the world which are also based on the Mathletics format. Champions The individual gold medal winners through the years are listed below: 2007 Results All Ages[9] 1Stefan L, Christian Alliance P.C.Lau Memorial International School, Hong Kong 2Kelvin H, Taunton School, United Kingdom 3Ana Catarina V, CLIP, Portugal 4Simone C, Newlands Intermediate, New Zealand 5Maoki G, International Christian Academy of Nagoya, Japan 6Shoaib Akram S, Beaconhouse School, Cambridge Branch, Pakistan 7Joshua S, Dulwich College, Shanghai, China 8Yiannis Z, The English School Nicosia, Cyprus 9Nicolae F, Mark Twain International School, Romania 10Ross R, Team Australia, Australia 2008 Results All Ages 1Tatiana D, Haileybury College, Australia 2Rock T, George Heriot's School, UK 3Kaya G, The McDonald College, Australia 4Chris T, The English College, UAE 5TK, Garden International School, Malaysia 6Joshua S, Dulwich College, Shanghai, China 7Abhishek C, International Pioneer School, Thailand 8Crystal L, Team Australia, Australia 9Pratyush G, Delia School of Canada, Hong Kong 10AS, Team Thailand, Thailand 2009 Results 5-8 Years9-13 Years14-18 Years 1N S, Thai Sikh International School, ThailandKaya G, Team Australia, AustraliaDavid A, Fraser Coast Anglican College, Australia 2Dushyant S, International Pioneers School, ThailandDavid M, Aloha College, Marbella, SpainM G, Izmir OzelI Isikkent lisesi, Turkey 3N K, International Pioneers School, ThailandShoaib A, Team Pakistan, PakistanCarlos D, Amman National School, Jordan 4Alexander B, Bechtel Elementary School, USADante M, Canterbury School, SpainThevaa C, ACS Ipoh School, Malaysia 5O C, St Paul's Convent School, Hong KongLarksana Y, Ontario International Institute, CanadaE K, Tevitol High School, Turkey 6Nico A, Bechtel Elementary School, USACaleb L, Australian International School, UAESaptarshi C, Bangladesh International School Riyadh, Saudi Arabia 7Rico C, Clearwater Bay School, Hong KongK H, Dalat International School,Sultan V, Prague British School, Czech 8Eric S, Clearwater Bay School, Hong KongM S, Indian International School, JapanMiguel B, Escola Secundaria Jorge Peixinho, Portugal 9Eda K, Galliard Primary School, UKWasif S, Gordon A Brown Middle School, CanadaTatiana D, Team Australia, Australia 10M N, Bechtel Elementary School, USAH K, British School of Bucharest, RomaniaJ S, Inti University College, Malaysia 2010 Results 5-8 Years9-13 Years14-18 Years 1Vivek R, Our Lady of Lourdes Park Lodge, Northern IrelandKaya G, The Southport School, AustraliaDavid A, Fraser Coast Anglican College, Australia 2Yizhen Y, Team Singapore, SingaporeCaleb L, Australian International School, UAETatiana D, Team Australia, Australia 3Yan Tung Jovanna Y, St Paul's Convent School, Hong KongBrody H, Team Australia, AustraliaArthur T, Team Hong Kong, Hong Kong 4Sik Chee Harriet C, St Paul's Convent School, Hong KongSharan M, Hamilton College, United KingdomKai Yuan Y, Wesley Methodist School, Malaysia 5Tien-erh H, Team Malaysia, MalaysiaSatvik T, St Georges School Cologne, GermanyFrancis L, Cempaka Schools, Malaysia 6Jimin J, Lake Highland Preparatory SchoolDavid M, Aloha College, Marbella, SpainJake C, Sha Tin College, Hong Kong 7Daniel Newton F, Master Brain Academy, UKSyed Ali R, Beaconhouse School, Middle Branch, PakistanByung hee C, Saint Louis School, USA 8Max W, Helen Wilson Public School, CanadaSai M, India International School, JapanLee Y, Cempaka Schools, Malaysia 9Aidan S, West Leeming Primary School, AustraliaHarish S, India International School, JapanEdwin See Jun H, Cempaka Schools, Malaysia 10Sum yin Tracy M, St Paul's Convent School, Hong KongNaunidh S, Thai Sikh International School, ThailandMohammed Shaan R, Slough Grammar School, London, U.K 2011 Results 4-7 Years8-10 Years11-13 Years14-18 Years 1Eric Z, Team Australia, AustraliaMason F, Team New Zealand, New ZealandKaya G, The Southport School, AustraliaDavid A, Fraser Coast Anglican College, Australia 2Vihangi R, Salcombe Prep School, England, U.KSai M, India International School, JapanDavid M, Aloha College, Marbella, SpainTham C, Team Malaysia, Malaysia 3Evan M, Stanley Bay School, New ZealandEdwin V, St Joseph's School, New ZealandSatvik R, St George's The English International, Cologne, GermanyTiger Z, Team Malaysia, Malaysia 4Andrey M, Laude San Pedro International College, SpainSachin Kumar M, Canadian International School, Hong KongHarish S, Team India, IndiaYeoh K, Team Malaysia, Malaysia 5Aditya C, Team United States, USAMuhammad Abdul Mannan, Thorncliffe PS, CanadaMoosa Feroze T, BeaconHouse School System, PakistanEdwin See Jun H, Cempaka Schools, Malaysia 6Zahid B, Team Pakistan, PakistanWillem E, Remarkables Primary School, New ZealandSharan M, The Glasgow Academy, UK Lim C, Cempaka Schools, Malaysia 7Michael Z, Holy Family Primary School, AustraliaGordon C, German Swiss International School, Hong KongAhsan A, Beaconhouse School Mandi Bahauddin, PakistanAaron T, Sha Tin College, Hong Kong 8Jovanka Vienna S, Bunda Mulia International School, IndonesiaThomas P, Goulburn Street Primary, AustraliaChong Seng K, Independent SchoolSiddharth P, Team United States, USA 9Alldon Garren Tan T, Rosyth School, SingaporeMax W, Team Canada, CanadaAaron H, Team Australia, AustraliaMalayandi P, Cempaka Schools, Malaysia 10Ali S, Team Pakistan, PakistanDaniel Newton F, Master Brain Academy, UKAnna S, British International School of Ljubljana, SloveniaKaan Aykurt, Team Turkey, Turkey 2012 Results 4-7 Years8-10 Years11-13 Years14-18 Years 1Yousuf M, Team Saudi Arabia, Saudi ArabiaDarshan S, The Indian Public School, IndiaMoosa Feroze T, Beaconhouse School System, PakistanKaya G, Team Australia, Australia 2Sandali R, Salcombe Prep School, England, U.KOliver P, Balwyn Primary School, AustraliaHusnain Ali Abid, FFC Grammar H/S School, PakistanOsama Shahid, Beaconhouse School Gujranwala, Pakistan 3Daksh C, Team India, IndiaThomas P, Goulburn Street Primary, AustraliaKarl H, Team Australia, AustraliaZhe W, Team United States, USA 4Joy J, Team India, IndiaRohith N, Team Japan, JapanDavid M, Aloha College, Marbella, SpainMohammed Shaan R, Slough Grammar School, London, UK 5Austin M, AHES School, USARishabh K, Team India, IndiaJoseph T, Team Australia, AustraliaAisya A, Cempaka Schools DH, Malaysia 6Douglas G, Pitt Island School, New ZealandDaniel Newton F, Master Brain Academy, UKSachin Kumar M, Canadian International School, Hong KongFrancis L, Cempaka Schools DH, Malaysia 7Joyel G, Ghyllside Primary School, UKEric Z, Palmerston District Primary School, AustraliaAaron H, Team Australia, AustraliaAngela M, Team Macedonia, Macedonia 8Alison K, Canberra Grammar School, AustraliaLeon H, Team Australia, AustraliaEdwin V, Team New Zealand, New ZealandAaron T, Sha Tin College, Hong Kong 9Archie G, Pitt Island School, New ZealandVihangi R, Salcombe Prep School, England, U.KSharan M, The Glasgow Academy, UKEdwin See Jun H, Cempaka Schools DH, Malaysia 10Leow Z, Sri Tenby School, MalaysiaKangan M, Australian International SchoolFilip Szary, Team England, UKBelinda C, Green Bay High School, New Zealand 2013 Results 4-7 Years[10] 8-10 Years[10]11-13 Years[10]14-18 Years[10] 1Sandali R, Salcombe Prep School, England, U.KRohith N, Global Indian International School, JapanKedar H, Liverpool Public School, AustraliaHusnain Ali A, FFC Grammar School, Pakistan 2Becky L, Undercliffe Public School, AustraliaVihangi R, Salcombe Prep School, England, UKMeet S, SN Kansagra, IndiaAaron H, Team Seaford, Australia 3Beykent Doga T, Beykent Doga College, TurkeyRohidh M, Riyadh, Saudi ArabiaMuhammad Abdul Mannan, Toronto, Canada Edwin See Jun H, Team Malaysia, Malaysia 4Martin E, Izmir Ata Koleji, TurkeyMartin E, Izmir Ata Koleji, TurkeyChoong M, Cempaka School CH, Malaysia Low C, Cempaka School CH, Malaysia 5Tuna Y, Izmir Ata Koleji, TurkeyYousuef M, Orbit International School (Khobar), Saudi ArabiaSachin Kumar M, Canadian International School, Hong KongHussain A, Bloomfield Hall Upper School, Pakistan 6Tapkac D, Izmir Ata Koleji, TurkeyAditya C, New Albany Elementary, Albany, USAFilip S, Team England, England Panayioti K, St Spyridon College, Australia 7Abeeha Saud K, Beaconhouse School, Mandi B, PakistanUgo Dos R, American International School of Bucharest, Romania Hassan Feroze T, Beaconhouse School Mandi B, PakistanHassan Ali A, FFC Grammar School, Pakistan 8Usman A, Millennium School Mirpur, Pakistan & Tuna C, Izmir Ata Koleji I, TurkeyDaniyal N, Australian International School, UAEKarl H, Riverina Anglican College, Australia & Willem E, Remarkables Primary School, New ZealandMohammed Shaan R, Team England, England, U.K 9Evan Manning, Team New Zealand, New ZealandApoorv Agrawal, Anubhuti School, India Danial B, Cempaka School CH, Malaysia 10Bahar F, Izmir Ata Koleji, TurkeyChloe Isabella Tsang, Chinese International School, Hong KongBenjamin H, McCarthy Catholic College, AustraliaYi Shuen L, Cempaka Schools DH, Malaysia 2015 Event The 2015 event was held on October 13-October 15, 2015. There were 10 ages categories: 1 each for grades K-8, and one for grades 9+. The game limit was dropped to 20 games per student. It is possible to play further, but these do not count to ones personal total, only the event total. 169 Million points were scored across Maths, Literacy and Science. Grade KGrade 1Grade 2Grade 3Grade 4Grade 5Grade 6Grade 7Grade 8Grade 9+ 1Jinansh D, Genius Kid, IndiaSarah R, Team PakistanAhmed Feroze T, Team PakistanMuhammad S, Beaconhouse School System, Mandi Bahauddin, PakistanAbeeha S, Beaconhouse School System, Mandi Bahauddin, PakistanAustin M, Steuart Weller, USABonnie L, Undercliffe Public School, AustraliaHashir Feroze T, Team PakistanDara H, Team AustraliaAli Saud K, Beaconhouse School System, Mandi Bahauddin, Pakistan 2Vilaxi S Geniud Kid, IndiaYashdeep S Genius Kid, IndiaBilge Kaan S Takev Karsiyaka, TurkeyEmmanuel A Ladybird Nursery/Primary School, NigeriaKainat F Beaconhouse School System, Mandi Bahauddin, PakistanFilbert Ephraim WMGC New Life Christian AcademySydny L Cempaka DamansaraIman F Team Pakistan, PakistanJayden L Cempaka DamansaraBenjamin H Team Australia Official National Mathletics Challenges leading up to World Maths Day Throughout the year Mathletics host several National Mathletics challenges in the lead up to World Maths Day. These challenges and the winners list are as follows: 2010 Results The American Math Challenge :Winner- Alek K, Haddonfields schools, Null. The Australian Maths Challenge :Winner- Parker C, Home Education, Queensland The Canadian Math Challenge :Winner- Shekar S, North Kipling Junior Middle School, ON. The European Schools Maths Challenge:Winner- Anna S, British International School of Ljubljana, Slovenia. The Middle East Schools Maths Challenge:Winner- Zakria Y, Australian International School, UAE. The NZ Maths Challenge :Winner- Vlad B, St Mary's School, Christchurch. The South African Maths Challenge :Winner- Jaden D, Wilton House, GT. The UK Four Nations Maths Challenge :Winner- Sharan Maiya, Glasgow Academy, Scotland. 2011 Results The American Math Challenge :Winner- Sayan Das, Team USA, Minnesota. The Australian Maths Challenge :Winner- Tatiana Devendranath, Team Australia, VIC. The Canadian Math Challenge :Winner- Tom.L, MPS, Etobicoke. The European Schools Maths Challenge:Winner- The Middle East Schools Maths Challenge:Winner- . The NZ Maths Challenge :Winner- Thomas Graydon, Pitt Island School. The Pakistan Maths Challenge: Winner- Dilsher A, The International School of Choueifat. The South African Maths Challenge :Winner- Jaden D, Team ZAF. The UK Four Nations Maths Challenge :Winner- Sharan Maiya, Glasgow Academy, Scotland, United Kingdom. 2012 Results The American Math Challenge :Winner- Zhe W, Team USA, Massachusetts The Latin American Math Challenge :Winner- Adriana Donis, Colegio Internacional Montessori, Guatemala The Australian Maths Challenge :Winner- Aaron Herrmann,, Seaford 6-12 School, South Australia The Canadian Math Challenge :Winner- Hanting C, Maywood Community School, Canada The European Schools Maths Challenge:Winner- Filip Szary, Team Poland The Middle East Schools Maths Challenge:Winner- Pushp raj P, MES Indian School, Qatar The NZ Maths Challenge :Winner- Willem Ebbinge, Remarkables Primary School, Otago The Pakistan Maths Challenge: Winner- Husnain Ali Abid, FFC Grammar H/S School, Punjab The South African Maths Challenge :Winner- Bradley P, Merrifield College, Eastern Cape The UK Four Nations Maths Challenge :Winner- Ryan Conlan, Team GBR, Scotland 2014 results The Nigerian Maths Challenge Winner Ayomide Adebanjo, Xplanter Private School, Lagos References 1. "Learn About 3P Learning - 15 Years of Edtech Excellence". 3P Learning. Retrieved 2023-01-03. 2. "Mathletics Announce World Maths Day 2018". influencing.com. Retrieved 2022-03-03. 3. "The Largest Online Maths Competition". Archived from the original on March 7, 2012. Retrieved March 22, 2012. 4. "Top ten facts about maths". Express. March 6, 2013. Retrieved January 16, 2014. 5. mathletics_admin (2021-04-22). "[Press Release] Mathletes get ready… World Maths Day is back! - Mathletics Japan" (in Japanese). Retrieved 2022-03-03. 6. Learning, 3P. "Mathletes get ready - World Maths Day is back". www.prnewswire.com (Press release). Retrieved 2022-03-03. 7. "Largest maths competition". Guinness World Records. 3 March 2010. Retrieved 2021-11-24. 8. "World Maths Day 2020". www.woodsprimaryschool.com. Retrieved 2022-03-03. 9. "History-2007". Archived from the original on 2013-04-08. Retrieved 2013-04-07. 10. "World Education Games 2013 Winners" (PDF). External links • Official website
Ranking A ranking is a relationship between a set of items such that, for any two items, the first is either "ranked higher than", "ranked lower than", or "ranked equal to" the second.[1] In mathematics, this is known as a weak order or total preorder of objects. It is not necessarily a total order of objects because two different objects can have the same ranking. The rankings themselves are totally ordered. For example, materials are totally preordered by hardness, while degrees of hardness are totally ordered. If two items are the same in rank it is considered a tie. By reducing detailed measures to a sequence of ordinal numbers, rankings make it possible to evaluate complex information according to certain criteria.[2] Thus, for example, an Internet search engine may rank the pages it finds according to an estimation of their relevance, making it possible for the user quickly to select the pages they are likely to want to see. Analysis of data obtained by ranking commonly requires non-parametric statistics. Strategies for handling ties It is not always possible to assign rankings uniquely. For example, in a race or competition two (or more) entrants might tie for a place in the ranking.[3] When computing an ordinal measurement, two (or more) of the quantities being ranked might measure equal. In these cases, one of the strategies below for assigning the rankings may be adopted. A common shorthand way to distinguish these ranking strategies is by the ranking numbers that would be produced for four items, with the first item ranked ahead of the second and third (which compare equal) which are both ranked ahead of the fourth.[4] These names are also shown below. Standard competition ranking ("1224" ranking) In competition ranking, items that compare equal receive the same ranking number, and then a gap is left in the ranking numbers. The number of ranking numbers that are left out in this gap is one less than the number of items that compared equal. Equivalently, each item's ranking number is 1 plus the number of items ranked above it. This ranking strategy is frequently adopted for competitions, as it means that if two (or more) competitors tie for a position in the ranking, the position of all those ranked below them is unaffected (i.e., a competitor only comes second if exactly one person scores better than them, third if exactly two people score better than them, fourth if exactly three people score better than them, etc.). Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 4 ("fourth"). This method is called "Low" by IBM SPSS[5] and "min" by the R programming language[6] in their methods to handle ties. Modified competition ranking ("1334" ranking) Sometimes, competition ranking is done by leaving the gaps in the ranking numbers before the sets of equal-ranking items (rather than after them as in standard competition ranking). The number of ranking numbers that are left out in this gap remains one less than the number of items that compared equal. Equivalently, each item's ranking number is equal to the number of items ranked equal to it or above it. This ranking ensures that a competitor only comes second if they score higher than all but one of their opponents, third if they score higher than all but two of their opponents, etc. Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 3 ("joint third"), C also gets ranking number 3 ("joint third") and D gets ranking number 4 ("fourth"). In this case, nobody would get ranking number 2 ("second") and that would be left as a gap. This method is called "High" by IBM SPSS[5] and "max" by the R programming language[6] in their methods to handle ties. Dense ranking ("1223" ranking) In dense ranking, items that compare equally receive the same ranking number, and the next items receive the immediately following ranking number. Equivalently, each item's ranking number is 1 plus the number of items ranked above it that are distinct with respect to the ranking order. Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B gets ranking number 2 ("joint second"), C also gets ranking number 2 ("joint second") and D gets ranking number 3 ("Third"). This method is called "Sequential" by IBM SPSS[5] and "dense" by the R programming language[7] in their methods to handle ties. Ordinal ranking ("1234" ranking) In ordinal ranking, all items receive distinct ordinal numbers, including items that compare equal. The assignment of distinct ordinal numbers to items that compare equal can be done at random, or arbitrarily, but it is generally preferable to use a system that is arbitrary but consistent, as this gives stable results if the ranking is done multiple times. An example of an arbitrary but consistent system would be to incorporate other attributes into the ranking order (such as alphabetical ordering of the competitor's name) to ensure that no two items exactly match. With this strategy, if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first") and D gets ranking number 4 ("fourth"), and either B gets ranking number 2 ("second") and C gets ranking number 3 ("third") or C gets ranking number 2 ("second") and B gets ranking number 3 ("third"). In computer data processing, ordinal ranking is also referred to as "row numbering". This method corresponds to the "first", "last", and "random" methods in the R programming language[6] to handle ties. Fractional ranking ("1 2.5 2.5 4" ranking) Items that compare equal receive the same ranking number, which is the mean of what they would have under ordinal rankings; equivalently, the ranking number of 1 plus the number of items ranked above it plus half the number of items equal to it. This strategy has the property that the sum of the ranking numbers is the same as under ordinal ranking. For this reason, it is used in computing Borda counts and in statistical tests (see below). Thus if A ranks ahead of B and C (which compare equal) which are both ranked ahead of D, then A gets ranking number 1 ("first"), B and C each get ranking number 2.5 (average of "joint second/third") and D gets ranking number 4 ("fourth"). Here is an example: Suppose you have the data set 1.0, 1.0, 2.0, 3.0, 3.0, 4.0, 5.0, 5.0, 5.0. The ordinal ranks are 1, 2, 3, 4, 5, 6, 7, 8, 9. For v = 1.0, the fractional rank is the average of the ordinal ranks: (1 + 2) / 2 = 1.5. In a similar manner, for v = 5.0, the fractional rank is (7 + 8 + 9) / 3 = 8.0. Thus the fractional ranks are: 1.5, 1.5, 3.0, 4.5, 4.5, 6.0, 8.0, 8.0, 8.0 This method is called "Mean" by IBM SPSS[5] and "average" by the R programming language[6] in their methods to handle ties. Statistics This section is an excerpt from Ranking (statistics).[edit] In statistics, ranking is the data transformation in which numerical or ordinal values are replaced by their rank when the data are sorted. For example, the numerical data 3.4, 5.1, 2.6, 7.3 are observed, the ranks of these data items would be 2, 3, 1 and 4 respectively. For example, the ordinal data hot, cold, warm would be replaced by 3, 1, 2. In these examples, the ranks are assigned to values in ascending order. (In some other cases, descending ranks are used.) Ranks are related to the indexed list of order statistics, which consists of the original dataset rearranged into ascending order. Sports This section is an excerpt from Standings (sports).[edit] In sports, standings, rankings, or league tables group teams of a particular league, conference, or division in a chart based on how well each is doing in a particular season of a sports league or competition. These lists are generally published in newspapers and other media, as well as the official web sites of the sports leagues and competitions. Education League tables are used to compare the academic achievements of different institutions. College and university rankings order institutions in higher education by combinations of factors. In addition to entire institutions, specific programs, departments, and schools are ranked. These rankings usually are conducted by magazines, newspapers, governments and academics. For example, league tables of British universities are published annually by The Guardian, The Independent, The Sunday Times, and The Times. The primary aim of these rankings is to inform potential applicants about British universities based on a range of criteria. Similarly, in countries like India, league tables are being developed and a popular magazine, Education World, published them based on data from TheLearningPoint.net. It is complained that the ranking of England's schools to rigid guidelines that fail to take into account wider social conditions actually makes failing schools even worse. This is because the most involved parents will then avoid such schools, leaving only the children of non-ambitious parents to attend.[8] Business In business, league tables list the leaders in investment banking activity, enabling people to quickly analyze financial data. Companies which collect this kind of data include Dealogic, whose league tables are rankings of investment banks in terms of the dollar volume of deals that investment banks work on; Bloomberg L.P., whose league tables provide an overview of top underwriters and legal advisers to securities deals, as well as fees netted from these transactions; and Thomson Reuters, whose league tables list the top financiers in a particular industry. Applications The rank methodology based on some specific indices is one of the most common systems used by policy makers and international organizations in order to assess the socio-economic context of the countries. Some notable examples include the Human Development Index (United Nations), Doing Business Index (World Bank), Corruption Perceptions Index (Transparency International), and Index of Economic Freedom (the Heritage Foundation). For instance, the Doing Business Indicator of the World Bank measures business regulations and their enforcement in 190 countries. Countries are ranked according to ten indicators that are synthesized to produce the final rank. Each indicator is composed of sub-indicators; for instance, the Registering Property Indicator is composed of four sub-indicators measuring time, procedures, costs, and quality of the land registration system. These kinds of ranks are based on subjective criteria for assigning the score. Sometimes, the adopted parameters may produce discrepancies with the empirical observations, therefore potential biases and paradox may emerge from the application of these criteria.[9] Other examples • In politics, rankings focus on the comparison of economic, social, environmental and governance performance of countries. • In relation to credit standing, the ranking of a security refers to where that particular security would stand in a wind up of the issuing company, i.e., its seniority in the company's capital structure. For instance, capital notes are subordinated securities; they would rank behind senior debt in a wind up. In other words, the holders of senior debt would be paid out before subordinated debt holders received any funds. • Search engines rank web pages by their expected relevance to a user's query using a combination of query-dependent and query-independent methods. Query-independent methods attempt to measure the estimated importance of a page, independent of any consideration of how well it matches the specific query. Query-independent ranking is usually based on link analysis; examples include the HITS algorithm, PageRank and TrustRank. Query-dependent methods attempt to measure the degree to which a page matches a specific query, independent of the importance of the page. Query-dependent ranking is usually based on heuristics that consider the number and locations of matches of the various query words on the page itself, in the URL or in any anchor text referring to the page. • In webometrics, it is possible to rank institutions according to their presence in the web (number of webpages) and the impact of these contents, such as the Webometrics Ranking of World Universities. • In video gaming, players may be given a ranking. To "rank up" is to achieve a higher ranking relative to other players, especially with strategies that do not depend on the player's skill. • The TrueSkill ranking system is a skill based ranking system for Xbox Live developed at Microsoft Research. • A bibliogram ranks common noun phrases in a piece of text. • In language, the status of an item (usually through what is known as "downranking" or "rank-shifting") in relation to the uppermost rank in a clause; for example, in the sentence "I want to eat the cake you made today", "eat" is on the uppermost rank, but "made" is downranked as part of the nominal group "the cake you made today"; this nominal group behaves as though it were a single noun (i.e., I want to eat it), and thus the verb within it ("made") is ranked differently from "eat". • Academic journals are sometimes ranked according to impact factor; the number of later articles that cite articles in a given journal. See also • League table • Ordinal data • Percentile rank • Rating (disambiguation) References 1. "Definition of RANKING". 2. Malara, Zbigniew; Miśko, Rafał; Sulich, Adam. "Wroclaw University of Technology graduates' career paths". {{cite journal}}: Cite journal requires \|journal= (help) 3. Sulich, Adam. "The young people's labour market and crisis of integration in European Union". Retrieved 2017-03-04. 4. "The Data School - How to Rank by Group in Alteryx - Part 1 - Standard Competition, Dense, Ordinal Ranking". www.thedataschool.co.uk. Retrieved 2023-07-23. 5. "Rank Cases: Ties". www.ibm.com. Retrieved 2023-07-23. 6. "rank function - RDocumentation". www.rdocumentation.org. Retrieved 2023-07-23. 7. "R: Fast Sample Ranks". search.r-project.org. Retrieved 2023-07-23. 8. Chris Roberts, Heavy Words Lightly Thrown: The Reason Behind Rhyme, Thorndike Press, 2006 (ISBN 0-7862-8517-6) 9. RIEDS, Italian Review of Economics Demography and Statistics (2014). "World Bank Doing Business Project and the statistical methods based on ranks: the paradox of the time indicator". Rieds - Rivista Italiana di Economia, Demografia e Statistica - the Italian Journal of Economic, Demographic and Statistical Studies. 68 (1): 79–86. External links Wikimedia Commons has media related to Rankings. Look up ranking in Wiktionary, the free dictionary. • RANKNUM, a Matlab function to compute the five types of ranks • Matlab Toolbox with functions to compute ranks • TrueSkill Ranking System • Ranking Library written in Ruby • List of Global Development Indexes and Rankings Statistics • Outline • Index Descriptive statistics Continuous data Center • Mean • Arithmetic • Arithmetic-Geometric • Cubic • Generalized/power • Geometric • Harmonic • Heronian • Heinz • Lehmer • Median • Mode Dispersion • Average absolute deviation • Coefficient of variation • Interquartile range • Percentile • Range • Standard deviation • Variance Shape • Central limit theorem • Moments • Kurtosis • L-moments • Skewness Count data • Index of dispersion Summary tables • Contingency table • Frequency distribution • Grouped data Dependence • Partial correlation • Pearson product-moment correlation • Rank correlation • Kendall's τ • Spearman's ρ • Scatter plot Graphics • Bar chart • Biplot • Box plot • Control chart • Correlogram • Fan chart • Forest plot • Histogram • Pie chart • Q–Q plot • Radar chart • Run chart • Scatter plot • Stem-and-leaf display • Violin plot Data collection Study design • Effect size • Missing data • Optimal design • Population • Replication • Sample size determination • Statistic • Statistical power Survey methodology • Sampling • Cluster • Stratified • Opinion poll • Questionnaire • Standard error Controlled experiments • Blocking • Factorial experiment • Interaction • Random assignment • Randomized controlled trial • Randomized experiment • Scientific control Adaptive designs • Adaptive clinical trial • Stochastic approximation • Up-and-down designs Observational studies • Cohort study • Cross-sectional study • Natural experiment • Quasi-experiment Statistical inference Statistical theory • Population • Statistic • Probability distribution • Sampling distribution • Order statistic • Empirical distribution • Density estimation • Statistical model • Model 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Student's t-test • F-test Goodness of fit • Chi-squared • G-test • Kolmogorov–Smirnov • Anderson–Darling • Lilliefors • Jarque–Bera • Normality (Shapiro–Wilk) • Likelihood-ratio test • Model selection • Cross validation • AIC • BIC Rank statistics • Sign • Sample median • Signed rank (Wilcoxon) • Hodges–Lehmann estimator • Rank sum (Mann–Whitney) • Nonparametric anova • 1-way (Kruskal–Wallis) • 2-way (Friedman) • Ordered alternative (Jonckheere–Terpstra) • Van der Waerden test Bayesian inference • Bayesian probability • prior • posterior • Credible interval • Bayes factor • Bayesian estimator • Maximum posterior estimator • Correlation • Regression analysis Correlation • Pearson product-moment • Partial correlation • Confounding variable • Coefficient of determination Regression analysis • Errors and residuals • Regression validation • Mixed effects models • Simultaneous equations models • Multivariate adaptive regression splines (MARS) Linear regression • Simple linear regression • 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Worldly cardinal In mathematical set theory, a worldly cardinal is a cardinal κ such that the rank Vκ is a model of Zermelo–Fraenkel set theory.[1] Relationship to inaccessible cardinals By Zermelo's theorem on inaccessible cardinals, every inaccessible cardinal is worldly. By Shepherdson's theorem, inaccessibility is equivalent to the stronger statement that (Vκ, Vκ+1) is a model of second order Zermelo-Fraenkel set theory.[2] Being worldly and being inaccessible are not equivalent; in fact, the smallest worldly cardinal has countable cofinality and therefore is a singular cardinal.[3] The following are in strictly increasing order, where ι is the least inaccessible cardinal: • The least worldly κ. • The least worldly κ and λ (κ<λ, and same below) with Vκ and Vλ satisfying the same theory. • The least worldly κ that is a limit of worldly cardinals (equivalently, a limit of κ worldly cardinals). • The least worldly κ and λ with Vκ ≺Σ2 Vλ (this is higher than even a κ-fold iteration of the above item). • The least worldly κ and λ with Vκ ≺ Vλ. • The least worldly κ of cofinality ω1 (corresponds to the extension of the above item to a chain of length ω1). • The least worldly κ of cofinality ω2 (and so on). • The least κ>ω with Vκ satisfying replacement for the language augmented with the (Vκ,∈) satisfaction relation. • The least κ inaccessible in Lκ(Vκ); equivalently, the least κ>ω with Vκ satisfying replacement for formulas in Vκ in the infinitary logic L∞,ω. • The least κ with a transitive model M⊂Vκ+1 extending Vκ satisfying Morse–Kelley set theory. • (not a worldly cardinal) The least κ with Vκ having the same Σ2 theory as Vι. • The least κ with Vκ and Vι having the same theory. • The least κ with Lκ(Vκ) and Lι(Vι) having the same theory. • (not a worldly cardinal) The least κ with Vκ and Vι having the same Σ2 theory with real parameters. • (not a worldly cardinal) The least κ with Vκ ≺Σ2 Vι. • The least κ with Vκ ≺ Vι. • The least infinite κ with Vκ and Vι satisfying the same L∞,ω statements that are in Vκ. • The least κ with a transitive model M⊂Vκ+1 extending Vκ and satisfying the same sentences with parameters in Vκ as Vι+1 does. • The least inaccessible cardinal ι. References 1. Hamkins (2014). 2. Kanamori (2003), Theorem 1.3, p. 19. 3. Kanamori (2003), Lemma 6.1, p. 57. • Hamkins, Joel David (2014), "A multiverse perspective on the axiom of constructibility", Infinity and truth, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 25, Hackensack, NJ: World Sci. Publ., pp. 25–45, arXiv:1210.6541, Bibcode:2012arXiv1210.6541H, MR 3205072 • Kanamori, Akihiro (2003), The Higher Infinite, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag External links • Worldly cardinal in Cantor's attic
Worldwide Online Olympiad Training The Worldwide Online Olympiad Training (WOOT) program was established in 2005 by Art of Problem Solving,[1] with sponsorship from Google and quantitative hedge fund giant D. E. Shaw & Co., in order to meet the needs of the world's top high school math students. Sponsorship allowed free enrollment for students of the Mathematical Olympiad Program (MOP). D.E. Shaw continued to sponsor enrollment of those students for the 2006-2007 year of WOOT. Program The focus on the WOOT program is taking already excellent pre-college students deeper into their studies of elementary mathematics, with a focus on proof-writing. • Numerous exams are given over the course of the program and graded by undergraduates at MIT and Harvard. Feedback on proofs is returned to students electronically within a couple weeks of exam submission. These tests are styled after the American Invitational Mathematics Examination (AIME) and the International Mathematics Olympiad (IMO) • Online classes are given throughout the school year on topics such as bridging ideas between different areas of math (using algebraic tactics on number theory problems), combinatorial geometry, inequalities (such as the Cauchy–Schwarz inequality), invariants, and proof writing. • Problem sets are given out on a private LaTeX-enabled WOOT phpbb message board where students post proofs, discuss problem solving tactics, and review each other's solutions. • A private, LaTeX-enabled chatroom allows students to discuss problems at any time. Students During the first year (2005–2006) of the WOOT program, a little over 100 students participated, over 90% of whom were among the fewer than 500 qualifiers for the 2006 United States of America Mathematics Olympiad (USAMO), including most of the competition's 12 "winners." Several participants from the United States and other countries won medals at the 2006 IMO held in Slovenia. Instructors WOOT students (WOOTers) are guided by veterans of national and international mathematics competitions such as IMO medalists, winners of the USAMO, a former Westinghouse competition winner, a Canadian Math Olympiad winner, perfect scorers on the AIME, perfect scorers on the American High School Mathematics Examination (now the American Mathematics Competitions), and a perfect scorer at the national MathCounts competition. Funding The first year of the program was sponsored by Google and D. E. Shaw & Co. Subsequent years have been sponsored by:[2] • Alameda Research • Citadel • Hudson River Trading • IMC Financial Markets • Jane Street Capital • Susquehanna International Group • Two Sigma References 1. "WOOT". Art of Problem Solving. Archived from the original on 2023-03-22. Retrieved 2023-03-22. 2. "WOOT: Worldwide Online Olympiad Training". Art of Problem Solving. Archived from the original on 2022-04-06. Retrieved 2023-03-22.{{cite web}}: CS1 maint: unfit URL (link)
Wright omega function In mathematics, the Wright omega function or Wright function,[note 1] denoted ω, is defined in terms of the Lambert W function as: $\omega (z)=W_{{\big \lceil }{\frac {\mathrm {Im} (z)-\pi }{2\pi }}{\big \rceil }}(e^{z}).$ Uses One of the main applications of this function is in the resolution of the equation z = ln(z), as the only solution is given by z = e−ω(π i). y = ω(z) is the unique solution, when $z\neq x\pm i\pi $ for x ≤ −1, of the equation y + ln(y) = z. Except on those two rays, the Wright omega function is continuous, even analytic. Properties The Wright omega function satisfies the relation $W_{k}(z)=\omega (\ln(z)+2\pi ik)$. It also satisfies the differential equation ${\frac {d\omega }{dz}}={\frac {\omega }{1+\omega }}$ wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation $\ln(\omega )+\omega =z$), and as a consequence its integral can be expressed as: $\int w^{n}\,dz={\begin{cases}{\frac {\omega ^{n+1}-1}{n+1}}+{\frac {\omega ^{n}}{n}}&{\mbox{if }}n\neq -1,\\\ln(\omega )-{\frac {1}{\omega }}&{\mbox{if }}n=-1.\end{cases}}$ Its Taylor series around the point $a=\omega _{a}+\ln(\omega _{a})$ takes the form : $\omega (z)=\sum _{n=0}^{+\infty }{\frac {q_{n}(\omega _{a})}{(1+\omega _{a})^{2n-1}}}{\frac {(z-a)^{n}}{n!}}$ where $q_{n}(w)=\sum _{k=0}^{n-1}{\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n+1\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }(-1)^{k}w^{k+1}$ in which ${\bigg \langle }\!\!{\bigg \langle }{\begin{matrix}n\\k\end{matrix}}{\bigg \rangle }\!\!{\bigg \rangle }$ is a second-order Eulerian number. Values ${\begin{array}{lll}\omega (0)&=W_{0}(1)&\approx 0.56714\\\omega (1)&=1&\\\omega (-1\pm i\pi )&=-1&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)+i\pi )&=-{\frac {1}{3}}&\\\omega (-{\frac {1}{3}}+\ln \left({\frac {1}{3}}\right)-i\pi )&=W_{-1}\left(-{\frac {1}{3}}e^{-{\frac {1}{3}}}\right)&\approx -2.237147028\\\end{array}}$ Plots • Plots of the Wright omega function on the complex plane • z = Re(ω(x + i y)) • z = Im(ω(x + i y)) • ω(x + i y) Notes 1. Not to be confused with the Fox–Wright function, also known as Wright function. References • "On the Wright ω function", Robert Corless and David Jeffrey
Writhe In knot theory, there are several competing notions of the quantity writhe, or $\operatorname {Wr} $. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amount of "coiling" of a mathematical knot (or any closed simple curve) in three-dimensional space and assumes real numbers as values. In both cases, writhe is a geometric quantity, meaning that while deforming a curve (or diagram) in such a way that does not change its topology, one may still change its writhe.[1] Writhe of link diagrams In knot theory, the writhe is a property of an oriented link diagram. The writhe is the total number of positive crossings minus the total number of negative crossings. A direction is assigned to the link at a point in each component and this direction is followed all the way around each component. For each crossing one comes across while traveling in this direction, if the strand underneath goes from right to left, the crossing is positive; if the lower strand goes from left to right, the crossing is negative. One way of remembering this is to use a variation of the right-hand rule. Positive crossing Negative crossing For a knot diagram, using the right-hand rule with either orientation gives the same result, so the writhe is well-defined on unoriented knot diagrams. The writhe of a knot is unaffected by two of the three Reidemeister moves: moves of Type II and Type III do not affect the writhe. Reidemeister move Type I, however, increases or decreases the writhe by 1. This implies that the writhe of a knot is not an isotopy invariant of the knot itself — only the diagram. By a series of Type I moves one can set the writhe of a diagram for a given knot to be any integer at all. Writhe of a closed curve Writhe is also a property of a knot represented as a curve in three-dimensional space. Strictly speaking, a knot is such a curve, defined mathematically as an embedding of a circle in three-dimensional Euclidean space, $\mathbb {R} ^{3}$. By viewing the curve from different vantage points, one can obtain different projections and draw the corresponding knot diagrams. Its writhe $\operatorname {Wr} $ (in the space curve sense) is equal to the average of the integral writhe values obtained from the projections from all vantage points.[2] Hence, writhe in this situation can take on any real number as a possible value.[1] In a paper from 1961,[3] Gheorghe Călugăreanu proved the following theorem: take a ribbon in $\mathbb {R} ^{3}$, let $\operatorname {Lk} $ be the linking number of its border components, and let $\operatorname {Tw} $ be its total twist. Then the difference $\operatorname {Lk} -\operatorname {Tw} $ depends only on the core curve of the ribbon,[2] and $\operatorname {Wr} =\operatorname {Lk} -\operatorname {Tw} $. In a paper from 1959,[4] Călugăreanu also showed how to calculate the writhe Wr with an integral. Let $C$ be a smooth, simple, closed curve and let $\mathbf {r} _{1}$ and $\mathbf {r} _{2}$ be points on $C$. Then the writhe is equal to the Gauss integral $\operatorname {Wr} ={\frac {1}{4\pi }}\int _{C}\int _{C}d\mathbf {r} _{1}\times d\mathbf {r} _{2}\cdot {\frac {\mathbf {r} _{1}-\mathbf {r} _{2}}{\left\|\mathbf {r} _{1}-\mathbf {r} _{2}\right\|^{3}}}$. Numerically approximating the Gauss integral for writhe of a curve in space Since writhe for a curve in space is defined as a double integral, we can approximate its value numerically by first representing our curve as a finite chain of $N$ line segments. A procedure that was first derived by Michael Levitt[5] for the description of protein folding and later used for supercoiled DNA by Konstantin Klenin and Jörg Langowski[6] is to compute $\operatorname {Wr} =\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\Omega _{ij}}{4\pi }}=2\sum _{i=2}^{N}\sum _{j<i}{\frac {\Omega _{ij}}{4\pi }}$, where $\Omega _{ij}/{4\pi }$ is the exact evaluation of the double integral over line segments $i$ and $j$; note that $\Omega _{ij}=\Omega _{ji}$ and $\Omega _{i,i+1}=\Omega _{ii}=0$.[6] To evaluate $\Omega _{ij}/{4\pi }$ for given segments numbered $i$ and $j$, number the endpoints of the two segments 1, 2, 3, and 4. Let $r_{pq}$ be the vector that begins at endpoint $p$ and ends at endpoint $q$. Define the following quantities:[6] $n_{1}={\frac {r_{13}\times r_{14}}{\left\|r_{13}\times r_{14}\right\|}},\;n_{2}={\frac {r_{14}\times r_{24}}{\left\|r_{14}\times r_{24}\right\|}},\;n_{3}={\frac {r_{24}\times r_{23}}{\left\|r_{24}\times r_{23}\right\|}},\;n_{4}={\frac {r_{23}\times r_{13}}{\left\|r_{23}\times r_{13}\right\|}}$ Then we calculate[6] $\Omega ^{}=\arcsin \left(n_{1}\cdot n_{2}\right)+\arcsin \left(n_{2}\cdot n_{3}\right)+\arcsin \left(n_{3}\cdot n_{4}\right)+\arcsin \left(n_{4}\cdot n_{1}\right).$ Finally, we compensate for the possible sign difference and divide by $4\pi $ to obtain[6] ${\frac {\Omega }{4\pi }}={\frac {\Omega ^{}}{4\pi }}{\text{sign}}\left(\left(r_{34}\times r_{12}\right)\cdot r_{13}\right).$ In addition, other methods to calculate writhe can be fully described mathematically and algorithmically, some of them outperform method above (which has quadratic computational complexity, by having linear complexity).[6] Applications in DNA topology DNA will coil when twisted, just like a rubber hose or a rope will, and that is why biomathematicians use the quantity of writhe to describe the amount a piece of DNA is deformed as a result of this torsional stress. In general, this phenomenon of forming coils due to writhe is referred to as DNA supercoiling and is quite commonplace, and in fact in most organisms DNA is negatively supercoiled.[1] Any elastic rod, not just DNA, relieves torsional stress by coiling, an action which simultaneously untwists and bends the rod. F. Brock Fuller shows mathematically[7] how the “elastic energy due to local twisting of the rod may be reduced if the central curve of the rod forms coils that increase its writhing number”. See also • DNA supercoiling • Linking number • Ribbon theory • Twist (mathematics) • Winding number References 1. Bates, Andrew (2005). DNA Topology. Oxford University Press. pp. 36–37. ISBN 978-0-19-850655-3. 2. Cimasoni, David (2001). "Computing the writhe of a knot". Journal of Knot Theory and Its Ramifications. 10 (387): 387–395. arXiv:math/0406148. doi:10.1142/S0218216501000913. MR 1825964. S2CID 15850269. 3. Călugăreanu, Gheorghe (1961). "Sur les classes d'isotopie des nœuds tridimensionnels et leurs invariants". Czechoslovak Mathematical Journal (in French). 11 (4): 588–625. doi:10.21136/CMJ.1961.100486. MR 0149378. 4. Călugăreanu, Gheorghe (1959). "L'intégrale de Gauss et l'analyse des nœuds tridimensionnels" (PDF). Revue de Mathématiques Pure et Appliquées (in French). 4: 5–20. MR 0131846. 5. Levitt, Michael (1986). "Protein Folding by Restrained Energy Minimization and Molecular Dynamics". Journal of Molecular Biology. 170 (3): 723–764. CiteSeerX 10.1.1.26.3656. doi:10.1016/s0022-2836(83)80129-6. PMID 6195346. 6. Klenin, Konstantin; Langowski, Jörg (2000). "Computation of writhe in modeling of supercoiled DNA". Biopolymers. 54 (5): 307–317. doi:10.1002/1097-0282(20001015)54:5<307::aid-bip20>3.0.co;2-y. PMID 10935971. 7. Fuller, F. Brock (1971). "The writhing number of a space curve". Proceedings of the National Academy of Sciences of the United States of America. 68 (4): 815–819. Bibcode:1971PNAS...68..815B. doi:10.1073/pnas.68.4.815. MR 0278197. PMC 389050. PMID 5279522. Further reading • Adams, Colin (2004), The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, ISBN 978-0-8218-3678-1 Knot theory (knots and links) Hyperbolic • Figure-eight (41) • Three-twist (52) • Stevedore (61) • 62 • 63 • Endless (74) • Carrick mat (818) • Perko pair (10161) • (−2,3,7) pretzel (12n242) • Whitehead (52 1 ) • Borromean rings (63 2 ) • L10a140 • Conway knot (11n34) Satellite • Composite knots • Granny • Square • Knot sum Torus • Unknot (01) • Trefoil (31) • Cinquefoil (51) • Septafoil (71) • Unlink (02 1 ) • Hopf (22 1 ) • Solomon's (42 1 ) Invariants • Alternating • Arf invariant • Bridge no. • 2-bridge • Brunnian • Chirality • Invertible • Crosscap no. • Crossing no. • Finite type invariant • Hyperbolic volume • Khovanov homology • Genus • Knot group • Link group • Linking no. • Polynomial • Alexander • Bracket • HOMFLY • Jones • Kauffman • Pretzel • Prime • list • Stick no. • Tricolorability • Unknotting no. and problem Notation and operations • Alexander–Briggs notation • Conway notation • Dowker–Thistlethwaite notation • Flype • Mutation • Reidemeister move • Skein relation • Tabulation Other • Alexander's theorem • Berge • Braid theory • Conway sphere • Complement • Double torus • Fibered • Knot • List of knots and links • Ribbon • Slice • Sum • Tait conjectures • Twist • Wild • Writhe • Surgery theory • Category • Commons
Wronskian In the mathematics of a square matrix, the Wronskian (or Wrońskian) is a determinant introduced by the Polish mathematician Józef Hoene-Wroński (1812). It is used in the study of differential equations, where it can sometimes show linear independence of a set of solutions. Differential equations Scope Fields • Natural sciences • Engineering • Astronomy • Physics • Chemistry • Biology • Geology Applied mathematics • Continuum mechanics • Chaos theory • Dynamical systems Social sciences • Economics • Population dynamics List of named differential equations Classification Types • Ordinary • Partial • Differential-algebraic • Integro-differential • Fractional • Linear • Non-linear By variable type • Dependent and independent variables • Autonomous • Coupled / Decoupled • Exact • Homogeneous / Nonhomogeneous Features • Order • Operator • Notation Relation to processes • Difference (discrete analogue) • Stochastic • Stochastic partial • Delay Solution Existence and uniqueness • Picard–Lindelöf theorem • Peano existence theorem • Carathéodory's existence theorem • Cauchy–Kowalevski theorem General topics • Initial conditions • Boundary values • Dirichlet • Neumann • Robin • Cauchy problem • Wronskian • Phase portrait • Lyapunov / Asymptotic / Exponential stability • Rate of convergence • Series / Integral solutions • Numerical integration • Dirac delta function Solution methods • Inspection • Method of characteristics • Euler • Exponential response formula • Finite difference (Crank–Nicolson) • Finite element • Infinite element • Finite volume • Galerkin • Petrov–Galerkin • Green's function • Integrating factor • Integral transforms • Perturbation theory • Runge–Kutta • Separation of variables • Undetermined coefficients • Variation of parameters People List • Isaac Newton • Gottfried Leibniz • Jacob Bernoulli • Leonhard Euler • Józef Maria Hoene-Wroński • Joseph Fourier • Augustin-Louis Cauchy • George Green • Carl David Tolmé Runge • Martin Kutta • Rudolf Lipschitz • Ernst Lindelöf • Émile Picard • Phyllis Nicolson • John Crank Definition The Wronskian of two differentiable functions f and g is $W(f,g)=fg'-gf'$. More generally, for n real- or complex-valued functions f1, …, fn, which are n – 1 times differentiable on an interval I, the Wronskian $W(f_{1},\ldots ,f_{n})$ is a function on $x\in I$ defined by $W(f_{1},\ldots ,f_{n})(x)=\det {\begin{bmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{bmatrix}}.$ This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the $(n-1)^{\text{th}}$ derivative, thus forming a square matrix. When the functions fi are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions fi are not known explicitly. (See below.) The Wronskian and linear independence If the functions fi are linearly dependent, then so are the columns of the Wronskian (since differentiation is a linear operation), and the Wronskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wronskian does not vanish identically. It may, however, vanish at isolated points.[1] A common misconception is that W = 0 everywhere implies linear dependence, but Peano (1889) pointed out that the functions x2 and \|x\| · x have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0.[lower-alpha 1] There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence. • Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.[3] • Bôcher (1901) gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of n functions is identically zero and the n Wronskians of n – 1 of them do not all vanish at any point then the functions are linearly dependent. • Wolsson (1989a) gave a more general condition that together with the vanishing of the Wronskian implies linear dependence. Over fields of positive characteristic p the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of xp and 1 is identically 0. Application to linear differential equations In general, for an $n$th order linear differential equation, if $(n-1)$ solutions are known, the last one can be determined by using the Wronskian. Consider the second order differential equation in Lagrange's notation: $y''=a(x)y'+b(x)y$ where $a(x)$, $b(x)$ are known, and y is the unknown function to be found. Let us call $y_{1},y_{2}$ the two solutions of the equation and form their Wronskian $W(x)=y_{1}y'_{2}-y_{2}y'_{1}$ Then differentiating $W(x)$ and using the fact that $y_{i}$ obey the above differential equation shows that $W'(x)=aW(x)$ Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved: $W(x)=C~e^{A(x)}$ where $A'(x)=a(x)$ and $C$ is a constant. Now suppose that we know one of the solutions, say $y_{2}$. Then, by the definition of the Wronskian, $y_{1}$ obeys a first order differential equation: $y'_{1}-{\frac {y'_{2}}{y_{2}}}y_{1}=-W(x)/y_{2}$ and can be solved exactly (at least in theory). The method is easily generalized to higher order equations. Generalized Wronskians For n functions of several variables, a generalized Wronskian is a determinant of an n by n matrix with entries Di(fj) (with 0 ≤ i < n), where each Di is some constant coefficient linear partial differential operator of order i. If the functions are linearly dependent then all generalized Wronskians vanish. As in the single variable case the converse is not true in general: if all generalized Wronskians vanish, this does not imply that the functions are linearly dependent. However, the converse is true in many special cases. For example, if the functions are polynomials and all generalized Wronskians vanish, then the functions are linearly dependent. Roth used this result about generalized Wronskians in his proof of Roth's theorem. For more general conditions under which the converse is valid see Wolsson (1989b). History The Wronskian was introduced by Józef Hoene-Wroński (1812) and given its current name by Thomas Muir (1882, Chapter XVIII). See also • Variation of parameters • Moore matrix, analogous to the Wronskian with differentiation replaced by the Frobenius endomorphism over a finite field. • Alternant matrix • Vandermonde matrix Notes 1. Peano published his example twice, because the first time he published it, an editor, Paul Mansion, who had written a textbook incorrectly claiming that the vanishing of the Wronskian implies linear dependence, added a footnote to Peano's paper claiming that this result is correct as long as neither function is identically zero. Peano's second paper pointed out that this footnote was nonsense.[2] Citations 1. Bender, Carl M.; Orszag, Steven A. (1999) [1978], Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory, New York: Springer, p. 9, ISBN 978-0-387-98931-0 2. Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. doi:10.4169/loci003642. Retrieved 2020-10-08. 3. Engdahl, Susannah; Parker, Adam (April 2011). "Peano on Wronskians: A Translation". Convergence. Mathematical Association of America. Section "On the Wronskian Determinant". doi:10.4169/loci003642. Retrieved 2020-10-08. The most famous theorem is attributed to Bocher, and states that if the Wronskian of $n$ analytic functions is zero, then the functions are linearly dependent ([B2], [BD]). [The citations 'B2' and 'BD' refer to Bôcher (1900–1901) and Bostan and Dumas (2010), respectively.] References • Bôcher, Maxime (1900–1901). "The Theory of Linear Dependence". Annals of Mathematics. Princeton University. 2 (1/4): 81–96. doi:10.2307/2007186. ISSN 0003-486X. JSTOR 2007186. • Bôcher, Maxime (1901), "Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence" (PDF), Transactions of the American Mathematical Society, Providence, R.I.: American Mathematical Society, 2 (2): 139–149, doi:10.2307/1986214, ISSN 0002-9947, JFM 32.0313.02, JSTOR 1986214 • Bostan, Alin; Dumas, Philippe (2010). "Wronskians and Linear Independence". American Mathematical Monthly. Taylor & Francis. 117 (8): 722–727. arXiv:1301.6598. doi:10.4169/000298910x515785. ISSN 0002-9890. JSTOR 10.4169/000298910x515785. S2CID 9322383. • Hartman, Philip (1964), Ordinary Differential Equations, New York: John Wiley & Sons, ISBN 978-0-89871-510-1, MR 0171038, Zbl 0125.32102 • Hoene-Wroński, Józef (1812), Réfutation de la théorie des fonctions analytiques de Lagrange, Paris • Muir, Thomas (1882), A Treatise on the Theorie of Determinants., Macmillan, JFM 15.0118.05 • Peano, Giuseppe (1889), "Sur le déterminant wronskien.", Mathesis (in French), IX: 75–76, 110–112, JFM 21.0153.01 • Rozov, N. Kh. (2001) [1994], "Wronskian", Encyclopedia of Mathematics, EMS Press • Wolsson, Kenneth (1989a), "A condition equivalent to linear dependence for functions with vanishing Wronskian", Linear Algebra and Its Applications, 116: 1–8, doi:10.1016/0024-3795(89)90393-5, ISSN 0024-3795, MR 0989712, Zbl 0671.15005 • Wolsson, Kenneth (1989b), "Linear dependence of a function set of m variables with vanishing generalized Wronskians", Linear Algebra and Its Applications, 117: 73–80, doi:10.1016/0024-3795(89)90548-X, ISSN 0024-3795, MR 0993032, Zbl 0724.15004 Matrix classes Explicitly constrained entries • Alternant • Anti-diagonal • Anti-Hermitian • Anti-symmetric • Arrowhead • Band • Bidiagonal • Bisymmetric • Block-diagonal • Block • Block tridiagonal • Boolean • Cauchy • Centrosymmetric • Conference • Complex Hadamard • Copositive • Diagonally dominant • Diagonal • Discrete Fourier Transform • Elementary • Equivalent • Frobenius • Generalized permutation • Hadamard • Hankel • Hermitian • Hessenberg • Hollow • Integer • Logical • Matrix unit • Metzler • Moore • Nonnegative • Pentadiagonal • Permutation • Persymmetric • Polynomial • Quaternionic • Signature • Skew-Hermitian • Skew-symmetric • Skyline • Sparse • Sylvester • Symmetric • Toeplitz • Triangular • Tridiagonal • Vandermonde • Walsh • Z Constant • Exchange • Hilbert • Identity • Lehmer • Of ones • Pascal • Pauli • Redheffer • Shift • Zero Conditions on eigenvalues or eigenvectors • Companion • Convergent • Defective • Definite • Diagonalizable • Hurwitz • Positive-definite • Stieltjes Satisfying conditions on products or inverses • Congruent • Idempotent or Projection • Invertible • Involutory • Nilpotent • Normal • Orthogonal • Unimodular • Unipotent • Unitary • Totally unimodular • Weighing With specific applications • Adjugate • Alternating sign • Augmented • Bézout • Carleman • Cartan • Circulant • Cofactor • Commutation • Confusion • Coxeter • Distance • Duplication and elimination • Euclidean distance • Fundamental (linear differential equation) • Generator • Gram • Hessian • Householder • Jacobian • Moment • Payoff • Pick • Random • Rotation • Seifert • Shear • Similarity • Symplectic • Totally positive • Transformation Used in statistics • Centering • Correlation • Covariance • Design • Doubly stochastic • Fisher information • Hat • Precision • Stochastic • Transition Used in graph theory • Adjacency • Biadjacency • Degree • Edmonds • Incidence • Laplacian • Seidel adjacency • Tutte Used in science and engineering • Cabibbo–Kobayashi–Maskawa • Density • Fundamental (computer vision) • Fuzzy associative • Gamma • Gell-Mann • Hamiltonian • Irregular • Overlap • S • State transition • Substitution • Z (chemistry) Related terms • Jordan normal form • Linear independence • Matrix exponential • Matrix representation of conic sections • Perfect matrix • Pseudoinverse • Row echelon form • Wronskian • Mathematics portal • List of matrices • Category:Matrices
Wu's method of characteristic set Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of the Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets.[1][2] Wu's method is powerful for mechanical theorem proving in elementary geometry, and provides a complete decision process for certain classes of problem. It has been used in research in his laboratory (KLMM, Key Laboratory of Mathematics Mechanization in Chinese Academy of Science) and around the world. The main trends of research on Wu's method concern systems of polynomial equations of positive dimension and differential algebra where Ritt's results have been made effective.[3][4] Wu's method has been applied in various scientific fields, like biology, computer vision, robot kinematics and especially automatic proofs in geometry.[5] Informal description Wu's method uses polynomial division to solve problems of the form: $\forall x,y,z,\dots I(x,y,z,\dots )\implies f(x,y,z,\dots )\,$ where f is a polynomial equation and I is a conjunction of polynomial equations. The algorithm is complete for such problems over the complex domain. The core idea of the algorithm is that you can divide one polynomial by another to give a remainder. Repeated division results in either the remainder vanishing (in which case the I implies f statement is true), or an irreducible remainder is left behind (in which case the statement is false). More specifically, for an ideal I in the ring k[x1, ..., xn] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I. Ritt characteristic set A Ritt characteristic set is a finite set of polynomials in triangular form of an ideal. This triangular set satisfies certain minimal condition with respect to the Ritt ordering, and it preserves many interesting geometrical properties of the ideal. However it may not be its system of generators. Notation Let R be the multivariate polynomial ring k[x1, ..., xn] over a field k. The variables are ordered linearly according to their subscript: x1 < ... < xn. For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role: p can be naturally regarded as a univariate polynomial in its main variable xk with coefficients in k[x1, ..., xk−1]. The degree of p as a univariate polynomial in its main variable is also called its main degree. Triangular set A set T of non-constant polynomials is called a triangular set if all polynomials in T have distinct main variables. This generalizes triangular systems of linear equations in a natural way. Ritt ordering For two non-constant polynomials p and q, we say p is smaller than q with respect to Ritt ordering and written as p <r q, if one of the following assertions holds: (1) the main variable of p is smaller than the main variable of q, that is, mvar(p) < mvar(q), (2) p and q have the same main variable, and the main degree of p is less than the main degree of q, that is, mvar(p) = mvar(q) and mdeg(p) < mdeg(q). In this way, (k[x1, ..., xn],<r) forms a well partial order. However, the Ritt ordering is not a total order: there exist polynomials p and q such that neither p <r q nor p >r q. In this case, we say that p and q are not comparable. The Ritt ordering is comparing the rank of p and q. The rank, denoted by rank(p), of a non-constant polynomial p is defined to be a power of its main variable: mvar(p)mdeg(p) and ranks are compared by comparing first the variables and then, in case of equality of the variables, the degrees. Ritt ordering on triangular sets A crucial generalization on Ritt ordering is to compare triangular sets. Let T = { t1, ..., tu} and S = { s1, ..., sv} be two triangular sets such that polynomials in T and S are sorted increasingly according to their main variables. We say T is smaller than S w.r.t. Ritt ordering if one of the following assertions holds 1. there exists k ≤ min(u, v) such that rank(ti) = rank(si) for 1 ≤ i < k and tk <r sk, 2. u > v and rank(ti) = rank(si) for 1 ≤ i ≤ v. Also, there exists incomparable triangular sets w.r.t Ritt ordering. Ritt characteristic set Let I be a non-zero ideal of k[x1, ..., xn]. A subset T of I is a Ritt characteristic set of I if one of the following conditions holds: 1. T consists of a single nonzero constant of k, 2. T is a triangular set and T is minimal w.r.t Ritt ordering in the set of all triangular sets contained in I. A polynomial ideal may possess (infinitely) many characteristic sets, since Ritt ordering is a partial order. Wu characteristic set The Ritt–Wu process, first devised by Ritt, subsequently modified by Wu, computes not a Ritt characteristic but an extended one, called Wu characteristic set or ascending chain. A non-empty subset T of the ideal ⟨F⟩ generated by F is a Wu characteristic set of F if one of the following condition holds 1. T = {a} with a being a nonzero constant, 2. T is a triangular set and there exists a subset G of ⟨F⟩ such that ⟨F⟩ = ⟨G⟩ and every polynomial in G is pseudo-reduced to zero with respect to T. Wu characteristic set is defined to the set F of polynomials, rather to the ideal ⟨F⟩ generated by F. Also it can be shown that a Ritt characteristic set T of ⟨F⟩ is a Wu characteristic set of F. Wu characteristic sets can be computed by Wu's algorithm CHRST-REM, which only requires pseudo-remainder computations and no factorizations are needed. Wu's characteristic set method has exponential complexity; improvements in computing efficiency by weak chains, regular chains, saturated chain were introduced[6] Decomposing algebraic varieties An application is an algorithm for solving systems of algebraic equations by means of characteristic sets. More precisely, given a finite subset F of polynomials, there is an algorithm to compute characteristic sets T1, ..., Te such that: $V(F)=W(T_{1})\cup \cdots \cup W(T_{e}),$ where W(Ti) is the difference of V(Ti) and V(hi), here hi is the product of initials of the polynomials in Ti. See also • Regular chain • Mathematics-Mechanization Platform References 1. Corrochano, Eduardo Bayro; Sobczyk, Garret, eds. (2001). Geometric algebra with applications in science and engineering. Boston, Mass: Birkhäuser. p. 110. ISBN 9780817641993. 2. P. Aubry, D. Lazard, M. Moreno Maza (1999). On the theories of triangular sets. Journal of Symbolic Computation, 28(1–2):105–124 3. Hubert, E. Factorisation Free Decomposition Algorithms in Differential Algebra. Journal of Symbolic Computation, (May 2000): 641–662. 4. Maple (software) package diffalg. 5. Chou, Shang-Ching; Gao, Xiao Shan; Zhang, Jing Zhong. Machine proofs in geometry. World Scientific, 1994. 6. Chou S C, Gao X S; Ritt–Wu's decomposition algorithm and geometry theorem proving. Proc of CADE, 10 LNCS, #449, Berlin, Springer Verlag, 1990 207–220. • P. Aubry, M. Moreno Maza (1999) Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods. J. Symb. Comput. 28(1–2): 125–154 • David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties, and Algorithms. 2007. • Hua-Shan, Liu (24 August 2005). "WuRittSolva: Implementation of Wu-Ritt Characteristic Set Method". Wolfram Library Archive. Wolfram. Retrieved 17 November 2012. • Heck, André (2003). Introduction to Maple (3. ed.). New York: Springer. pp. 105, 508. ISBN 9780387002309. • Ritt, J. (1966). Differential Algebra. New York, Dover Publications. • Dongming Wang (1998). Elimination Methods. Springer-Verlag, Wien, Springer-Verlag • Dongming Wang (2004). Elimination Practice, Imperial College Press, London ISBN 1-86094-438-8 • Wu, W. T. (1984). Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci., 4, 207–35 • Wu, W. T. (1987). A zero structure theorem for polynomial equations solving. MM Research Preprints, 1, 2–12 • Xiaoshan, Gao; Chunming, Yuan; Guilin, Zhang (2009). "Ritt-Wu's characteristic set method for ordinary difference polynomial systems with arbitrary ordering". Acta Mathematica Scientia. 29 (4): 1063–1080. CiteSeerX 10.1.1.556.9549. doi:10.1016/S0252-9602(09)60086-2. External links • wsolve Maple package • The Characteristic Set Method
Wu–Yang dictionary In topology and high energy physics, the Wu–Yang dictionary refers to the mathematical identification that allows to translate back and forth between the concepts of gauge theory and those of differential geometry. It was devised by Tai Tsun Wu and C. N. Yang in 1975 when studying the relation between electromagnetism and fiber bundle theory.[1] This dictionary has been credited as bringing mathematics and theoretical physics closer together.[2] A crucial example of the success of the dictionary is that it allowed to understand Paul Dirac's monopole quantization in terms of Hopf fibrations.[3] History In 1975, theoretical physicists Tsun Wu and C. N. Yang working in Stony Brook University, published a paper on the mathematical framework of electromagnetism and the Aharonov–Bohm effect in terms of fiber bundles. A year later, mathematician Isadore Singer came to visit and brought a copy back to the University of Oxford.[2][4][5] Singer showed the paper to Michael Atiyah and other mathematicians, sparking a close collaboration between physicists and mathematicians.[2] Yang also recounts a conversation that he had with one of the mathematicians that founded fiber bundle theory, Shiing-Shen Chern:[2] In 1975, impressed with the fact that gauge fields are connections on fiber bundles, I drove to the house of Shiing-Shen Chern in El Cerrito, near Berkeley. (I had taken courses with him in the early 1940s when he was a young professor and I an undergraduate student at the National Southwest Associated University in Kunming, China. That was before fiber bundles had become important in differential geometry and before Chern had made history with his contributions to the generalized Gauss–Bonnet theorem and the Chern classes.) We had much to talk about: friends, relatives, China. When our conversation turned to fiber bundles, I told him that I had finally learned from Jim Simons the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added ‘this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere.’ He immediately protested, ‘No, no. These concepts were not dreamed up. They were natural and real.' Description Summarized version The Wu-Yang dictionary relates terms in particle physics with terms in mathematics, specifically fiber bundle theory. Many versions and generalization of the dictionary exist. Here is an example of a dictionary, which puts each physics term next to its mathematical analogue:[6] Physics Mathematics Potential Connection Field tensor (interaction) Curvature Field tensor-potential relation Structural equation Gauge transformation Change of bundle coordinates Gauge group Structure group Original version for electromagnetism Wu and Yang considered the description of an electron traveling around a cylinder in the presence of a magnetic field inside the cylinder (outside the cylinder the field vanishes i.e. $f_{\mu \nu }=0$). According to the Aharonov–Bohm effect, the interference patterns shift by a factor $\exp(-i\Omega /\Omega _{0})$, where $\Omega $ is the magnetic flux and $\Omega _{0}$ is the magnetic flux quantum. For two different fluxes a and b, the results are identical if $\Omega _{a}-\Omega _{b}=N\Omega _{0}$, where $N$ is an integer. We define the operator $S_{ab}$ as the operator that brings the electron wave function from one configuration to the other $\psi _{b}=S_{ba}\psi _{a}$. For an electron that takes a path from point P to point Q, we define the phase factor as $\Phi _{PQ}=\exp \left(-{\frac {i}{\Omega _{0}}}\int _{P}^{Q}A_{\mu }\mathrm {d} x^{\mu }\right)$, where $A_{\mu }$ is the electromagnetic four-potential. For the case of a SU2 gauge field, we can make the substitution $A_{\mu }=ib_{\mu }^{k}X_{k}$, where $X_{k}=-i\sigma _{k}/2$ are the generators of SU2, $\sigma _{k}$ are the Pauli matrices. Under these concepts, Wu and Yang showed the relation between the language of gauge theory and fiber bundles, was codified in following dictionary:[2][7][8] Wu–Yang dictionary (1975) Gauge field terminology Bundle terminology Gauge (or global gauge) Principal coordinate fiber bundle Gauge type Principal fiber bundle Gauge potential $b_{\mu }^{k}$ Connection on principal fiber bundle $S_{ba}$ Transition function Phase factor $\Phi _{QP}$ Parallel displacement Field strength $f_{\mu \nu }^{k}$ Curvature Source $J_{\mu }^{k}$ ? Electromagnetism Connection in a U1(1) bundle Isotopic spin gauge field Connection in a SU2 bundle Dirac's monopole quantization Classification in a U1(1) bundle according to first Chern class Electromagnetism without monopole Connection on a trivial a U1(1) bundle Electromagnetism with monopole Connection on a nontrivial a U1(1) bundle See also • 't Hooft–Polyakov monopole • Wu–Yang monopole References 1. Wu, Tai Tsun; Yang, Chen Ning (1975-12-15). "Concept of nonintegrable phase factors and global formulation of gauge fields". Physical Review D. 12 (12): 3845–3857. doi:10.1103/PhysRevD.12.3845. ISSN 0556-2821. 2. Poo, Mu-ming; Chao, Alexander Wu (2020-01-01). "Conversation with Chen-Ning Yang: reminiscence and reflection". National Science Review. 7 (1): 233–236. doi:10.1093/nsr/nwz113. ISSN 2095-5138. PMC 8288855. PMID 34692035. 3. Woit, Peter (5 April 2008). "Stony Brook Dialogues in Mathematics and Physics". Not even wrong blog. Retrieved 2023-03-14. 4. Wells, Raymond O'Neil; Weyl, Hermann (1988). The Mathematical Heritage of Hermann Weyl. American Mathematical Soc. ISBN 978-0-8218-1482-6. 5. Freed, Daniel S. (2021). "Isadore Singer Transcended Mathematical Boundaries". Quanta Magazine. 6. Zeidler, Eberhard (2008-09-03). Quantum Field Theory II: Quantum Electrodynamics: A Bridge between Mathematicians and Physicists. Springer Science & Business Media. ISBN 978-3-540-85377-0. 7. Boi, Luciano (2004). "Geometrical and topological foundations of theoretical physics: from gauge theories to string program". International Journal of Mathematics and Mathematical Sciences. 2004 (34): 1777–1836. doi:10.1155/S0161171204304400. ISSN 0161-1712. 8. Wells, Raymond O'Neil; Weyl, Hermann (1988). The Mathematical Heritage of Hermann Weyl. American Mathematical Soc. ISBN 978-0-8218-1482-6.
Cynthia Wyels Cynthia Jean Wyels is an American mathematician whose interests include linear algebra, combinatorics, and mathematics education, and who is known for her research in graph pebbling and radio coloring of graphs. She is a professor of mathematics at California State University Channel Islands (CSUCI) in Camarillo, California,[1] where she also co-directs the Alliance for Minority Participation.[2] Education and Career Wyels did her undergraduate studies at Pomona College, and earned a master's degree from the University of Michigan.[1] She completed her Ph.D. in mathematics from the University of California, Santa Barbara in 1994; her dissertation, Isomorphism Problems In A Matrix Setting, was supervised by Morris Newman.[1][3] She has taught mathematics at Weber State University and the United States Military Academy, and was chair of mathematics at California Lutheran University before moving to CSUCI.[1][4] Awards In 2012, Wyels was a winner of the Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics, given by the Mathematical Association of America to recognize teaching excellence that extends beyond a single institution. Her award citation particularly recognized her mentorship of Mexican and first-generation college students through the Research Experiences for Undergraduates program and through personal donations to education in Mexico, and her foundation of a mentorship program at CSUCI.[5] In 2017, the Society for the Advancement of Chicanos/Hispanics and Native Americans in Science gave Wyels their distinguished mentor award.[6] She received the CSUCI UndocuAlly of the Year award in 2017-18.[7] References 1. "Cynthia J. Wyels", Faculty Profiles, California State University Channel Islands, retrieved 2018-04-29 2. Estrada, Roxanne (January 26, 2012), "Educator's passion for math a real plus", Thousand Oaks Acorn 3. Cynthia Wyels at the Mathematics Genealogy Project 4. Bush, Sharon Raiford (November 29, 2015), Mathematics Professor Helps Students Excel In Societal Areas, CBS Los Angeles 5. "MAA Prizes Presented in Boston" (PDF), Notices of the American Mathematical Society, 59 (5): 680–683, May 2012 6. Gazette staff (November 6, 2017), "CSUCI Professor of Mathematics wins national mentoring award", The Fillmore Gazette 7. "Cynthia J. Wyels - Faculty Biographies- CSU Channel Islands". ciapps.csuci.edu. Retrieved 2020-10-22. Authority control: Academics • MathSciNet • Mathematics Genealogy Project
Navier–Stokes equations The Navier–Stokes equations (/nævˈjeɪ stoʊks/ nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes). Claude-Louis Navier George Gabriel Stokes Part of a series on Continuum mechanics $J=-D{\frac {d\varphi }{dx}}$ Fick's laws of diffusion Laws Conservations • Mass • Momentum • Energy Inequalities • Clausius–Duhem (entropy) Solid mechanics • Deformation • Elasticity • linear • Plasticity • Hooke's law • Stress • Finite strain • Infinitesimal strain • Compatibility • Bending • Contact mechanics • frictional • Material failure theory • Fracture mechanics Fluid mechanics Fluids • Statics · Dynamics • Archimedes' principle · Bernoulli's principle • Navier–Stokes equations • Poiseuille equation · Pascal's law • Viscosity • (Newtonian · non-Newtonian) • Buoyancy · Mixing · Pressure Liquids • Adhesion • Capillary action • Chromatography • Cohesion (chemistry) • Surface tension Gases • Atmosphere • Boyle's law • Charles's law • Combined gas law • Fick's law • Gay-Lussac's law • Graham's law Plasma Rheology • Viscoelasticity • Rheometry • Rheometer Smart fluids • Electrorheological • Magnetorheological • Ferrofluids Scientists • Bernoulli • Boyle • Cauchy • Charles • Euler • Fick • Gay-Lussac • Graham • Hooke • Newton • Navier • Noll • Pascal • Stokes • Truesdell The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density.[1] They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable). The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics. The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in the domain. This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample.[2][3] Flow velocity The solution of the equations is a flow velocity. It is a vector field—to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is usually studied in three spatial dimensions and one time dimension, although two (spatial) dimensional and steady-state cases are often used as models, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various trajectories. In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time. General continuum equations See also: Cauchy momentum equation § Conservation form The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is ${\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {g} .$ By setting the Cauchy stress tensor $ {\boldsymbol {\sigma }}$ to be the sum of a viscosity term $ {\boldsymbol {\tau }}$ (the deviatoric stress) and a pressure term $ -p\mathbf {I} $ (volumetric stress), we arrive at Cauchy momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {g} $ where • $ {\frac {\mathrm {D} }{\mathrm {D} t}}$ is the material derivative, defined as $ {\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla $, • $ \rho $ is the (mass) density, • $ \mathbf {u} $ is the flow velocity, • $ \nabla \cdot \,$ is the divergence, • $ p$ is the pressure, • $ t$ is time, • $ {\boldsymbol {\tau }}$ is the deviatoric stress tensor, which has order 2, • $ \mathbf {g} $ represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electrostatic accelerations, and so on. In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the Euler equations. Assuming conservation of mass we can use the mass continuity equation (or simply continuity equation), ${\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \,\mathbf {u} )=0$ to arrive at the conservation form of the equations of motion. This is often written:[4] Cauchy momentum equation (conservation form) ${\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {g} $ where $ \otimes $ is the outer product: $\mathbf {u} \otimes \mathbf {v} =\mathbf {u} \mathbf {v} ^{\mathrm {T} }.$ The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity). All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation. By expressing the deviatoric (shear) stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below. Convective acceleration See also: Cauchy momentum equation § Convective acceleration A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle. Compressible flow Remark: here, the deviatoric stress tensor is denoted $ {\boldsymbol {\sigma }}$ (instead of $ {\boldsymbol {\tau }}$ as it was in the general continuum equations and in the incompressible flow section). The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:[5] • the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient $ \nabla \mathbf {u} $. • the stress is linear in this variable: $ {\boldsymbol {\sigma }}\left(\nabla \mathbf {u} \right)=\mathbf {C} :\left(\nabla \mathbf {u} \right)$ :\left(\nabla \mathbf {u} \right)} , where $ \mathbf {C} $ is the fourth-order tensor representing the constant of proportionality, called the viscosity or elasticity tensor, and : is the double-dot product. • the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $ \mathbf {V} $ is an isotropic tensor; furthermore, since the stress tensor is symmetric, by Helmholtz decomposition it can be expressed in terms of two scalar Lamé parameters, the second viscosity $ \lambda $ and the dynamic viscosity $ \mu $, as it is usual in linear elasticity: Linear stress constitutive equation (expression used for elastic solid) ${\boldsymbol {\sigma }}=\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}$ where $ \mathbf {I} $ is the identity tensor, $ {\boldsymbol {\varepsilon }}\left(\nabla \mathbf {u} \right)\equiv {\frac {1}{2}}\nabla \mathbf {u} +{\frac {1}{2}}\left(\nabla \mathbf {u} \right)^{T}$ is the rate-of-strain tensor and $ \nabla \cdot \mathbf {u} $ is the divergence (i.e. rate of expansion) of the flow. So this decomposition can be explicitly defined as: ${\boldsymbol {\sigma }}=\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right).$ Since the trace of the rate-of-strain tensor in three dimensions is: $\operatorname {tr} ({\boldsymbol {\varepsilon }})=\nabla \cdot \mathbf {u} .$ The trace of the stress tensor in three dimensions becomes: $\operatorname {tr} ({\boldsymbol {\sigma }})=(3\lambda +2\mu )\nabla \cdot \mathbf {u} .$ So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:[6] ${\boldsymbol {\sigma }}=\left(\lambda +{\tfrac {2}{3}}\mu \right)\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} +\mu \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{\mathrm {T} }-{\tfrac {2}{3}}\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} \right)$ Introducing the bulk viscosity $ \zeta $, $\zeta \equiv \lambda +{\tfrac {2}{3}}\mu ,$ we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:[5] Linear stress constitutive equation (expression used for fluids) ${\boldsymbol {\sigma }}=\zeta (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]$ Both second viscosity $ \zeta $ and dynamic viscosity $ \mu $ need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state.[7] The most general of the Navier–Stokes equations become Navier–Stokes momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]+\zeta (\nabla \cdot \mathbf {u} )\mathbf {I} \right\}+\rho \mathbf {g} .$ Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. For instance, in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion. In some cases, the second viscosity $ \zeta $ can be assumed to be constant in which case, the effect of the volume viscosity $ \zeta $ is that the mechanical pressure is not equivalent to the thermodynamic pressure:[8] as demonstrated below. $\nabla \cdot (\nabla \cdot \mathbf {u} )\mathbf {I} =\nabla (\nabla \cdot \mathbf {u} ),$ ${\bar {p}}\equiv p-\zeta \,\nabla \cdot \mathbf {u} ,$ However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves,[9] where second viscosity coefficient becomes important) by explicitly assuming $ \zeta =0$. The assumption of setting $ \zeta =0$ is called as the Stokes hypothesis.[10] The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory,;[11] for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become Navier–Stokes momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\rho \mathbf {g} .$ If the dynamic viscosity μ is also assumed to be constant, the equations can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor $ \nabla \mathbf {u} $ is $ \nabla ^{2}\mathbf {u} $ and the divergence of tensor $ \left(\nabla \mathbf {u} \right)^{\mathrm {T} }$ is $ \nabla \left(\nabla \cdot \mathbf {u} \right)$, one finally arrives to the compressible (most general) Navier–Stokes momentum equation:[12] Navier–Stokes momentum equation (convective form) $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} .$ where $ {\frac {\mathrm {D} }{\mathrm {D} t}}$ is the material derivative. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation: Navier–Stokes momentum equation (conservation form) ${\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla p+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} .$ Bulk viscosity is assumed to be constant, otherwise it should not be taken out of the last derivative. The convective acceleration term can also be written as $\mathbf {u} \cdot \nabla \mathbf {u} =(\nabla \times \mathbf {u} )\times \mathbf {u} +{\tfrac {1}{2}}\nabla \mathbf {u} ^{2},$ where the vector $ (\nabla \times \mathbf {u} )\times \mathbf {u} $ is known as the Lamb vector. For the special case of an incompressible flow, the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with $ \nabla \cdot \mathbf {u} =0$.[13] Incompressible flow The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:[5] • the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient $ \nabla \mathbf {u} $. • the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $ {\boldsymbol {\tau }}$ is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity $ \mu $: Stokes' stress constitutive equation (expression used for incompressible elastic solids) ${\boldsymbol {\tau }}=2\mu {\boldsymbol {\varepsilon }}$ where ${\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\mathbf {\nabla u} +\mathbf {\nabla u} ^{\mathrm {T} }\right)$ is the rate-of-strain tensor. So this decomposition can be made explicit as:[5] Stokes's stress constitutive equation (expression used for incompressible viscous fluids) ${\boldsymbol {\tau }}=\mu \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{\mathrm {T} }\right)$ Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state.[7] The divergence of the deviatoric stress is given by: $\nabla \cdot {\boldsymbol {\tau }}=2\mu \nabla \cdot {\boldsymbol {\varepsilon }}=\mu \nabla \cdot \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{\mathrm {T} }\right)=\mu \,\nabla ^{2}\mathbf {u} $ because $ \nabla \cdot \mathbf {u} =0$ for an incompressible fluid. Incompressibility rules out density and pressure waves like sound or shock waves, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.[14] the incompressible Navier–Stokes equations are best visualized by dividing for the density:[15] Incompressible Navier–Stokes equations (convective form) ${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {g} .$ If the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, $ \rho =\rho _{0}$, then we have Incompressible Navier–Stokes equations (convective form) ${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-\nabla \left({\frac {p}{\rho _{0}}}\right)+\mathbf {g} ,$ where $ \nu ={\frac {\mu }{\rho _{0}}}$ is called the kinematic viscosity. A laminar flow example Velocity profile (laminar flow): $u_{x}=u(y),\quad u_{y}=0,\quad u_{z}=0$ for the x-direction, simplify the Navier–Stokes equation: $0=-{\frac {\mathrm {d} P}{\mathrm {d} x}}+\mu \left({\frac {\mathrm {d} ^{2}u}{\mathrm {d} y^{2}}}\right)$ Integrate twice to find the velocity profile with boundary conditions y = h, u = 0, y = −h, u = 0: $u={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}y^{2}+Ay+B$ From this equation, substitute in the two boundary conditions to get two equations: ${\begin{aligned}0&={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}+Ah+B\\0&={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}-Ah+B\end{aligned}}$ Add and solve for B: $B=-{\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}$ Substitute and solve for A: $A=0$ Finally this gives the velocity profile: $u={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}\left(y^{2}-h^{2}\right)$ It is well worth observing the meaning of each term (compare to the Cauchy momentum equation): $\overbrace {{\vphantom {\frac {}{}}}\underbrace {\frac {\partial \mathbf {u} }{\partial t}} _{\text{Variation}}+\underbrace {{\vphantom {\frac {}{}}}(\mathbf {u} \cdot \nabla )\mathbf {u} } _{\text{Divergence}}} ^{\text{Inertia (per volume)}}=\overbrace {{\vphantom {\frac {\partial }{\partial }}}\underbrace {{\vphantom {\frac {}{}}}-\nabla w} _{\begin{smallmatrix}{\text{Internal}}\\{\text{source}}\end{smallmatrix}}+\underbrace {{\vphantom {\frac {}{}}}\nu \nabla ^{2}\mathbf {u} } _{\text{Diffusion}}} ^{\text{Divergence of stress}}+\underbrace {{\vphantom {\frac {}{}}}\mathbf {g} } _{\begin{smallmatrix}{\text{External}}\\{\text{source}}\end{smallmatrix}}.$ The higher-order term, namely the shear stress divergence $ \nabla \cdot {\boldsymbol {\tau }}$, has simply reduced to the vector Laplacian term $ \mu \nabla ^{2}\mathbf {u} $.[16] This Laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that – for a Newtonian fluid – viscosity operates as a diffusion of momentum, in much the same way as the heat conduction. In fact neglecting the convection term, incompressible Navier–Stokes equations lead to a vector diffusion equation (namely Stokes equations), but in general the convection term is present, so incompressible Navier–Stokes equations belong to the class of convection–diffusion equations. In the usual case of an external field being a conservative field: $\mathbf {g} =-\nabla \varphi $ by defining the hydraulic head: $h\equiv w+\varphi $ one can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field: ${\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-\nabla h.$ The incompressible Navier–Stokes equations with conservative external field is the fundamental equation of hydraulics. The domain for these equations is commonly a 3 or less dimensional Euclidean space, for which an orthogonal coordinate reference frame is usually set to explicit the system of scalar partial differential equations to be solved. In 3-dimensional orthogonal coordinate systems are 3: Cartesian, cylindrical, and spherical. Expressing the Navier–Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems. But for the higher order terms (the two coming from the divergence of the deviatoric stress that distinguish Navier–Stokes equations from Euler equations) some tensor calculus is required for deducing an expression in non-cartesian orthogonal coordinate systems. The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations, ${\begin{aligned}{\frac {\partial \mathbf {u} }{\partial t}}&=\Pi ^{S}\left(-(\mathbf {u} \cdot \nabla )\mathbf {u} +\nu \,\nabla ^{2}\mathbf {u} \right)+\mathbf {f} ^{S}\\\rho ^{-1}\,\nabla p&=\Pi ^{I}\left(-(\mathbf {u} \cdot \nabla )\mathbf {u} +\nu \,\nabla ^{2}\mathbf {u} \right)+\mathbf {f} ^{I}\end{aligned}}$ where $ \Pi ^{S}$ and $ \Pi ^{I}$ are solenoidal and irrotational projection operators satisfying $ \Pi ^{S}+\Pi ^{I}-1$ and $ \mathbf {f} ^{S}$ and $ \mathbf {f} ^{I}$ are the non-conservative and conservative parts of the body force. This result follows from the Helmholtz theorem (also known as the fundamental theorem of vector calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation. The explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem: $\Pi ^{S}\,\mathbf {F} (\mathbf {r} )={\frac {1}{4\pi }}\nabla \times \int {\frac {\nabla ^{\prime }\times \mathbf {F} (\mathbf {r} ')}{\|\mathbf {r} -\mathbf {r} '\|}}\,\mathrm {d} V',\quad \Pi ^{I}=1-\Pi ^{S}$ with a similar structure in 2D. Thus the governing equation is an integro-differential equation similar to Coulomb and Biot–Savart law, not convenient for numerical computation. An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation,[17] is given by, $\left(\mathbf {w} ,{\frac {\partial \mathbf {u} }{\partial t}}\right)=-{\bigl (}\mathbf {w} ,\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} {\bigr )}-\nu \left(\nabla \mathbf {w} :\nabla \mathbf {u} \right)+\left(\mathbf {w} ,\mathbf {f} ^{S}\right)$ :\nabla \mathbf {u} \right)+\left(\mathbf {w} ,\mathbf {f} ^{S}\right)} for divergence-free test functions $ \mathbf {w} $ satisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. The discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section. There one will be able to address the question "How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?". The absence of pressure forces from the governing velocity equation demonstrates that the equation is not a dynamic one, but rather a kinematic equation where the divergence-free condition serves the role of a conservation equation. This all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition. Strong form Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density $ \rho $ in a domain $\Omega \subset \mathbb {R} ^{d}\quad (d=2,3)$ with boundary $\partial \Omega =\Gamma _{D}\cup \Gamma _{N},$ being $ \Gamma _{D}$ and $ \Gamma _{N}$ portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied ($ \Gamma _{D}\cap \Gamma _{N}=\emptyset $):[18] ${\begin{cases}\rho {\dfrac {\partial \mathbf {u} }{\partial t}}+\rho (\mathbf {u} \cdot \nabla )\mathbf {u} -\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u} ,p)=\mathbf {f} &{\text{ in }}\Omega \times (0,T)\\\nabla \cdot \mathbf {u} =0&{\text{ in }}\Omega \times (0,T)\\\mathbf {u} =\mathbf {g} &{\text{ on }}\Gamma _{D}\times (0,T)\\{\boldsymbol {\sigma }}(\mathbf {u} ,p){\hat {\mathbf {n} }}=\mathbf {h} &{\text{ on }}\Gamma _{N}\times (0,T)\\\mathbf {u} (0)=\mathbf {u} _{0}&{\text{ in }}\Omega \times \{0\}\end{cases}}$ $ \mathbf {u} $ is the fluid velocity, $ p$ the fluid pressure, $ \mathbf {f} $ a given forcing term, ${\hat {\mathbf {n} }}$ the outward directed unit normal vector to $ \Gamma _{N}$, and $ {\boldsymbol {\sigma }}(\mathbf {u} ,p)$ the viscous stress tensor defined as:[18] ${\boldsymbol {\sigma }}(\mathbf {u} ,p)=-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} ).$ Let $ \mu $ be the dynamic viscosity of the fluid, $ \mathbf {I} $ the second-order identity tensor and $ {\boldsymbol {\varepsilon }}(\mathbf {u} )$ the strain-rate tensor defined as:[18] ${\boldsymbol {\varepsilon }}(\mathbf {u} )={\frac {1}{2}}\left(\left(\nabla \mathbf {u} \right)+\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right).$ The functions $ \mathbf {g} $ and $ \mathbf {h} $ are given Dirichlet and Neumann boundary data, while $ \mathbf {u} _{0}$ is the initial condition. The first equation is the momentum balance equation, while the second represents the mass conservation, namely the continuity equation. Assuming constant dynamic viscosity, using the vectorial identity $\nabla \cdot \left(\nabla \mathbf {f} \right)^{\mathrm {T} }=\nabla (\nabla \cdot \mathbf {f} )$ and exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as:[18] ${\begin{aligned}\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u} ,p)&=\nabla \cdot \left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} )\right)\\&=-\nabla p+2\mu \nabla \cdot {\boldsymbol {\varepsilon }}(\mathbf {u} )\\&=-\nabla p+2\mu \nabla \cdot \left[{\tfrac {1}{2}}\left(\left(\nabla \mathbf {u} \right)+\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right)\right]\\&=-\nabla p+\mu \left(\Delta \mathbf {u} +\nabla \cdot \left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right)\\&=-\nabla p+\mu {\bigl (}\Delta \mathbf {u} +\nabla \underbrace {(\nabla \cdot \mathbf {u} )} _{=0}{\bigr )}=-\nabla p+\mu \,\Delta \mathbf {u} .\end{aligned}}$ Moreover, note that the Neumann boundary conditions can be rearranged as:[18] ${\boldsymbol {\sigma }}(\mathbf {u} ,p){\hat {\mathbf {n} }}=\left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} )\right){\hat {\mathbf {n} }}=-p{\hat {\mathbf {n} }}+\mu {\frac {\partial {\boldsymbol {u}}}{\partial {\hat {\mathbf {n} }}}}.$ Weak form In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation[18] $\rho {\frac {\partial \mathbf {u} }{\partial t}}-\mu \Delta \mathbf {u} +\rho (\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\mathbf {f} $ multiply it for a test function $ \mathbf {v} $, defined in a suitable space $ V$, and integrate both members with respect to the domain $ \Omega $:[18] $\int \limits _{\Omega }\rho {\frac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} -\int \limits _{\Omega }\mu \Delta \mathbf {u} \cdot \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} +\int \limits _{\Omega }\nabla p\cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} $ Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:[18] ${\begin{aligned}-\int \limits _{\Omega }\mu \Delta \mathbf {u} \cdot \mathbf {v} &=\int _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} -\int \limits _{\partial \Omega }\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}\cdot \mathbf {v} \\\int \limits _{\Omega }\nabla p\cdot \mathbf {v} &=-\int \limits _{\Omega }p\nabla \cdot \mathbf {v} +\int \limits _{\partial \Omega }p\mathbf {v} \cdot {\hat {\mathbf {n} }}\end{aligned}}$ Using these relations, one gets:[18] $\int \limits _{\Omega }\rho {\dfrac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} +\int \limits _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} -\int \limits _{\Omega }p\nabla \cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} +\int \limits _{\partial \Omega }\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} \quad \forall \mathbf {v} \in V.$ In the same fashion, the continuity equation is multiplied for a test function q belonging to a space $ Q$ and integrated in the domain $ \Omega $:[18] $\int \limits _{\Omega }q\nabla \cdot \mathbf {u} =0.\quad \forall q\in Q.$ The space functions are chosen as follows: ${\begin{aligned}V=\left[H_{0}^{1}(\Omega )\right]^{d}&=\left\{\mathbf {v} \in \left[H^{1}(\Omega )\right]^{d}:\quad \mathbf {v} =\mathbf {0} {\text{ on }}\Gamma _{D}\right\},\\Q&=L^{2}(\Omega )\end{aligned}}$ Considering that the test function v vanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:[18] $\int \limits _{\partial \Omega }\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} =\underbrace {\int \limits _{\Gamma _{D}}\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} } _{\mathbf {v} =\mathbf {0} {\text{ on }}\Gamma _{D}\ }+\int \limits _{\Gamma _{N}}\underbrace {{\vphantom {\int \limits _{\Gamma _{N}}}}\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)} _{=\mathbf {h} {\text{ on }}\Gamma _{N}}\cdot \mathbf {v} =\int \limits _{\Gamma _{N}}\mathbf {h} \cdot \mathbf {v} .$ Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as:[18] ${\begin{aligned}&{\text{find }}\mathbf {u} \in L^{2}\left(\mathbb {R} ^{+}\;\left[H^{1}(\Omega )\right]^{d}\right)\cap C^{0}\left(\mathbb {R} ^{+}\;\left[L^{2}(\Omega )\right]^{d}\right){\text{ such that: }}\\[5pt]&\quad {\begin{cases}\displaystyle \int \limits _{\Omega }\rho {\dfrac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} +\int \limits _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} -\int \limits _{\Omega }p\nabla \cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} +\int \limits _{\Gamma _{N}}\mathbf {h} \cdot \mathbf {v} \quad \forall \mathbf {v} \in V,\\\displaystyle \int \limits _{\Omega }q\nabla \cdot \mathbf {u} =0\quad \forall q\in Q.\end{cases}}\end{aligned}}$ Discrete velocity With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is $\left(\mathbf {w} _{i},{\frac {\partial \mathbf {u} _{j}}{\partial t}}\right)=-{\bigl (}\mathbf {w} _{i},\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} _{j}{\bigr )}-\nu \left(\nabla \mathbf {w} _{i}:\nabla \mathbf {u} _{j}\right)+\left(\mathbf {w} _{i},\mathbf {f} ^{S}\right).$ It is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' theorem. Discussion will be restricted to 2D in the following. We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the plate-bending literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions, ${\begin{aligned}\nabla \varphi &=\left({\frac {\partial \varphi }{\partial x}},\,{\frac {\partial \varphi }{\partial y}}\right)^{\mathrm {T} },\\[5pt]\nabla \times \varphi &=\left({\frac {\partial \varphi }{\partial y}},\,-{\frac {\partial \varphi }{\partial x}}\right)^{\mathrm {T} }.\end{aligned}}$ Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements. Taking the curl of the scalar stream function elements gives divergence-free velocity elements.[19][20] The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces. Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces. Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions. The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations. Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D. Pressure recovery Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is, $(\mathbf {g} _{i},\nabla p)=-\left(\mathbf {g} _{i},\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} _{j}\right)-\nu \left(\nabla \mathbf {g} _{i}:\nabla \mathbf {u} _{j}\right)+\left(\mathbf {g} _{i},\mathbf {f} ^{I}\right)$ where the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of interest. In this case, one can use scalar Hermite elements for the pressure. For the test/weight functions $ \mathbf {g} _{i}$ one would choose the irrotational vector elements obtained from the gradient of the pressure element. Non-inertial frame of reference The rotating frame of reference introduces some interesting pseudo-forces into the equations through the material derivative term. Consider a stationary inertial frame of reference $ K$ , and a non-inertial frame of reference $ K'$, which is translating with velocity $ \mathbf {U} (t)$ and rotating with angular velocity $ \Omega (t)$ with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes Navier–Stokes momentum equation in non-inertial frame $\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla {\bar {p}}+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} -\rho \left[2\mathbf {\Omega } \times \mathbf {u} +\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} )+{\frac {\mathrm {d} \mathbf {U} }{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {x} \right].$ Here $ \mathbf {x} $ and $ \mathbf {u} $ are measured in the non-inertial frame. The first term in the parenthesis represents Coriolis acceleration, the second term is due to centrifugal acceleration, the third is due to the linear acceleration of $ K'$ with respect to $ K$ and the fourth term is due to the angular acceleration of $ K'$ with respect to $ K$. Other equations The Navier–Stokes equations are strictly a statement of the balance of momentum. To fully describe fluid flow, more information is needed, how much depending on the assumptions made. This additional information may include boundary data (no-slip, capillary surface, etc.), conservation of mass, balance of energy, and/or an equation of state. Continuity equation for incompressible fluid Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, given in its most general form as: ${\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0$ or, using the substantive derivative: ${\frac {\mathrm {D} \rho }{\mathrm {D} t}}+\rho (\nabla \cdot \mathbf {u} )=0.$ For incompressible fluid, density along the line of flow remains constant over time, ${\frac {\mathrm {D} \rho }{\mathrm {D} t}}=0.$ Therefore divergence of velocity is always zero: $\nabla \cdot \mathbf {u} =0.$ Stream function for incompressible 2D fluid Taking the curl of the incompressible Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with $ u_{z}=0$ and no dependence of anything on $ z$), where the equations reduce to: ${\begin{aligned}\rho \left({\frac {\partial u_{x}}{\partial t}}+u_{x}{\frac {\partial u_{x}}{\partial x}}+u_{y}{\frac {\partial u_{x}}{\partial y}}\right)&=-{\frac {\partial p}{\partial x}}+\mu \left({\frac {\partial ^{2}u_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{x}}{\partial y^{2}}}\right)+\rho g_{x}\\\rho \left({\frac {\partial u_{y}}{\partial t}}+u_{x}{\frac {\partial u_{y}}{\partial x}}+u_{y}{\frac {\partial u_{y}}{\partial y}}\right)&=-{\frac {\partial p}{\partial y}}+\mu \left({\frac {\partial ^{2}u_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{y}}{\partial y^{2}}}\right)+\rho g_{y}.\end{aligned}}$ Differentiating the first with respect to $ y$, the second with respect to $ x$ and subtracting the resulting equations will eliminate pressure and any conservative force. For incompressible flow, defining the stream function $ \psi $ through $u_{x}={\frac {\partial \psi }{\partial y}};\quad u_{y}=-{\frac {\partial \psi }{\partial x}}$ results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation: ${\frac {\partial }{\partial t}}\left(\nabla ^{2}\psi \right)+{\frac {\partial \psi }{\partial y}}{\frac {\partial }{\partial x}}\left(\nabla ^{2}\psi \right)-{\frac {\partial \psi }{\partial x}}{\frac {\partial }{\partial y}}\left(\nabla ^{2}\psi \right)=\nu \nabla ^{4}\psi $ where $ \nabla ^{4}$ is the 2D biharmonic operator and $ \nu $ is the kinematic viscosity, $ \nu ={\frac {\mu }{p}}$. We can also express this compactly using the Jacobian determinant: ${\frac {\partial }{\partial t}}\left(\nabla ^{2}\psi \right)+{\frac {\partial \left(\psi ,\nabla ^{2}\psi \right)}{\partial (y,x)}}=\nu \nabla ^{4}\psi .$ This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero. In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function. The incompressible Navier–Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest. Properties Nonlinearity The Navier–Stokes equations are nonlinear partial differential equations in the general case and so remain in almost every real situation.[21][22] In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model. The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.[23] Turbulence Turbulence is the time-dependent chaotic behaviour seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.[24] The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or direct numerical simulation. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged Navier–Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart–Allmaras, k–ω, k–ε, and SST models, which add a variety of additional equations to bring closure to the RANS equations. Large eddy simulation (LES) can also be used to solve these equations numerically. This approach is computationally more expensive—in time and in computer memory—than RANS, but produces better results because it explicitly resolves the larger turbulent scales. Applicability Together with supplemental equations (for example, conservation of mass) and well-formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations. The Navier–Stokes equations assume that the fluid being studied is a continuum (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. For example, capillarity of internal layers in fluids appears for flow with high gradients.[25] For large Knudsen number of the problem, the Boltzmann equation may be a suitable replacement.[26] Failing that, one may have to resort to molecular dynamics or various hybrid methods.[27] Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually written for Newtonian fluids where the viscosity model is linear; truly general models for the flow of other kinds of fluids (such as blood) do not exist.[28] Application to specific problems The Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by surface tension. Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by scale analysis to further simplify the problem. Parallel flow Assume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) boundary value problem is: ${\frac {\mathrm {d} ^{2}u}{\mathrm {d} y^{2}}}=-1;\quad u(0)=u(1)=0.$ The boundary condition is the no slip condition. This problem is easily solved for the flow field: $u(y)={\frac {y-y^{2}}{2}}.$ From this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate. Radial flow Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the radial flow between parallel plates; this involves convection and thus non-linearity. The velocity field may be represented by a function f(z) that must satisfy: ${\frac {\mathrm {d} ^{2}f}{\mathrm {d} z^{2}}}+Rf^{2}=-1;\quad f(-1)=f(1)=0.$ This ordinary differential equation is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The nonlinear term makes this a very difficult problem to solve analytically (a lengthy implicit solution may be found which involves elliptic integrals and roots of cubic polynomials). Issues with the actual existence of solutions arise for $ R>1.41$ (approximately; this is not √2), the parameter $ R$ being the Reynolds number with appropriately chosen scales.[29] This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.[29] Convection A type of natural convection that can be described by the Navier–Stokes equation is the Rayleigh–Bénard convection. It is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility. Exact solutions of the Navier–Stokes equations Some exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are Poiseuille flow, Couette flow and the oscillatory Stokes boundary layer. But also, more interesting examples, solutions to the full non-linear equations, exist, such as Jeffery–Hamel flow, Von Kármán swirling flow, stagnation point flow, Landau–Squire jet, and Taylor–Green vortex.[30][31][32] Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers. Under additional assumptions, the component parts can be separated.[33] A two-dimensional example For example, in the case of an unbounded planar domain with two-dimensional — incompressible and stationary — flow in polar coordinates (r,φ), the velocity components (ur,uφ) and pressure p are:[34] ${\begin{aligned}u_{r}&={\frac {A}{r}},\\u_{\varphi }&=B\left({\frac {1}{r}}-r^{{\frac {A}{\nu }}+1}\right),\\p&=-{\frac {A^{2}+B^{2}}{2r^{2}}}-{\frac {2B^{2}\nu r^{\frac {A}{\nu }}}{A}}+{\frac {B^{2}r^{\left({\frac {2A}{\nu }}+2\right)}}{{\frac {2A}{\nu }}+2}}\end{aligned}}$ where A and B are arbitrary constants. This solution is valid in the domain r ≥ 1 and for A < −2ν. In Cartesian coordinates, when the viscosity is zero (ν = 0), this is: ${\begin{aligned}\mathbf {v} (x,y)&={\frac {1}{x^{2}+y^{2}}}{\begin{pmatrix}Ax+By\\Ay-Bx\end{pmatrix}},\\p(x,y)&=-{\frac {A^{2}+B^{2}}{2\left(x^{2}+y^{2}\right)}}\end{aligned}}$ A three-dimensional example For example, in the case of an unbounded Euclidean domain with three-dimensional — incompressible, stationary and with zero viscosity (ν = 0) — radial flow in Cartesian coordinates (x,y,z), the velocity vector v and pressure p are: ${\begin{aligned}\mathbf {v} (x,y,z)&={\frac {A}{x^{2}+y^{2}+z^{2}}}{\begin{pmatrix}x\\y\\z\end{pmatrix}},\\p(x,y,z)&=-{\frac {A^{2}}{2\left(x^{2}+y^{2}+z^{2}\right)}}.\end{aligned}}$ There is a singularity at x = y = z = 0. A three-dimensional steady-state vortex solution A steady-state example with no singularities comes from considering the flow along the lines of a Hopf fibration. Let $ r$ be a constant radius of the inner coil. One set of solutions is given by:[35] ${\begin{aligned}\rho (x,y,z)&={\frac {3B}{r^{2}+x^{2}+y^{2}+z^{2}}}\\p(x,y,z)&={\frac {-A^{2}B}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{3}}}\\\mathbf {u} (x,y,z)&={\frac {A}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{2}}}{\begin{pmatrix}2(-ry+xz)\\2(rx+yz)\\r^{2}-x^{2}-y^{2}+z^{2}\end{pmatrix}}\\g&=0\\\mu &=0\end{aligned}}$ for arbitrary constants $ A$ and $ B$. This is a solution in a non-viscous gas (compressible fluid) whose density, velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium problem because that refers to incompressible fluids where $ \rho $ is a constant, and neither does it deal with the uniqueness of the Navier–Stokes equations with respect to any turbulence properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the Pythagorean quadruple parametrization. Other choices of density and pressure are possible with the same velocity field: Other choices of density and pressure Another choice of pressure and density with the same velocity vector above is one where the pressure and density fall to zero at the origin and are highest in the central loop at z = 0, x2 + y2 = r2: ${\begin{aligned}\rho (x,y,z)&={\frac {20B\left(x^{2}+y^{2}\right)}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{3}}}\\p(x,y,z)&={\frac {-A^{2}B}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{4}}}+{\frac {-4A^{2}B\left(x^{2}+y^{2}\right)}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{5}}}.\end{aligned}}$ In fact in general there are simple solutions for any polynomial function f where the density is: $\rho (x,y,z)={\frac {1}{r^{2}+x^{2}+y^{2}+z^{2}}}f\left({\frac {x^{2}+y^{2}}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{2}}}\right).$ Viscous three-dimensional periodic solutions Two examples of periodic fully-three-dimensional viscous solutions are described in.[36] These solutions are defined on a three-dimensional torus $\mathbb {T} ^{3}=[0,L]^{3}$ and are characterized by positive and negative helicity respectively. The solution with positive helicity is given by: ${\begin{aligned}u_{x}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(kx-\pi /3)\cos(ky+\pi /3)\sin(kz+\pi /2)-\cos(kz-\pi /3)\sin(kx+\pi /3)\sin(ky+\pi /2)\,\right]e^{-3\nu k^{2}t}\\u_{y}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(ky-\pi /3)\cos(kz+\pi /3)\sin(kx+\pi /2)-\cos(kx-\pi /3)\sin(ky+\pi /3)\sin(kz+\pi /2)\,\right]e^{-3\nu k^{2}t}\\u_{z}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(kz-\pi /3)\cos(kx+\pi /3)\sin(ky+\pi /2)-\cos(ky-\pi /3)\sin(kz+\pi /3)\sin(kx+\pi /2)\,\right]e^{-3\nu k^{2}t}\end{aligned}}$ where $k=2\pi /L$ is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is $U_{0}^{2}/2$ at $t=0$. The pressure field is obtained from the velocity field as $p=p_{0}-\rho _{0}\\|{\boldsymbol {u}}\\|^{2}/2$ (where $p_{0}$ and $\rho _{0}$ are reference values for the pressure and density fields respectively). Since both the solutions belong to the class of Beltrami flow, the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by $\omega ={\sqrt {3}}\,k\,{\boldsymbol {u}}$. These solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green Taylor–Green vortex. Wyld diagrams Wyld diagrams are bookkeeping graphs that correspond to the Navier–Stokes equations via a perturbation expansion of the fundamental continuum mechanics. Similar to the Feynman diagrams in quantum field theory, these diagrams are an extension of Keldysh's technique for nonequilibrium processes in fluid dynamics. In other words, these diagrams assign graphs to the (often) turbulent phenomena in turbulent fluids by allowing correlated and interacting fluid particles to obey stochastic processes associated to pseudo-random functions in probability distributions.[37] Representations in 3D Note that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. $ \partial _{x}u$ means the partial derivative of $ u$ with respect to $ x$, and $ \partial _{y}^{2}f_{\theta }$ means the second-order partial derivative of $ f_{\theta }$ with respect to $ y$. A 2022 paper provides a less costly, dynamical and recurrent solution of the Navier-Stokes equation for 3D turbulent fluid flows. On suitably short time scales, the dynamics of turbulence is deterministic.[38] Cartesian coordinates From the general form of the Navier–Stokes, with the velocity vector expanded as $ \mathbf {u} =(u_{x},u_{y},u_{z})$, sometimes respectively named $ u$, $ v$, $ w$, we may write the vector equation explicitly, ${\begin{aligned}x:\ &\rho \left({\partial _{t}u_{x}}+u_{x}\,{\partial _{x}u_{x}}+u_{y}\,{\partial _{y}u_{x}}+u_{z}\,{\partial _{z}u_{x}}\right)\\&\quad =-\partial _{x}p+\mu \left({\partial _{x}^{2}u_{x}}+{\partial _{y}^{2}u_{x}}+{\partial _{z}^{2}u_{x}}\right)+{\frac {1}{3}}\mu \ \partial _{x}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{x}\\\end{aligned}}$ ${\begin{aligned}y:\ &\rho \left({\partial _{t}u_{y}}+u_{x}{\partial _{x}u_{y}}+u_{y}{\partial _{y}u_{y}}+u_{z}{\partial _{z}u_{y}}\right)\\&\quad =-{\partial _{y}p}+\mu \left({\partial _{x}^{2}u_{y}}+{\partial _{y}^{2}u_{y}}+{\partial _{z}^{2}u_{y}}\right)+{\frac {1}{3}}\mu \ \partial _{y}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{y}\\\end{aligned}}$ ${\begin{aligned}z:\ &\rho \left({\partial _{t}u_{z}}+u_{x}{\partial _{x}u_{z}}+u_{y}{\partial _{y}u_{z}}+u_{z}{\partial _{z}u_{z}}\right)\\&\quad =-{\partial _{z}p}+\mu \left({\partial _{x}^{2}u_{z}}+{\partial _{y}^{2}u_{z}}+{\partial _{z}^{2}u_{z}}\right)+{\frac {1}{3}}\mu \ \partial _{z}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{z}.\end{aligned}}$ Note that gravity has been accounted for as a body force, and the values of $ g_{x}$, $ g_{y}$, $ g_{z}$ will depend on the orientation of gravity with respect to the chosen set of coordinates. The continuity equation reads: $\partial _{t}\rho +\partial _{x}(\rho u_{x})+\partial _{y}(\rho u_{y})+\partial _{z}(\rho u_{z})=0.$ When the flow is incompressible, $ \rho $ does not change for any fluid particle, and its material derivative vanishes: $ {\frac {\mathrm {D} \rho }{\mathrm {D} t}}=0$. The continuity equation is reduced to: $\partial _{x}u_{x}+\partial _{y}u_{y}+\partial _{z}u_{z}=0.$ Thus, for the incompressible version of the Navier–Stokes equation the second part of the viscous terms fall away (see Incompressible flow). This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is still a nonlinear system of partial differential equations for which solutions are difficult to obtain. Cylindrical coordinates A change of variables on the Cartesian equations will yield[14] the following momentum equations for $ r$, $ \phi $, and $ z$[39] ${\begin{aligned}r:\ &\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{r}}+u_{z}{\partial _{z}u_{r}}-{\frac {u_{\varphi }^{2}}{r}}\right)\\&\quad =-{\partial _{r}p}\\&\qquad +\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{r}}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{r}}+{\partial _{z}^{2}u_{r}}-{\frac {u_{r}}{r^{2}}}-{\frac {2}{r^{2}}}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{r}\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{r}\\[8px]\end{aligned}}$ ${\begin{aligned}\varphi :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{\varphi }}+u_{z}{\partial _{z}u_{\varphi }}+{\frac {u_{r}u_{\varphi }}{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r}}\ \partial _{r}\left(r{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{\varphi }}+{\partial _{z}^{2}u_{\varphi }}+{\frac {2}{r^{2}}}{\partial _{\varphi }u_{r}}-{\frac {u_{\varphi }}{r^{2}}}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\varphi }\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}$ :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{\varphi }}+u_{z}{\partial _{z}u_{\varphi }}+{\frac {u_{r}u_{\varphi }}{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r}}\ \partial _{r}\left(r{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{\varphi }}+{\partial _{z}^{2}u_{\varphi }}+{\frac {2}{r^{2}}}{\partial _{\varphi }u_{r}}-{\frac {u_{\varphi }}{r^{2}}}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\varphi }\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}} ${\begin{aligned}z:\ &\rho \left({\partial _{t}u_{z}}+u_{r}{\partial _{r}u_{z}}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{z}}+u_{z}{\partial _{z}u_{z}}\right)\\&\quad =-{\partial _{z}p}\\&\qquad +\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{z}}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{z}}+{\partial _{z}^{2}u_{z}}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{z}\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{z}.\end{aligned}}$ The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressure gradient). The continuity equation is: ${\partial _{t}\rho }+{\frac {1}{r}}\partial _{r}\left(\rho ru_{r}\right)+{\frac {1}{r}}{\partial _{\varphi }\left(\rho u_{\varphi }\right)}+{\partial _{z}\left(\rho u_{z}\right)}=0.$ This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very common case is axisymmetric flow with the assumption of no tangential velocity ($ u_{\phi }=0$), and the remaining quantities are independent of $ \phi $: ${\begin{aligned}\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+u_{z}{\partial _{z}u_{r}}\right)&=-{\partial _{r}p}+\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{r}}\right)+{\partial _{z}^{2}u_{r}}-{\frac {u_{r}}{r^{2}}}\right)+\rho g_{r}\\\rho \left({\partial _{t}u_{z}}+u_{r}{\partial _{r}u_{z}}+u_{z}{\partial _{z}u_{z}}\right)&=-{\partial _{z}p}+\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{z}}\right)+{\partial _{z}^{2}u_{z}}\right)+\rho g_{z}\\{\frac {1}{r}}\partial _{r}\left(ru_{r}\right)+{\partial _{z}u_{z}}&=0.\end{aligned}}$ Spherical coordinates \|In spherical coordinates, the $ r$, $ \phi $, and $ \theta $ momentum equations are[14] (note the convention used: $ \theta $ is polar angle, or colatitude,[40] $ 0\leq \theta \leq \pi $): ${\begin{aligned}r:\ &\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{r}}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{r}}-{\frac {u_{\varphi }^{2}+u_{\theta }^{2}}{r}}\right)\\&\quad =-{\partial _{r}p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{r}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{r}}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{r}}\right)-2{\frac {u_{r}+{\partial _{\theta }u_{\theta }}+u_{\theta }\cot \theta }{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{r}\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{r}\\[8px]\end{aligned}}$ ${\begin{aligned}\varphi :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\varphi }}+{\frac {u_{r}u_{\varphi }+u_{\varphi }u_{\theta }\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r\sin \theta }}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\varphi }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\varphi }}\right)+{\frac {2\sin \theta {\partial _{\varphi }u_{r}}+2\cos \theta {\partial _{\varphi }u_{\theta }}-u_{\varphi }}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r\sin \theta }}\partial _{\varphi }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}$ :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\varphi }}+{\frac {u_{r}u_{\varphi }+u_{\varphi }u_{\theta }\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r\sin \theta }}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\varphi }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\varphi }}\right)+{\frac {2\sin \theta {\partial _{\varphi }u_{r}}+2\cos \theta {\partial _{\varphi }u_{\theta }}-u_{\varphi }}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r\sin \theta }}\partial _{\varphi }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}} ${\begin{aligned}\theta :\ &\rho \left({\partial _{t}u_{\theta }}+u_{r}{\partial _{r}u_{\theta }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\theta }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\theta }}+{\frac {u_{r}u_{\theta }-u_{\varphi }^{2}\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\theta }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\theta }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\theta }}\right)+{\frac {2}{r^{2}}}{\partial _{\theta }u_{r}}-{\frac {u_{\theta }+2\cos \theta {\partial _{\varphi }u_{\varphi }}}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\theta }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\theta }.\end{aligned}}$ :\ &\rho \left({\partial _{t}u_{\theta }}+u_{r}{\partial _{r}u_{\theta }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\theta }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\theta }}+{\frac {u_{r}u_{\theta }-u_{\varphi }^{2}\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\theta }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\theta }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\theta }}\right)+{\frac {2}{r^{2}}}{\partial _{\theta }u_{r}}-{\frac {u_{\theta }+2\cos \theta {\partial _{\varphi }u_{\varphi }}}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\theta }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\theta }.\end{aligned}}} Mass continuity will read: ${\partial _{t}\rho }+{\frac {1}{r^{2}}}\partial _{r}\left(\rho r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }(\rho u_{\varphi })}+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(\sin \theta \rho u_{\theta }\right)=0.$ These equations could be (slightly) compacted by, for example, factoring $ {\frac {1}{r^{2}}}$ from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities. Navier–Stokes equations use in games The Navier–Stokes equations are used extensively in video games in order to model a wide variety of natural phenomena. Simulations of small-scale gaseous fluids, such as fire and smoke, are often based on the seminal paper "Real-Time Fluid Dynamics for Games"[41] by Jos Stam, which elaborates one of the methods proposed in Stam's earlier, more famous paper "Stable Fluids"[42] from 1999. Stam proposes stable fluid simulation using a Navier–Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian advection scheme, as first proposed in 1992. More recent implementations based upon this work run on the game systems graphics processing unit (GPU) as opposed to the central processing unit (CPU) and achieve a much higher degree of performance.[43][44] Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass. An introduction to interactive fluid simulation can be found in the 2007 ACM SIGGRAPH course, Fluid Simulation for Computer Animation.[45] See also • Adhémar Jean Claude Barré de Saint-Venant • Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations • Boltzmann equation • Cauchy momentum equation • Cauchy stress tensor • Chapman–Enskog theory • Churchill–Bernstein equation • Coandă effect • Computational fluid dynamics • Continuum mechanics • Convection–diffusion equation • Derivation of the Navier–Stokes equations • Einstein–Stokes equation • Euler equations • Hagen–Poiseuille flow from the Navier–Stokes equations • Millennium Prize Problems • Non-dimensionalization and scaling of the Navier–Stokes equations • Pressure-correction method • Primitive equations • Rayleigh–Bénard convection • Reynolds transport theorem • Stokes equations • Supersonic flow over a flat plate • Vlasov equation Citations 1. McLean, Doug (2012). "Continuum Fluid Mechanics and the Navier-Stokes Equations". 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Bibcode:1989STIN...9123418S. 30. Wang, C. Y. (1991), "Exact solutions of the steady-state Navier–Stokes equations", Annual Review of Fluid Mechanics, 23: 159–177, Bibcode:1991AnRFM..23..159W, doi:10.1146/annurev.fl.23.010191.001111 31. Landau & Lifshitz (1987) pp. 75–88. 32. Ethier, C. R.; Steinman, D. A. (1994), "Exact fully 3D Navier–Stokes solutions for benchmarking", International Journal for Numerical Methods in Fluids, 19 (5): 369–375, Bibcode:1994IJNMF..19..369E, doi:10.1002/fld.1650190502 33. http://www.claudino.webs.com/Navier%20Stokes%20Equations.pps 34. Ladyzhenskaya, O. A. (1969), The Mathematical Theory of viscous Incompressible Flow (2nd ed.), p. preface, xi 35. Kamchatno, A. M. (1982), Topological solitons in magnetohydrodynamics (PDF), archived (PDF) from the original on 2016-01-28 36. Antuono, M. (2020), "Tri-periodic fully three-dimensional analytic solutions for the Navier–Stokes equations", Journal of Fluid Mechanics, 890, Bibcode:2020JFM...890A..23A, doi:10.1017/jfm.2020.126, S2CID 216463266 37. McComb, W. D. (2008), Renormalization methods: A guide for beginners, Oxford University Press, pp. 121–128, ISBN 978-0-19-923652-7 38. Georgia Institute of Technology (August 29, 2022). "Physicists uncover new dynamical framework for turbulence". Proceedings of the National Academy of Sciences of the United States of America. Phys.org. 119 (34): e2120665119. doi:10.1073/pnas.2120665119. PMC 9407532. PMID 35984901. S2CID 251693676. 39. de' Michieli Vitturi, Mattia, Navier–Stokes equations in cylindrical coordinates, retrieved 2016-12-26 40. Eric W. Weisstein (2005-10-26), Spherical Coordinates, MathWorld, retrieved 2008-01-22 41. Stam, Jos (2003), Real-Time Fluid Dynamics for Games (PDF), S2CID 9353969, archived from the original (PDF) on 2020-08-05 42. Stam, Jos (1999), Stable Fluids (PDF), archived (PDF) from the original on 2019-07-15 43. Harris, Mark J. (2004), "38", GPUGems - Fast Fluid Dynamics Simulation on the GPU 44. Sander, P.; Tatarchuck, N.; Mitchell, J.L. (2007), "9.6", ShaderX5 - Explicit Early-Z Culling for Efficient Fluid Flow Simulation, pp. 553–564 45. Robert Bridson; Matthias Müller-Fischer. "Fluid Simulation for Computer Animation". www.cs.ubc.ca. General references • Acheson, D. J. (1990), Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series, Oxford University Press, ISBN 978-0-19-859679-0 • Batchelor, G. K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 978-0-521-66396-0 • Currie, I. G. (1974), Fundamental Mechanics of Fluids, McGraw-Hill, ISBN 978-0-07-015000-3 • V. Girault and P. A. Raviart. Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986. • Landau, L. D.; Lifshitz, E. M. (1987), Fluid mechanics, vol. Course of Theoretical Physics Volume 6 (2nd revised ed.), Pergamon Press, ISBN 978-0-08-033932-0, OCLC 15017127 • Polyanin, A. D.; Kutepov, A. M.; Vyazmin, A. V.; Kazenin, D. A. (2002), Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London, ISBN 978-0-415-27237-7 • Rhyming, Inge L. (1991), Dynamique des fluides, Presses polytechniques et universitaires romandes • Smits, Alexander J. (2014), A Physical Introduction to Fluid Mechanics, Wiley, ISBN 0-47-1253499 • Temam, Roger (1984): Navier–Stokes Equations: Theory and Numerical Analysis, ACM Chelsea Publishing, ISBN 978-0-8218-2737-6 • White, Frank M. (2006), Viscous Fluid Flow, McGraw-Hill, ISBN 978-0-07-124493-0 External links • Simplified derivation of the Navier–Stokes equations • Three-dimensional unsteady form of the Navier–Stokes equations Glenn Research Center, NASA Topics in continuum mechanics Divisions • Solid mechanics • Fluid mechanics • Acoustics • Vibrations • Rigid body dynamics Laws and Definitions Laws • Conservation of mass • Conservation of momentum • Navier-Stokes • Bernoulli • Poiseuille • Archimedes • Pascal • Conservation of energy • Entropy inequality Definitions • Stress • Cauchy stress • Stress measures • Deformation • Small strain • Antiplane shear • Large strain • Compatibility Solid and Structural mechanics Solids • Elasticity • linear • Hooke's law • Transverse isotropy • Orthotropy • hyperelasticity • Membrane elasticity • Equation of state • Hugoniot • JWL • hypoelasticity • Cauchy elasticity • Viscoelasticity • Creep • Concrete creep • Plasticity • Rock mass plasticity • Viscoplasticity • Yield criterion • Bresler-Pister • Contact mechanics • Frictionless • Frictional Material failure theory • Drucker stability • Material failure theory • Fatigue • Fracture mechanics • J-integral • Compact tension specimen • Damage mechanics • Johnson-Holmquist Structures • Bending • Bending moment • Bending of plates • Sandwich theory Fluid mechanics Fluids • Fluid statics • Fluid dynamics • Navier–Stokes equations • Bernoulli's principle • Poiseuille equation • Buoyancy • Viscosity • Newtonian • Non-Newtonian • Archimedes' principle • Pascal's law • Pressure • Liquids • Surface tension • Capillary action Gases • Atmosphere • Boyle's law • Charles's law • Gay-Lussac's law • Combined gas law Plasma Acoustics • Acoustic theory • Aeroacoustics Rheology • Viscoelasticity • Smart fluids • Magnetorheological • Electrorheological • Ferrofluids • Rheometry • Rheometer Scientists • Bernoulli • Boyle • Cauchy • Charles • Euler • Gay-Lussac • Hooke • Pascal • Newton • Navier • Stokes Awards • Eringen Medal • William Prager Medal Authority control National • Spain • France • BnF data • Germany • Israel • United States • Czech Republic Other • IdRef
Willem Abraham Wythoff Willem Abraham Wythoff, born Wijthoff (Dutch pronunciation: [ʋɛithɔf]), (6 October 1865 – 21 May 1939) was a Dutch mathematician. Willem Abraham Wythoff Born Willem Abraham Wijthoff (1865-10-06)6 October 1865 Amsterdam Died21 May 1939(1939-05-21) (aged 73) Amsterdam NationalityDutch Alma materUniversity of Amsterdam Known forWythoff's game, Wythoff construction, Wythoff symbol Scientific career FieldsMathematics Doctoral advisorDiederik Korteweg Biography Wythoff was born in Amsterdam to Anna C. F. Kerkhoven and Abraham Willem Wijthoff,[1] who worked in a sugar refinery.[2] He studied at the University of Amsterdam, and earned his Ph.D. in 1898 under the supervision of Diederik Korteweg.[3] Contributions Wythoff is known in combinatorial game theory and number theory for his study of Wythoff's game, whose solution involves the Fibonacci numbers.[2] The Wythoff array, a two-dimensional array of numbers related to this game and to the Fibonacci sequence, is also named after him.[4][5] In geometry, Wythoff is known for the Wythoff construction of uniform tilings and uniform polyhedra and for the Wythoff symbol used as a notation for these geometric objects. Selected publications • Wythoff, W. A. (1905–1907), "A modification of the game of nim", Nieuw Archief voor Wiskunde, 2: 199–202. • Wythoff, W. A. (1918), "A relation between the polytopes of the C600-family", Proceedings of the Section of Sciences, Koninklijke Akademie van Wetenschappen te Amsterdam, 20: 966–970, Bibcode:1918KNAB...20..966W. References 1. "Gezinsblad van Willem Abraham Wijthoff". www.humanitarisme.nl. Retrieved Oct 12, 2022. 2. Stakhov, Alexey; Stakhov, Alekseĭ Petrovich; Olsen, Scott Anthony (2009), The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science, K & E Series on Knots and Everything, vol. 22, World Scientific, pp. 129–130, ISBN 9789812775825. 3. Willem Abraham Wythoff at the Mathematics Genealogy Project 4. Kimberling, Clark (1995), "The Zeckendorf array equals the Wythoff array" (PDF), Fibonacci Quarterly, 33 (1): 3–8. 5. 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. External links Media related to Willem Wijthoff at Wikimedia Commons • Kimberling, Clark, Willem Abraham Wythoff (1865–1939) number-theorist Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
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.
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.
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".
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 Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Japan • Czech Republic • Netherlands • Poland Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie • Trove Other • SNAC • IdRef
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 International • ISNI • VIAF National • Poland
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)
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
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 thechacha20-poly1305@openssh.com 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 Widely used ciphers • A5/1 • A5/2 • ChaCha • Crypto-1 • E0 • RC4 eSTREAM Portfolio Software • HC-256 • Rabbit • Salsa20 • SOSEMANUK Hardware • Grain • MICKEY • Trivium Other ciphers • Achterbahn (stream cipher) • F-FCSR • FISH • ISAAC • MUGI • ORYX • Panama • Phelix • Pike • Py • QUAD • Scream • SEAL • SNOW • SOBER • SOBER-128 • VEST • VMPC • WAKE Generators • shrinking generator • self-shrinking generator • alternating step generator Theory • block ciphers in stream mode • shift register • LFSR • NLFSR • T-function • IV Attacks • correlation attack • correlation immunity • stream cipher attacks Cryptographic hash functions and message authentication codes • List • Comparison • Known attacks Common functions • MD5 (compromised) • SHA-1 (compromised) • SHA-2 • SHA-3 • BLAKE2 SHA-3 finalists • BLAKE • Grøstl • JH • Skein • Keccak (winner) Other functions • BLAKE3 • CubeHash • ECOH • FSB • Fugue • GOST • HAS-160 • HAVAL • Kupyna • LSH • Lane • MASH-1 • MASH-2 • MD2 • MD4 • MD6 • MDC-2 • N-hash • RIPEMD • RadioGatún • SIMD • SM3 • SWIFFT • Shabal • Snefru • Streebog • Tiger • VSH • Whirlpool Password hashing/ key stretching functions • Argon2 • Balloon • bcrypt • Catena • crypt • LM hash • Lyra2 • Makwa • PBKDF2 • scrypt • yescrypt General purpose key derivation functions • HKDF • KDF1/KDF2 MAC functions • CBC-MAC • DAA • GMAC • HMAC • NMAC • OMAC/CMAC • PMAC • Poly1305 • SipHash • UMAC • VMAC Authenticated encryption modes • CCM • ChaCha20-Poly1305 • CWC • EAX • GCM • IAPM • OCB Attacks • Collision attack • Preimage attack • Birthday attack • Brute-force attack • Rainbow table • Side-channel attack • Length extension attack Design • Avalanche effect • Hash collision • Merkle–Damgård construction • Sponge function • HAIFA construction Standardization • CAESAR Competition • CRYPTREC • NESSIE • NIST hash function competition • Password Hashing Competition Utilization • Hash-based cryptography • Merkle tree • Message authentication • Proof of work • Salt • Pepper Cryptography General • History of cryptography • Outline of cryptography • Cryptographic protocol • Authentication protocol • Cryptographic primitive • Cryptanalysis • Cryptocurrency • Cryptosystem • Cryptographic nonce • Cryptovirology • Hash function • Cryptographic hash function • Key derivation function • Digital signature • Kleptography • Key (cryptography) • Key exchange • Key generator • Key schedule • Key stretching • Keygen • Cryptojacking malware • Ransomware • Random number generation • Cryptographically secure pseudorandom number generator (CSPRNG) • Pseudorandom noise (PRN) • Secure channel • Insecure channel • Subliminal channel • Encryption • Decryption • End-to-end encryption • Harvest now, decrypt later • Information-theoretic security • Plaintext • Codetext • Ciphertext • Shared secret • Trapdoor function • Trusted timestamping • Key-based routing • Onion routing • Garlic routing • Kademlia • Mix network Mathematics • Cryptographic hash function • Block cipher • Stream cipher • Symmetric-key algorithm • Authenticated encryption • Public-key cryptography • Quantum key distribution • Quantum cryptography • Post-quantum cryptography • Message authentication code • Random numbers • Steganography • Category
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
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.
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
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 52) 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 Authority control: National • France • BnF data • Germany • Israel • United States • Japan • Czech Republic
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 Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Czech Republic Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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. Authority control International • ISNI • VIAF National • Catalonia • Germany • Israel • United States Academics • DBLP • MathSciNet • Mathematics Genealogy Project • ORCID • Scopus • zbMATH Other • IdRef
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.
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.
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
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".
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
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.
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 Authority control International • VIAF Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
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). Authority control: Academics • MathSciNet • Scopus
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 Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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 Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Italy • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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 Authority control International • VIAF National • Germany Academics • DBLP • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • ResearcherID • Scopus
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 Authority control: Academics • Google Scholar • MathSciNet • Mathematics Genealogy Project
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 Authority control: Academics • ORCID • ResearcherID
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.
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. Authority control International • ISNI • VIAF National • Norway • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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 Authority control National • Germany Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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. 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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
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.
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.
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.
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
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
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.
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
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 Authority control International • ISNI • VIAF • 2 National • France • BnF data • Germany • Israel • United States Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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 Authority control International • ISNI • VIAF National • Germany • Israel • United States Academics • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID • ResearcherID • zbMATH Other • IdRef
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) Mathematics portal Authority control International • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Sweden • Croatia • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • ORCID • zbMATH Other • IdRef
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 Authority control International • ISNI • VIAF National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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 Fellows • Robert Ainsworth • Ross J. 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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.
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.
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
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
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
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.
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. Authority control International • VIAF National • Germany • United States • Netherlands • Poland
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". Authority control International • ISNI • VIAF National • France • BnF data • Catalonia • Israel • United States • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
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. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States Academics • CiNii • ORCID • Scopus Other • IdRef
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
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 • P-symmetry • T-symmetry • Lorentz symmetry • Poincaré symmetry • Gauge symmetry • Explicit symmetry breaking • Spontaneous symmetry breaking • Noether charge • Topological charge Tools • Anomaly • Background field method • BRST quantization • Correlation function • Crossing • Effective action • Effective field theory • Expectation value • Feynman diagram • Lattice field theory • LSZ reduction formula • Partition function • Propagator • Quantization • Regularization • Renormalization • Vacuum state • Wick's theorem Equations • Dirac equation • Klein–Gordon equation • Proca equations • Wheeler–DeWitt equation • Bargmann–Wigner equations Standard Model • Quantum electrodynamics • Electroweak interaction • Quantum chromodynamics • Higgs mechanism Incomplete theories • String theory • Supersymmetry • Technicolor • Theory of everything • Quantum gravity Scientists • Anderson • Anselm • Bargmann • Becchi • Belavin • Berezin • Bethe • Bjorken • Bleuer • Bogoliubov • Brodsky • Brout • Buchholz • Cachazo • Callan • Coleman • Dashen • DeWitt • Dirac • Doplicher • Dyson • Englert • Faddeev • Fadin • Fermi • Feynman • Fierz • Fock • Frampton • Fritzsch • Fröhlich • Fredenhagen • Furry • Glashow • Gelfand • Gell-Mann • Goldstone • Gribov • Gross • Gupta • Guralnik • Haag • Heisenberg • Hepp • Higgs • Hagen • 't Hooft • Ivanenko • Jackiw • Jona-Lasinio • Jordan • Jost • Källén • Kendall • Kinoshita • Klebanov • Kontsevich • Kuraev • Landau • Lee • Lehmann • Leutwyler • Lipatov • Łopuszański • Low • Lüders • Maiani • Majorana • Maldacena • Migdal • Mills • Møller • Naimark • Nambu • Neveu • Nishijima • Oehme • Oppenheimer • Osterwalder • Parisi • Pauli • Peskin • Polyakov • Pomeranchuk • Popov • Proca • Rubakov • Ruelle • Salam • Schrader • Schwarz • Schwinger • Segal • Seiberg • Semenoff • Shifman • Shirkov • Skyrme • Stora • Stueckelberg • Sudarshan • Symanzik • Thirring • Tomonaga • 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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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.
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). Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States Academics • zbMATH People • Deutsche Biographie Other • IdRef
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 Authority control International • FAST • ISNI • VIAF National • Germany • United States • Australia • Netherlands Academics • zbMATH People • Trove Other • IdRef
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 • Axiomatic QFT • Conformal field theory • Lattice field theory • Noncommutative QFT • Gauge theory • QFT in curved spacetime • String theory • Supergravity • Thermal QFT • Topological QFT • Two-dimensional conformal field theory Models Regular • Born–Infeld • Euler–Heisenberg • Ginzburg–Landau • Non-linear sigma • Proca • Quantum electrodynamics • Quantum chromodynamics • Quartic interaction • Scalar electrodynamics • Scalar chromodynamics • Soler • Yang–Mills • Yang–Mills–Higgs • Yukawa Low dimensional • 2D Yang–Mills • Bullough–Dodd • Gross–Neveu • Schwinger • Sine-Gordon • Thirring • Thirring–Wess • Toda Conformal • 2D free massless scalar • Liouville • Minimal • Polyakov • Wess–Zumino–Witten 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 • Chern–Simons Particle theory • Chiral • Fermi • MSSM • Nambu–Jona-Lasinio • NMSSM • Standard Model • Stueckelberg Related • Casimir effect • Cosmic string • History • Loop quantum gravity • Loop quantum cosmology • On shell and off shell • Quantum chaos • Quantum dynamics • Quantum foam • Quantum fluctuations • links • Quantum gravity • links • Quantum hadrodynamics • Quantum hydrodynamics • Quantum information • Quantum information science • links • Quantum logic • Quantum thermodynamics See also: Template:Quantum mechanics topics
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
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 Authority control International • VIAF National • Germany Academics • Google Scholar • MathSciNet • Mathematics Genealogy Project • zbMATH
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.
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
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.
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.
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
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.
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)
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.
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 References 1. Tian, Gang (1997). "Kähler–Einstein metrics with positive scalar curvature". Inventiones Mathematicae. 130 (1): 1–37. Bibcode:1997InMat.130....1T. doi:10.1007/s002220050176. MR 1471884. S2CID 122529381. 2. Donaldson, Simon K. (2002). 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II: Limits with cone angle less than 2π". Journal of the American Mathematical Society. 28: 199–234. arXiv:1212.4714. doi:10.1090/S0894-0347-2014-00800-6. S2CID 119140033. 27. Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2014). "Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof". Journal of the American Mathematical Society. 28: 235–278. arXiv:1302.0282. doi:10.1090/S0894-0347-2014-00801-8. S2CID 119575364. 28. Tian, Gang (2015). "K-Stability and Kähler-Einstein Metrics". Communications on Pure and Applied Mathematics. 68 (7): 1085–1156. arXiv:1211.4669. doi:10.1002/cpa.21578. S2CID 119303358. 29. Tian, Gang (2015). "Corrigendum: K-Stability and Kähler-Einstein Metrics". Communications on Pure and Applied Mathematics. 68 (11): 2082–2083. doi:10.1002/cpa.21612. S2CID 119666069. 30. "2019 Oswald Veblen Prize in Geometry to Xiuxiong Chen, Simon Donaldson, and Song Sun". American Mathematical Society. 2018-11-19. Retrieved 2019-04-09. 31. Simon Donaldson "For the new revolutionary invariants of four-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties." 32. Breakthrough Prize in Mathematics 2021 33. Székelyhidi, Gábor (2016). "The partial 𝐶⁰-estimate along the continuity method". Journal of the American Mathematical Society. 29 (2): 537–560. doi:10.1090/jams/833. 34. Datar, Ved; Székelyhidi, Gábor (2016). "Kähler–Einstein metrics along the smooth continuity method". Geometric and Functional Analysis. 26 (4): 975–1010. arXiv:1506.07495. doi:10.1007/s00039-016-0377-4. S2CID 253643887. 35. Chen, Xiuxiong; Sun, Song; Wang, Bing (2018). "Kähler–Ricci flow, Kähler–Einstein metric, and K–stability". Geometry & Topology. 22 (6): 3145–3173. doi:10.2140/gt.2018.22.3145. MR 3858762. S2CID 5667938. 36. Berman, Robert; Boucksom, Sébastien; Jonsson, Mattias (2021). "A variational approach to the Yau–Tian–Donaldson conjecture". Journal of the American Mathematical Society. 34 (3): 605–652. arXiv:1509.04561. doi:10.1090/jams/964. MR 4334189. S2CID 119323049. 37. Donaldson, Simon K. (2005). "Interior estimates for solutions of Abreu's equation". Collectanea Mathematica. 56 (2): 103–142. arXiv:math/0407486. Zbl 1085.53063. 38. Donaldson, S. K. (2008). "Extremal metrics on toric surfaces: A continuity method". Journal of Differential Geometry. 79 (3): 389–432. doi:10.4310/jdg/1213798183. 39. Donaldson, Simon K. (2009). "Constant Scalar Curvature Metrics on Toric Surfaces". Geometric and Functional Analysis. 19: 83–136. arXiv:0805.0128. doi:10.1007/s00039-009-0714-y. S2CID 17765416. 40. Stoppa, Jacopo (2009). "K-stability of constant scalar curvature Kähler manifolds". Advances in Mathematics. 221 (4): 1397–1408. doi:10.1016/j.aim.2009.02.013. S2CID 6554854. 41. Arezzo, Claudio; Pacard, Frank (2006). "Blowing up and desingularizing constant scalar curvature Kähler manifolds". Acta Mathematica. 196 (2): 179–228. doi:10.1007/s11511-006-0004-6. S2CID 14605574. 42. Deligne, P.; Mumford, D. (1969). "The irreducibility of the space of curves of given genus". Publications Mathématiques de l'IHÉS. 36: 75–109. doi:10.1007/BF02684599. 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."
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
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
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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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
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 • Birkhoff, Garrett (1935). "On the structure of abstract algebras". Proc. Camb. Phil. Soc. 31 (4): 433–454. Bibcode:1935PCPS...31..433B. doi:10.1017/s0305004100013463. ISSN 0008-1981. S2CID 121173630. • Boole, George (2003) [1854]. An Investigation of the Laws of Thought. Prometheus Books. ISBN 978-1-59102-089-9. • Dwinger, Philip (1971). Introduction to Boolean algebras. Würzburg: Physica Verlag. • Gaifman, Haim (1964). "Infinite Boolean Polynomials, I". 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ISSN 0002-9947. JSTOR 1989664. • Tarski, Alfred (1983). Logic, Semantics, Metamathematics, Corcoran, J., ed. Hackett. 1956 1st edition edited and translated by J. H. Woodger, Oxford Uni. Press. Includes English translations of the following two articles: • Tarski, Alfred (1929). "Sur les classes closes par rapport à certaines opérations élémentaires". Fundamenta Mathematicae. 16: 195–97. ISSN 0016-2736. • Tarski, Alfred (1935). "Zur Grundlegung der Booleschen Algebra, I". Fundamenta Mathematicae. 24: 177–98. doi:10.4064/fm-24-1-177-198. ISSN 0016-2736. • Vladimirov, D.A. (1969). булевы алгебры (Boolean algebras, in Russian, German translation Boolesche Algebren 1974). Nauka (German translation Akademie-Verlag). References 1. "The Mathematics of Boolean Algebra". The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University. 2022. 2. "Chapter 1 Boolean algebras". Hausdorff Gaps and Limits. Studies in Logic and the Foundations of Mathematics. Vol. 132. 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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.
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 Authority control International • ISNI • VIAF National • Israel • United States Academics • zbMATH Other • IdRef
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.

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