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https://en.wikipedia.org/wiki/Arithmetic%20for%20Parents | Arithmetic for Parents (Sumizdat, 2007, ) is a book about mathematics education aimed at parents and teachers.
The author, Ron Aharoni, is a professor of mathematics at the Technion; he wrote the book based on his experiences teaching elementary mathematics to Israeli schoolchildren.
The book was originally written in Hebrew and was translated to English, Portuguese and Dutch.
References
Elementary arithmetic
Israeli non-fiction books
Books about mathematics education |
https://en.wikipedia.org/wiki/Minimum-weight%20triangulation | In computational geometry and computer science, the minimum-weight triangulation problem is the problem of finding a triangulation of minimal total edge length. That is, an input polygon or the convex hull of an input point set must be subdivided into triangles that meet edge-to-edge and vertex-to-vertex, in such a way as to minimize the sum of the perimeters of the triangles. The problem is NP-hard for point set inputs, but may be approximated to any desired degree of accuracy. For polygon inputs, it may be solved exactly in polynomial time. The minimum weight triangulation has also sometimes been called the optimal triangulation.
History
The problem of minimum weight triangulation of a point set was posed by , who suggested its application to the construction of triangulated irregular network models of land countours, and used a greedy heuristic to approximate it. conjectured that the minimum weight triangulation always coincided with the Delaunay triangulation, but this was quickly disproved by , and indeed showed that the weights of the two triangulations can differ by a linear factor.
The minimum-weight triangulation problem became notorious when included it in a list of open problems in their book on NP-completeness, and many subsequent authors published partial results on it. Finally, showed it to be NP-hard, and showed that accurate approximations to it can be constructed efficiently.
Complexity
The weight of a triangulation of a set of points in the Euclidean plane is defined as the sum of lengths of its edges. Its decision variant is the problem of deciding whether there exists a triangulation of weight less than a given weight; it was proven to be NP-hard by . Their proof is by reduction from PLANAR-1-IN-3-SAT, a special case of the Boolean satisfiability problem in which a 3-CNF whose graph is planar is accepted when it has a truth assignment that satisfies exactly one literal in each clause. The proof uses complex gadgets, and involves computer assistance to verify the correct behavior of these gadgets.
It is not known whether the minimum-weight triangulation decision problem is NP-complete, since this depends on the known open problem whether the sum of radicals may be computed in polynomial time. However, Mulzer and Rote remark that the problem is NP-complete if the edge weights are rounded to integer values.
Although NP-hard, the minimum weight triangulation may be constructed in subexponential time by a dynamic programming algorithm that considers all possible simple cycle separators of points within the triangulation, recursively finds the optimal triangulation on each side of the cycle, and chooses the cycle separator leading to the smallest total weight. The total time for this method is .
Approximation
Several authors have proven results relating the minimum weight triangulation to other triangulations in terms of the approximation ratio, the worst-case ratio of the total edge length of the alternative triangula |
https://en.wikipedia.org/wiki/Quasi-homogeneous%20polynomial | In algebra, a multivariate polynomial
is quasi-homogeneous or weighted homogeneous, if there exist r integers , called weights of the variables, such that the sum is the same for all nonzero terms of . This sum is the weight or the degree of the polynomial.
The term quasi-homogeneous comes from the fact that a polynomial is quasi-homogeneous if and only if
for every in any field containing the coefficients.
A polynomial is quasi-homogeneous with weights if and only if
is a homogeneous polynomial in the . In particular, a homogeneous polynomial is always quasi-homogeneous, with all weights equal to 1.
A polynomial is quasi-homogeneous if and only if all the belong to the same affine hyperplane. As the Newton polytope of the polynomial is the convex hull of the set the quasi-homogeneous polynomials may also be defined as the polynomials that have a degenerate Newton polytope (here "degenerate" means "contained in some affine hyperplane").
Introduction
Consider the polynomial , which is not homogeneous. However, if instead of considering we use the pair to test homogeneity, then
We say that is a quasi-homogeneous polynomial of type
, because its three pairs of exponents , and all satisfy the linear equation . In particular, this says that the Newton polytope of lies in the affine space with equation inside .
The above equation is equivalent to this new one: . Some authors prefer to use this last condition and prefer to say that our polynomial is quasi-homogeneous of type .
As noted above, a homogeneous polynomial of degree is just a quasi-homogeneous polynomial of type ; in this case all its pairs of exponents will satisfy the equation .
Definition
Let be a polynomial in variables with coefficients in a commutative ring . We express it as a finite sum
We say that is quasi-homogeneous of type , , if there exists some such that
whenever .
References
Commutative algebra
Algebraic geometry |
https://en.wikipedia.org/wiki/Tadeusz%20Iwaniec | Tadeusz Iwaniec (born October 9, 1947 in Elbląg) is a Polish-American mathematician, and since 1996 John Raymond French Distinguished Professor of Mathematics at Syracuse University.
He and mathematician Henryk Iwaniec are twin brothers.
Awards and honors
Iwaniec was given the Prize of the President of the Polish Academy of Sciences, 1980, the Alfred Jurzykowski Award in Mathematics in 1997, the Prix 2001 Institut Henri-Poincaré Gauthier-Villars, and the 2009 Sierpinski Medal of the Polish Mathematical Society and Warsaw University. In 1998 he was elected as a foreign member of the Academia di Scienze Fisiche e Matematiche, Italy and in 2012 as a foreign member of the Finnish Academy of Science and Letters.
References
Polish mathematicians
21st-century American mathematicians
Polish emigrants to the United States
People from Elbląg
Polish twins
1947 births
Living people
Syracuse University faculty
Mathematicians from New York (state)
20th-century American mathematicians |
https://en.wikipedia.org/wiki/Ascending%20chain%20condition%20on%20principal%20ideals | In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals (abbreviated to ACCP) is satisfied if there is no infinite strictly ascending chain of principal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.
The counterpart descending chain condition may also be applied to these posets, however there is currently no need for the terminology "DCCP" since such rings are already called left or right perfect rings. (See below.)
Noetherian rings (e.g. principal ideal domains) are typical examples, but some important non-Noetherian rings also satisfy (ACCP), notably unique factorization domains and left or right perfect rings.
Commutative rings
It is well known that a nonzero nonunit in a Noetherian integral domain factors into irreducibles. The proof of this relies on only (ACCP) not (ACC), so in any integral domain with (ACCP), an irreducible factorization exists. (In other words, any integral domains with (ACCP) are atomic. But the converse is false, as shown in .) Such a factorization may not be unique; the usual way to establish uniqueness of factorizations uses Euclid's lemma, which requires factors to be prime rather than just irreducible. Indeed, one has the following characterization: let A be an integral domain. Then the following are equivalent.
A is a UFD.
A satisfies (ACCP) and every irreducible of A is prime.
A is a GCD domain satisfying (ACCP).
The so-called Nagata criterion holds for an integral domain A satisfying (ACCP): Let S be a multiplicatively closed subset of A generated by prime elements. If the localization S−1A is a UFD, so is A. (Note that the converse of this is trivial.)
An integral domain A satisfies (ACCP) if and only if the polynomial ring A[t] does. The analogous fact is false if A is not an integral domain.
An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) if and only if it is a principal ideal domain.
The ring Z+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of principal ideals
is non-terminating.
Noncommutative rings
In the noncommutative case, it becomes necessary to distinguish the right ACCP from left ACCP. The former only requires the poset of ideals of the form xR to satisfy the ascending chain condition, and the latter only examines the poset of ideals of the form Rx.
A theorem of Hyman Bass in now known as "Bass' Theorem P" showed that the descending chain condition on principal left ideals of a ring R is equivalent to R being a right perfect ring. D. Jonah showed in that there is a side-switching connection between the ACCP and perfect rings. It was shown th |
https://en.wikipedia.org/wiki/Verena%20Huber-Dyson | Verena Esther Huber-Dyson (May 6, 1923 – March 12, 2016) was a Swiss-American mathematician, known for her work on group theory and formal logic. She has been described as a "brilliant mathematician", who did research on the interface between algebra and logic, focusing on undecidability in group theory. At the time of her death, she was emeritus faculty in the philosophy department of the University of Calgary, Alberta.
Life and career
Family and early life
Huber-Dyson was born Verena Esther Huber in Naples, Italy, on May 6, 1923. Her parents, Karl (Charles) Huber (1893–1946) and Berthy Ryffel (1899–1945), were Swiss nationals who raised Verena and her sister Adelheid ("Heidi", 1925–1987) in Athens, Greece, where the girls attended the German-speaking Deutsche Schule, or German School of Athens, until forced to return to Switzerland in 1940 by the war.
Charles Huber, who had managed the Middle Eastern operations of Bühler AG, a Swiss food-process engineering firm, began working for the International Committee of the Red Cross (ICRC), monitoring the treatment of prisoners of war in internment camps. As the ICRC delegate to India and Ceylon, he was responsible for Italian prisoners held in British camps, but also visited German and Allied camps in Europe. In 1945-46 he served as an ICRC delegate to the United States, which he described to Verena as a place she "definitely ought to experience at length and in depth but just as definitely ought not to settle in."
She studied mathematics, with minors in physics and philosophy, at the University of Zurich, where she obtained her Ph.D. in mathematics in 1947 with a thesis in finite group theory under the supervision of Andreas Speiser.
Children
Verena married Hans-Georg Haefeli, a fellow mathematician, in 1942, and was divorced in 1948. Her first daughter, Katarina Halm (née Halm), was born in 1945.
She subsequently married Freeman Dyson in Ann Arbor, Michigan, on August 11, 1950. They had two children together, Esther Dyson (born July 14, 1951, in Zurich) and George Dyson (born 1953, Ithaca, New York), and divorced in 1958.
Career
Huber-Dyson accepted a postdoctoral fellow appointment at the Institute for Advanced Study in Princeton in 1948, where she worked on group theory and formal logic. She also began teaching at Goucher College near Baltimore during this time.
She moved to California with her daughter Katarina, began teaching at San Jose State University in 1959, and then joined Alfred Tarski's Group in Logic and the Methodology of Science at the University of California, Berkeley.
Huber-Dyson taught at San Jose State University, the University of Zürich, Monash University, as well as at University of California, Berkeley, Adelphi University, University of California, Los Angeles, and the University of Illinois at Chicago, in mathematics and in philosophy departments. She accepted a position in the philosophy department of the University of Calgary in 1973, becoming emerita in 1988 |
https://en.wikipedia.org/wiki/K-frame | In linear algebra, a branch of mathematics, a k-frame is an ordered set of k linearly independent vectors in a vector space; thus k ≤ n, where n is the dimension of the space, and if k = n an n-frame is precisely an ordered basis.
If the vectors are orthogonal, or orthonormal, the frame is called an orthogonal frame, or orthonormal frame, respectively.
Properties
The set of k-frames (particularly the set of orthonormal k-frames) in a given space X is known as the Stiefel manifold, and denoted Vk(X).
A k-frame defines a parallelotope (a generalized parallelepiped); the volume can be computed via the Gram determinant.
See also
Frame (linear algebra)
Frame of a vector space
Riemannian geometry
Orthonormal frame
Moving frame
Linear algebra |
https://en.wikipedia.org/wiki/Irreducible%20ideal | In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.
Examples
Every prime ideal is irreducible. Let and be ideals of a commutative ring , with neither one contained in the other. Then there exist and , where neither is in but the product is. This proves that a reducible ideal is not prime. A concrete example of this are the ideals and contained in . The intersection is , and is not a prime ideal.
Every irreducible ideal of a Noetherian ring is a primary ideal, and consequently for Noetherian rings an irreducible decomposition is a primary decomposition.
Every primary ideal of a principal ideal domain is an irreducible ideal.
Every irreducible ideal is primal.
Properties
An element of an integral domain is prime if and only if the ideal generated by it is a non-zero prime ideal. This is not true for irreducible ideals; an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in for the ideal since it is not the intersection of two strictly greater ideals.
An ideal I of a ring R can be irreducible only if the algebraic set it defines is irreducible (that is, any open subset is dense) for the Zariski topology, or equivalently if the closed space of spec R consisting of prime ideals containing I is irreducible for the spectral topology. The converse does not hold; for example the ideal of polynomials in two variables with vanishing terms of first and second order is not irreducible.
If k is an algebraically closed field, choosing the radical of an irreducible ideal of a polynomial ring over k is exactly the same as choosing an embedding of the affine variety of its Nullstelle in the affine space.
See also
Irreducible module
Irreducible space
Laskerian ring
References
Ring theory
Algebraic topology |
https://en.wikipedia.org/wiki/Symmetric%20set | In mathematics, a nonempty subset of a group is said to be symmetric if it contains the inverses of all of its elements.
Definition
In set notation a subset of a group is called if whenever then the inverse of also belongs to
So if is written multiplicatively then is symmetric if and only if where
If is written additively then is symmetric if and only if where
If is a subset of a vector space then is said to be a if it is symmetric with respect to the additive group structure of the vector space; that is, if which happens if and only if
The of a subset is the smallest symmetric set containing and it is equal to The largest symmetric set contained in is
Sufficient conditions
Arbitrary unions and intersections of symmetric sets are symmetric.
Any vector subspace in a vector space is a symmetric set.
Examples
In examples of symmetric sets are intervals of the type with and the sets and
If is any subset of a group, then and are symmetric sets.
Any balanced subset of a real or complex vector space is symmetric.
See also
References
R. Cristescu, Topological vector spaces, Noordhoff International Publishing, 1977.
Group theory |
https://en.wikipedia.org/wiki/Duane%20H.%20Cooper | Duane H. Cooper (August 21, 1923 in Gibson City, Illinois – April 4, 1995) was a physicist, who made early investigations regarding
the intricate geometry of the phonograph stylus-groove interface.
He earned a Bachelor of Science and Ph.D. degree with honors in physics in 1950 and 1955 from the California Institute of Technology.
In 1954, Dr Cooper joined the Coordinated Sciences Laboratory at the University of Illinois, where
he became a research professor. He developed a unified treatment of phonograph tracking and tracing distortion by utilizing a skew transformation. In the late 1960s and early 70s, Dr Cooper contributed significantly to the theory of surround sound
multi-channel stereo.
Dr Cooper served as president of the Audio Engineering Society (AES) from 1975-1976. He was named an AES Fellow in 1966, and received the AES Silver and Gold Medals.
References
Obituary AES Journal July 1995 Archived
1923 births
1995 deaths
20th-century American physicists
California Institute of Technology alumni
University of Illinois faculty
People from Gibson City, Illinois |
https://en.wikipedia.org/wiki/Selim%20Sadak | Selim Sadak, (born 1954 in İdil, Şırnak) is a Turkish politician of Kurdish origin.
Background
Selim Sadak graduated from the Mathematics department of Diyarbakır Eğitim Enstitüsü. He then worked as a freelancer in Kurdish, English and Arabic.
He is married and has 10 children.
Political career
In the 1991 Turkey Parliamentary general election, he joined Leyla Zana, Mahmut Alınak, Hatip Dicle, Orhan Doğan, Ahmet Türk, Sırrı Sakık and Sedat Yurtdaş in the Social Democratic Populist Party (SHP) and was elected as the member of parliament for Şırnak in the 19th Parliament of Turkey. In March 1994, the parliamentary immunity was lifted from Selim Sadak, Orhan Dogan, Hatip Dicle and Leyla Zana. On 16 June 1994 the Democracy Party (DEP) was closed down by the Turkish Constitutional Court, and Selim Sadak, along with other members of the party, were put in prison. Based on a decision by the State Security Court he was sentenced to 15 years in prison in 1994. In April 2004 the European parliament condemned the imprisonment of Sadak and hoped for the quashing of the sentence in a resolution. He was released in June 2004 following a decision of Turkey's Appeal Court.
Later career and prosecution
Following his release from prison, Sadak toured the countryside testing the possibilities of founding a new party. At the time he saw it difficult a Kurdish party could be formed without referencing Öcalan. Sadak along with Leyla Zana and Hatip Dicle set up the Democratic Society Party (DTP) and was elected mayor of Siirt under the DTP, receiving 49,4% of the votes in the local elections in March 2009.
In December 2009, however, Turkey banned the DTP due to alleged links with the PKK and Selim Sadak as well as Ahmet Türk, Aysel Tuğluk, Leyla Zana and Nurettin Demirtaş, were banned from politics for 5 years. In 2010 he was dismissed by the Interior Ministry as Mayor of Siirt after a sentence of 10 months imprisonment from 2008 got confirmed, a decision he objected. Then the Council of State Administrative Trials Board General Council overruled the decision of the Interior Ministry to dismiss Sadak and ruled he can stay Mayor of Siirt until the end of his term.
On 26 April 2010 he was sentenced of 1 year imprisonment because he used the word Kurdistan and on the 13 April to 10 months imprisonment for a photo depicted in a calendar of 2010.
In August 2011 he was sentenced in Siirt Criminal Court to five months in prison, a sentence which was later converted to a fine.
On 7 October 2011 he was sentenced in Diyarbakir court to one year and eight months prison for terrorist propaganda in relation to a speech that he made in 2007.
References
External links
Selim Sadak web site
Living people
1954 births
Democracy Party (Turkey) politicians
Democratic Society Party politicians
Deputies of Şırnak
Turkish prisoners and detainees
Turkish Kurdish politicians
Amnesty International prisoners of conscience held by Turkey
People expelled from public office
People's La |
https://en.wikipedia.org/wiki/Plactic%20monoid | In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It was discovered by (who called it the tableau algebra), using an operation given by in his study of the longest increasing subsequence of a permutation.
It was named the "monoïde plaxique" by , who allowed any totally ordered alphabet in the definition. The etymology of the word "plaxique" is unclear; it may refer to plate tectonics ("tectonique des plaques" in French), as elementary relations that generate the equivalence allow conditional commutation of generator symbols: they can sometimes slide across each other (in apparent analogy to tectonic plates), but not freely.
Definition
The plactic monoid over some totally ordered alphabet (often the positive integers) is the monoid with the following presentation:
The generators are the letters of the alphabet
The relations are the elementary Knuth transformations yzx ≡ yxz whenever x < y ≤ z and xzy ≡ zxy whenever x ≤ y < z.
Knuth equivalence
Two words are called Knuth equivalent if they represent the same element of the plactic monoid, or in other words if one can be obtained from the other by a sequence of elementary Knuth transformations.
Several properties are preserved by Knuth equivalence.
If a word is a reverse lattice word, then so is any word Knuth equivalent to it.
If two words are Knuth equivalent, then so are the words obtained by removing their rightmost maximal elements, as are the words obtained by removing their leftmost minimal elements.
Knuth equivalence preserves the length of the longest nondecreasing subsequence, and more generally preserves the maximum of the sum of the lengths of k disjoint non-decreasing subsequences for any fixed k.
Correspondence with semistandard Young tableaux
Every word is Knuth equivalent to the word of a unique semistandard Young tableau (this means that each row is non-decreasing and each column is strictly increasing) over the same ordered alphabet, where the tableau may be read by rows or by columns. So the elements of the plactic monoid can be identified with the semistandard Young tableaux, which therefore also form a monoid.
Multiplying the word of a semistandard Young tableau to the left with a generator is equivalent to Schensted insertion into the Young tableau. In row order, the word of the tableau is equivalent to a product of increasingly longer nondecreasing sequences of generators. The new generator may be inserted in its proper place by either appending it if it is larger, and otherwise by repeatedly applying the plactic relations to move the out of sequence element to the next row. In the latter case, the out of order element replaces the leftmost entry larger than it in each row, and the displaced element is then inserted in the next row.
Since Schensted insertion preserves Young tableaux, this gives an inductive proof that |
https://en.wikipedia.org/wiki/Pieri%27s%20formula | In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function.
In terms of Schur functions sλ indexed by partitions λ, it states that
where hr is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by adding r elements, no two in the same column.
By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule
for multiplying an elementary symmetric polynomial with a Schur polynomial:
The sum is now taken over all partitions λ obtained from μ by adding r elements, no two in the same row.
Pieri's formula implies Giambelli's formula. The Littlewood–Richardson rule is a generalization of Pieri's formula
giving the product of any two Schur functions. Monk's formula is an analogue of Pieri's formula for flag manifolds.
References
Symmetric functions |
https://en.wikipedia.org/wiki/Giambelli%27s%20formula | In mathematics, Giambelli's formula, named after Giovanni Giambelli, expresses Schubert classes as determinants in terms of special Schubert classes.
It states
where σλ is the Schubert class of a partition λ.
Giambelli's formula may be derived as a consequence of Pieri's formula. The Porteous formula is a generalization to morphisms of vector bundles over a variety.
In the theory of symmetric functions, the same identity, known as the first Jacobi-Trudi identity expresses Schur functions as determinants in terms of complete symmetric functions. There is also the dual second Jacobi-Trudi identity which expresses Schur functions as determinants in terms of elementary symmetric functions. The corresponding identity also holds for Schubert classes.
There is another Giambelli identity, expressing Schur functions as determinants of matrices whose entries are Schur functions corresponding to hook partitions contained within the same Young diagram. This too is valid for Schubert classes, as are all Schur function identities. For instance, hook partition Schur functions can be expressed bilinearly in terms of elementary and complete symmetric functions, and Schubert classes satisfy these same relations.
See also
Schubert calculus - includes examples
References
Symmetric functions |
https://en.wikipedia.org/wiki/Fibonomial%20coefficient | In mathematics, the Fibonomial coefficients or Fibonacci-binomial coefficients are defined as
where n and k are non-negative integers, 0 ≤ k ≤ n, Fj is the j-th Fibonacci number and n!F is the nth Fibonorial, i.e.
where 0!F, being the empty product, evaluates to 1.
Special values
The Fibonomial coefficients are all integers. Some special values are:
Fibonomial triangle
The Fibonomial coefficients are similar to binomial coefficients and can be displayed in a triangle similar to Pascal's triangle. The first eight rows are shown below.
The recurrence relation
implies that the Fibonomial coefficients are always integers.
The fibonomial coefficients can be expressed in terms of the Gaussian binomial coefficients and the golden ratio :
Applications
Dov Jarden proved that the Fibonomials appear as coefficients of an equation involving powers of consecutive Fibonacci numbers, namely Jarden proved that given any generalized Fibonacci sequence , that is, a sequence that satisfies for every then
for every integer , and every nonnegative integer .
References
Ewa Krot, An introduction to finite fibonomial calculus, Institute of Computer Science, Bia lystok University, Poland.
Dov Jarden, Recurring Sequences (second edition 1966), pages 30–33.
Fibonacci numbers
Factorial and binomial topics
Triangles of numbers |
https://en.wikipedia.org/wiki/Nicholas%20Polson | Nicholas Polson (born 7 May 1963) is a British statistician who is a professor of econometrics and statistics at the University of Chicago Booth School of Business. His works are primarily in Bayesian statistics, Markov chain Monte Carlo and Sequential Monte Carlo, (aka Particle filter). Polson was educated at Worcester College, Oxford University and the University of Nottingham where his PhD supervisor was Adrian Smith.
Polson is the co-author (with James Scott) of the book AIQ: How People and Machines Are Smarter Together (2018), about the key ideas that played a role in the historical development of artificial intelligence.
Selected publications
Eraker, B., M. Johannes and N.G. Polson, "The Impact of Jumps in Volatility in Returns," (2003) Journal of Finance, 58, 3, 1269–1300.
Carlin, B.P., N.G. Polson and D.S. Stoffer, "A Monte Carlo Approach to Non-Normal and Non-Linear State Space Modelling" (1992) Journal of the American Statistical Association, 87, 493–500.
References
Nicholas Polson at University of Chicago Booth School of Business
Nicholas Polson's Research page
Article : Do leveraged funds deliver long term?
Book chapter : Handbook of Financial Time Series
British statisticians
20th-century British mathematicians
21st-century British mathematicians
Fellows of the American Statistical Association
Living people
1963 births
Alumni of the University of Nottingham |
https://en.wikipedia.org/wiki/Diffiety | In mathematics, a diffiety () is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.
Intuitive definition
In algebraic geometry the main objects of study (varieties) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of polynomials), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraic ideal generated by the initial set of polynomials.
When dealing with differential equations, apart from applying algebraic operations as above, one has also the option to differentiate the starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal, one needs therefore to work with a differential ideal.
An elementary diffiety will consist therefore of the infinite prolongation of a differential equation , together with an extra structure provided by a special distribution. Elementary diffieties play the same role in the theory of differential equations as affine algebraic varieties do in the theory of algebraic equations. Accordingly, just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one defines a (non-elementary) diffiety as an object that locally looks like an elementary diffiety.
Formal definition
The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.
Jet spaces of submanifolds
For instance, for one recovers just points in and for one recovers the Grassmannian of -dimensional subspaces of . More generally, all the projections are fibre bundles.
As a particular case, when has a structure of fibred manifold over an -dimensional manifold , one can consider submanifolds of given by the graphs of local sections of . Then the notion of jet of submanifolds boils down to the standard notion of jet of sections, and the jet bundle turns out to be an open and dense subset of .
Prolongations of submanifolds
The -jet prolongation of a submanifold is
The map is a smooth embedding and its image , called the prolongation of the submanifold , is a submanifold of diffeomorphic to .
Cartan distribution on jet spaces
A space of the fo |
https://en.wikipedia.org/wiki/Richard%20Laver | Richard Joseph Laver (October 20, 1942 – September 19, 2012) was an American mathematician, working in set theory.
Biography
Laver received his PhD at the University of California, Berkeley in 1969, under the supervision of Ralph McKenzie, with a thesis on Order Types and Well-Quasi-Orderings. The largest part of his career he spent as Professor and later Emeritus Professor at the University of Colorado at Boulder.
Richard Laver died in Boulder, CO, on September 19, 2012 after a long illness.
Research contributions
Among Laver's notable achievements some are the following.
Using the theory of better-quasi-orders, introduced by Nash-Williams, (an extension of the notion of well-quasi-ordering), he proved Fraïssé's conjecture (now Laver's theorem): if (A0,≤),(A1,≤),...,(Ai,≤), are countable ordered sets, then for some i<j (Ai,≤) isomorphically embeds into (Aj,≤). This also holds if the ordered sets are countable unions of scattered ordered sets.
He proved the consistency of the Borel conjecture, i.e., the statement that every strong measure zero set is countable. This important independence result was the first when a forcing (see Laver forcing), adding a real, was iterated with countable support iteration. This method was later used by Shelah to introduce proper and semiproper forcing.
He proved the existence of a Laver function for supercompact cardinals. With the help of this, he proved the following result. If κ is supercompact, there is a κ-c.c. forcing notion (P, ≤) such that after forcing with (P, ≤) the following holds: κ is supercompact and remains supercompact in any forcing extension via a κ-directed closed forcing. This statement, known as the indestructibility result, is used, for example, in the proof of the consistency of the proper forcing axiom and variants.
Laver and Shelah proved that it is consistent that the continuum hypothesis holds and there are no ℵ2-Suslin trees.
Laver proved that the perfect subtree version of the Halpern–Läuchli theorem holds for the product of infinitely many trees. This solved a longstanding open question.
Laver started investigating the algebra that j generates where j:Vλ→Vλ is some elementary embedding. This algebra is the free left-distributive algebra on one generator. For this he introduced Laver tables.
He also showed that if V[G] is a (set-)forcing extension of V, then V is a class in V[G].
Notes and references
External links
Set theorists
20th-century American mathematicians
21st-century American mathematicians
University of Colorado faculty
1942 births
2012 deaths |
https://en.wikipedia.org/wiki/Oriel%20school | Oriel school may refer to:
Oriel High School, Crawley, England
Ormiston Venture Academy, Great Yarmouth, England, formerly called Oriel Grammar School, Oriel High School and Oriel Specialist Maths and Computing College
See also
Oriel (disambiguation) |
https://en.wikipedia.org/wiki/Ernie%20Moser | Ernie Moser (born April 30, 1949) is a Canadian former professional ice hockey right winger who was drafted 9th overall in the 1969 NHL Amateur Draft by the Toronto Maple Leafs.
Career statistics
External links
1949 births
Canadian ice hockey right wingers
Flint Generals (IHL) players
Ice hockey people from Saskatchewan
Living people
Muskegon Mohawks players
National Hockey League first-round draft picks
Springfield Indians players
Toronto Maple Leafs draft picks
Tulsa Oilers (1964–1984) players
People from Rural Municipality Happyland No. 231, Saskatchewan
Canadian expatriate ice hockey players in the United States |
https://en.wikipedia.org/wiki/Hitchin%E2%80%93Thorpe%20inequality | In differential geometry the Hitchin–Thorpe inequality is a relation which restricts the topology of 4-manifolds that carry an Einstein metric.
Statement of the Hitchin–Thorpe inequality
Let M be a closed, oriented, four-dimensional smooth manifold. If there exists a Riemannian metric on M which is an Einstein metric, then
where is the Euler characteristic of and is the signature of .
This inequality was first stated by John Thorpe in a footnote to a 1969 paper focusing on manifolds of higher dimension. Nigel Hitchin then rediscovered the inequality, and gave a complete characterization of the equality case in 1974; he found that if is an Einstein manifold for which equality in the Hitchin-Thorpe inequality is obtained, then the Ricci curvature of is zero; if the sectional curvature is not identically equal to zero, then is a Calabi–Yau manifold whose universal cover is a K3 surface.
Already in 1961, Marcel Berger showed that the Euler characteristic is always non-negative.
Proof
Let be a four-dimensional smooth Riemannian manifold which is Einstein. Given any point of , there exists a -orthonormal basis of the tangent space such that the curvature operator , which is a symmetric linear map of into itself, has matrix
relative to the basis . One has that is zero and that is one-fourth of the scalar curvature of at . Furthermore, under the conditions and , each of these six functions is uniquely determined and defines a continuous real-valued function on .
According to Chern-Weil theory, if is oriented then the Euler characteristic and signature of can be computed by
Equipped with these tools, the Hitchin-Thorpe inequality amounts to the elementary observation
Failure of the converse
A natural question to ask is whether the Hitchin–Thorpe inequality provides a sufficient condition for the existence of Einstein metrics. In 1995, Claude LeBrun and
Andrea Sambusetti independently showed that the answer is no: there exist infinitely many non-homeomorphic compact, smooth, oriented 4-manifolds that carry no Einstein metrics but nevertheless satisfy
LeBrun's examples are actually simply connected, and the relevant obstruction depends on the smooth structure of the manifold. By contrast, Sambusetti's obstruction only applies to 4-manifolds with infinite fundamental group, but the volume-entropy estimate he uses to prove non-existence only depends on the homotopy type of the manifold.
Footnotes
References
Einstein manifolds
Geometric inequalities
Einstein manifold |
https://en.wikipedia.org/wiki/Schreier%20domain | In abstract algebra, a Schreier domain, named after Otto Schreier, is an integrally closed domain where every nonzero element is primal; i.e., whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. An integral domain is said to be pre-Schreier if every nonzero element is primal. A GCD domain is an example of a Schreier domain. The term "Schreier domain" was introduced by P. M. Cohn in 1960s. The term "pre-Schreier domain" is due to Muhammad Zafrullah.
In general, an irreducible element is primal if and only if it is a prime element. Consequently, in a Schreier domain, every irreducible is prime. In particular, an atomic Schreier domain is a unique factorization domain; this generalizes the fact that an atomic GCD domain is a UFD.
References
Cohn, P.M., Bezout rings and their subrings, 1968.
Zafrullah, Muhammad, On a property of pre-Schreier domains, 1987.
Ring theory |
https://en.wikipedia.org/wiki/Arithmetical%20ring | In algebra, a commutative ring R is said to be arithmetical (or arithmetic) if any of the following equivalent conditions hold:
The localization of R at is a uniserial ring for every maximal ideal of R.
For all ideals , and ,
For all ideals , and ,
The last two conditions both say that the lattice of all ideals of R is distributive.
An arithmetical domain is the same thing as a Prüfer domain.
References
External links
Ring theory |
https://en.wikipedia.org/wiki/Big%20O%20in%20probability%20notation | The order in probability notation is used in probability theory and statistical theory in direct parallel to the big-O notation that is standard in mathematics. Where the big-O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in probability.
Definitions
Small o: convergence in probability
For a set of random variables Xn and a corresponding set of constants an (both indexed by n, which need not be discrete), the notation
means that the set of values Xn/an converges to zero in probability as n approaches an appropriate limit.
Equivalently, Xn = op(an) can be written as Xn/an = op(1),
i.e.
for every positive ε.
Big O: stochastic boundedness
The notation
means that the set of values Xn/an is stochastically bounded. That is, for any ε > 0, there exists a finite M > 0 and a finite N > 0 such that
Comparison of the two definitions
The difference between the definitions is subtle. If one uses the definition of the limit, one gets:
Big :
Small :
The difference lies in the : for stochastic boundedness, it suffices that there exists one (arbitrary large) to satisfy the inequality, and is allowed to be dependent on (hence the ). On the other hand, for convergence, the statement has to hold not only for one, but for any (arbitrary small) . In a sense, this means that the sequence must be bounded, with a bound that gets smaller as the sample size increases.
This suggests that if a sequence is , then it is , i.e. convergence in probability implies stochastic boundedness. But the reverse does not hold.
Example
If is a stochastic sequence such that each element has finite variance, then
(see Theorem 14.4-1 in Bishop et al.)
If, moreover, is a null sequence for a sequence of real numbers, then converges to zero in probability by Chebyshev's inequality, so
References
Mathematical notation
Probability theory
Statistical theory
Convergence (mathematics) |
https://en.wikipedia.org/wiki/Real-valued%20function | In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions.
Algebraic structure
Let be the set of all functions from a set to real numbers . Because is a field, may be turned into a vector space and a commutative algebra over the reals with the following operations:
– vector addition
– additive identity
– scalar multiplication
– pointwise multiplication
These operations extend to partial functions from to with the restriction that the partial functions and are defined only if the domains of and have a nonempty intersection; in this case, their domain is the intersection of the domains of and .
Also, since is an ordered set, there is a partial order
on which makes a partially ordered ring.
Measurable
The σ-algebra of Borel sets is an important structure on real numbers. If has its σ-algebra and a function is such that the preimage of any Borel set belongs to that σ-algebra, then is said to be measurable. Measurable functions also form a vector space and an algebra as explained above in .
Moreover, a set (family) of real-valued functions on can actually define a σ-algebra on generated by all preimages of all Borel sets (or of intervals only, it is not important). This is the way how σ-algebras arise in (Kolmogorov's) probability theory, where real-valued functions on the sample space are real-valued random variables.
Continuous
Real numbers form a topological space and a complete metric space. Continuous real-valued functions (which implies that is a topological space) are important in theories of topological spaces and of metric spaces. The extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist.
The concept of metric space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of continuous functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a special topological space.
Continuous functions also form a vector space and an algebra as explained above in , and are a subclass of measurable functions because any topological space has the σ-algebra generated by open (or closed) sets.
Smooth
Real numbers are used as the codomain to define smooth functions. A domain of a real smooth function can be the real coordinate space (which yields a real multivariable function), a topological vector space, an open subset of them, or a smooth manifold.
Spaces of smooth functions also are vector spaces and algebras as explained |
https://en.wikipedia.org/wiki/Keisuke%20Shimizu | is a Japanese professional footballer who plays as a goalkeeper for J1 League club Cerezo Osaka.
Club statistics
Honours and awards
Team
J. League Cup - 2008
References
External links
Profile at Cerezo Osaka
1988 births
Living people
Association football people from Hyōgo Prefecture
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Oita Trinita players
Giravanz Kitakyushu players
Avispa Fukuoka players
Kyoto Sanga FC players
Cerezo Osaka players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Extremum%20estimator | In statistics and econometrics, extremum estimators are a wide class of estimators for parametric models that are calculated through maximization (or minimization) of a certain objective function, which depends on the data. The general theory of extremum estimators was developed by .
Definition
An estimator is called an extremum estimator, if there is an objective function such that
where Θ is the parameter space. Sometimes a slightly weaker definition is given:
where op(1) is the variable converging in probability to zero. With this modification doesn't have to be the exact maximizer of the objective function, just be sufficiently close to it.
The theory of extremum estimators does not specify what the objective function should be. There are various types of objective functions suitable for different models, and this framework allows us to analyse the theoretical properties of such estimators from a unified perspective. The theory only specifies the properties that the objective function has to possess, and so selecting a particular objective function only requires verifying that those properties are satisfied.
Consistency
If the parameter space Θ is compact and there is a limiting function Q0(θ) such that: converges to Q0(θ) in probability uniformly over Θ, and the function Q0(θ) is continuous and has a unique maximum at θ = θ0 then is consistent for θ0.
The uniform convergence in probability of means that
The requirement for Θ to be compact can be replaced with a weaker assumption that the maximum of Q0 was well-separated, that is there should not exist any points θ that are distant from θ0 but such that Q0(θ) were close to Q0(θ0). Formally, it means that for any sequence {θi} such that , it should be true that .
Asymptotic normality
Assuming that consistency has been established and the derivatives of the sample satisfy some other conditions, the extremum estimator converges to an asymptotically Normal distribution.
Examples
See also
M-estimators
Notes
References
Estimator |
https://en.wikipedia.org/wiki/2005%20FC%20Seoul%20season |
Pre-season
Pre-season match results
Competitions
Overview
K League
FA Cup
League Cup
Match reports and match highlights
Fixtures and Results at FC Seoul Official Website
Season statistics
K League records
2005 season's league position was decided by aggregate points, because this season had first stage and second stage.
All competitions records
Attendance records
Season total attendance is K League Regular Season, League Cup, FA Cup, AFC Champions League in the aggregate and friendly match attendance is not included.
K League season total attendance is K League Regular Season and League Cup in the aggregate.
Squad statistics
Goals
Assists
Coaching staff
Players
Team squad
All players registered for the 2005 season are listed.
(Out)
(Conscripted)
(Conscripted)
(Conscripted)
(Conscripted)
(Conscripted)
Out on loan & military service
In: Transferred from other teams in the middle of season.
Out: Transferred to other teams in the middle of season.
Discharged: Transferred from Gwangju Sangmu and Police FC for military service after end of season. (Not registered in 2005 season.)
Conscripted: Transferred to Gwangju Sangmu and Police FC for military service after end of season.
Transfers
In
Rookie Free Agent
Out
Loan & Military service
Tactics
Tactical analysis
Starting eleven and formation
This section shows the most used players for each position considering a 3-5-2 formation.
Substitutes
See also
FC Seoul
References
FC Seoul 2005 Matchday Magazines
External links
FC Seoul Official Website
2005
Seoul |
https://en.wikipedia.org/wiki/Symbolic-numeric%20computation | In mathematics and computer science, symbolic-numeric computation is the use of software that combines symbolic and numeric methods to solve problems.
Background
Computational Algebraic Geometry
References
External links
Professional organizations
ACM SIGSAM: Special Interest Group in Symbolic and Algebraic Manipulation
Computer algebra
Numerical analysis
Computational science |
https://en.wikipedia.org/wiki/Alexandru%20Froda | Alexandru Froda (July 16, 1894 – October 7, 1973) was a Romanian mathematician with contributions in the field of mathematical analysis, algebra, number theory and rational mechanics. In his 1929 thesis he proved what is now known as Froda's theorem.
Life
Alexandru Froda was born in Bucharest in 1894. In 1927 he graduated from the University of Sciences (now the Faculty of Mathematics of the University of Bucharest). He received his Ph.D. from the University of Paris in 1929 under the direction of Émile Borel.
Froda was elected president of the Romanian Mathematical Society in 1946. In 1948 he became professor in the Faculty of Mathematics and Physics of the University of Bucharest.
Work
Froda's major contribution was in the field of mathematical analysis. His first important result was concerned with the set of discontinuities of a real-valued function of a real variable. In this theorem Froda proves that the set of simple discontinuities of a real-valued function of a real variable is at most countable.
In a paper from 1936 he proved a necessary and sufficient condition for a function to be measurable. In the theory of algebraic equations, Froda proved a method of solving algebraic equations having complex coefficients.
In 1929 Dimitrie Pompeiu conjectured that any continuous function of two real variables defined on the entire plane is constant if the integral over any circle in the plane is constant. In the same year Froda proved that, in the case that the conjecture is true, the condition that the function is defined on the whole plane is indispensable. Later it was shown that the conjecture is not true in general.
In 1907 Pompeiu constructed an example of a continuous function with a nonzero derivative which has a zero in every interval. Using this result Froda finds a new way of looking at an older problem posed by Mikhail Lavrentyev in 1925, namely whether there is a function of two real variables such that the ordinary differential equation has at least two solutions passing through every point in the plane.
In the theory of numbers, beside rational triangles he also proved several conditions for a real number, which is the limit of a rational convergent sequence, to be irrational, extending a previous result of Viggo Brun from 1910.
In 1937 Froda independently noticed and proved the case of the Borsuk–Ulam theorem. He died in Bucharest in 1973.
See also
Froda's theorem
References
20th-century Romanian mathematicians
Scientists from Bucharest
Academic staff of the University of Bucharest
1894 births
1969 deaths
University of Bucharest alumni
University of Paris alumni
Romanian expatriates in France |
https://en.wikipedia.org/wiki/Omega%20ratio | The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the probability weighted ratio of gains versus losses for some threshold return target. The ratio is an alternative for the widely used Sharpe ratio and is based on information the Sharpe ratio discards.
Omega is calculated by creating a partition in the cumulative return distribution in order to create an area of losses and an area for gains relative to this threshold.
The ratio is calculated as:
where is the cumulative probability distribution function of the returns and is the target return threshold defining what is considered a gain versus a loss. A larger ratio indicates that the asset provides more gains relative to losses for some threshold and so would be preferred by an investor. When is set to zero the gain-loss-ratio by Bernardo and Ledoit arises as a special case.
Comparisons can be made with the commonly used Sharpe ratio which considers the ratio of return versus volatility. The Sharpe ratio considers only the first two moments of the return distribution whereas the Omega ratio, by construction, considers all moments.
Optimization of the Omega ratio
The standard form of the Omega ratio is a non-convex function, but it is possible to optimize a transformed version using linear programming. To begin with, Kapsos et al. show that the Omega ratio of a portfolio is:If we are interested in maximizing the Omega ratio, then the relevant optimization problem to solve is:The objective function is still non-convex, so we have to make several more modifications. First, note that the discrete analogue of the objective function is:For sampled asset class returns, let and . Then the discrete objective function becomes:With these substitutions, we have been able to transform the non-convex optimization problem into an instance of linear-fractional programming. Assuming that the feasible region is non-empty and bounded, it is possible to transform a linear-fractional program into a linear program. Conversion from a linear-fractional program to a linear program gives us the final form of the Omega ratio optimization problem:where are the respective lower and upper bounds for the portfolio weights. To recover the portfolio weights, normalize the values of so that their sum is equal to 1.
See also
Modern portfolio theory
Post-modern portfolio theory
Sharpe ratio
Sortino ratio
Upside potential ratio
References
External links
How good an investment is property?
Financial ratios
Financial models
Financial risk modeling
Investment indicators
Linear programming |
https://en.wikipedia.org/wiki/Steven%20Roman | Steven Roman is a mathematician, currently Emeritus Professor of Mathematics at California State University, Fullerton and Visiting Professor of Mathematics at University of California, Irvine. He is one of the main developers of umbral calculus. He has written about 40 books on mathematics and computer programming.
Professor Roman's books have been translated into Portuguese, French, Korean, Chinese, Russian, Polish, Bulgarian, Czech and Spanish.
Computer programming books
Concepts of Object-Oriented Programming with Visual Basic, Springer-Verlag.
Access Database Design and Programming, Third Edition, O'Reilly and Associates.
Understanding Personal Computer Hardware, Springer-Verlag.
Writing Word Macros, O'Reilly and Associates.
Writing Visual Basic Add-Ins, O'Reilly and Associates.
Writing Excel Macros with VBA, Second Edition, O'Reilly and Associates.
Win32 API Programming with Visual Basic, O'Reilly and Associates.
VB .NET Language in a Nutshell, O'Reilly and Associates.
VB .NET Language Pocket Reference, O'Reilly and Associates.
Mathematics books
The Umbral Calculus, Pure and Applied Mathematics Vol. 111, Academic Press, 1984.
An Introduction to Linear Algebra with Applications, Second edition, 1988, Saunders College Publishing.
An Introduction to Discrete Mathematics, Second edition, 1989, Saunders College Publishing.
College Algebra and Trigonometry, Saunders 1987.
College Algebra, Saunders 1987.
Precalculus, Saunders 1987.
Coding and Information Theory, the Springer-Verlag, Graduate Texts in Mathematics Vol. 134, 1992.
Advanced Linear Algebra, Springer-Verlag, Graduate Texts in Mathematics Vol. 135, 1992.
Field Theory, Springer-Verlag, Graduate Texts in Mathematics Vol. 158, 1995.
Introduction to Coding and Information Theory, Springer-Verlag, Undergraduate Texts in Mathematics, Springer-Verlag, 1996.
Lattices and Ordered Sets, Springer-Verlag, 2008.
Introduction to the Mathematics of Finance: Arbitrage and Option Pricing, Springer, 2012.
Fundamentals of Group Theory, An Advanced Approach, Birkhauser, 2012.
An Introduction to Catalan Numbers, Birkhauser, 2015.
An Introduction to the language of Category Theory, Birkhauser, 2017.
References
External links
Website: www.sroman.com
YouTube channel
20th-century American mathematicians
21st-century American mathematicians
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Chinese%20monoid | In mathematics, the Chinese monoid is a monoid generated by a totally ordered alphabet with the relations cba = cab = bca for every a ≤ b ≤ c. An algorithm similar to Schensted's algorithm yields characterisation of the equivalence classes and a cross-section theorem. It was discovered by during their classification of monoids with growth similar to that of the plactic monoid, and studied in detail by Julien Cassaigne, Marc Espie, Daniel Krob, Jean-Christophe Novelli, and Florent Hivert in 2001.
The Chinese monoid has a regular language cross-section
and hence polynomial growth of dimension .
The Chinese monoid equivalence class of a permutation is the preimage of an involution under the map where denotes the product in the Iwahori-Hecke algebra with .
See also
Plactic monoid
References
Combinatorics
Semigroup theory |
https://en.wikipedia.org/wiki/Expected%20value%20%28disambiguation%29 | Expected value is a term used in probability theory and statistics. It may also refer to:
Physics
Expectation value (quantum mechanics), the probabilistic expected value of the result (measurement) of an experiment
Decision theory and quantitative policy analysis
Expected value of perfect information, the price that one would be willing to pay in order to gain access to perfect information
Expected value of sample information, the expected increase in utility that a decision-maker could obtain from gaining access to a sample of additional observations before making a decision
Expected value of including uncertainty, the expected difference in the value of a decision based on a probabilistic analysis versus a decision based on an analysis that ignores uncertainty
Business
Expected commercial value, also known as estimated commercial value, the prospect-weighted value for a project with unclear conclusions
See also
Expected (disambiguation) |
https://en.wikipedia.org/wiki/Lizhen%20Ji | Lizhen Ji (Chinese: 季理真; born 1964), is a Chinese-American mathematician. He is a professor of mathematics at the University of Michigan, Ann Arbor.
Biography
April 1964, Ji was born in Wenzhou, Zhejiang Province, China. Ji graduated BS from Hangzhou University (previous and current Zhejiang University) in Hangzhou in 1984. From 1984 to 1985, Ji was a master student at the Department of Mathematics of Hangzhou University. Ji went to United States to continue his study in 1985, and in 1987 Ji obtained MS from the Department of Mathematics of the University of California, San Diego. In 1991, Ji obtained PhD from the Northeastern University (doctoral advisors: R. Mark Goresky and Shing-Tung Yau).
From 1991 to 1994, Ji was C.L.E. Moore instructor at the Department of Mathematics of MIT. From 1994 to 1995, Ji was a member of the Institute for Advanced Study School of Mathematics in Princeton, New Jersey. From 1995 to 1999, Ji was an assistant professor at the Department of Mathematics, University of Michigan (UM). From 1999 to 2005, Ji was an associate professor at the same department. In 2005, Ji was promoted to full professor at UM.
Awards
From 1998 to 2001, Ji was an Alfred P. Sloan Research Fellow. Ji received the Silver Morningside Medal of Mathematics in 2007. Ji was a Simons Fellow in 2014.
Publications
Besides academic papers, Ji has also published or co-written many influential books in mathematics, including:
Compactifications of Symmetric and Locally Symmetric Spaces; by Armand Borel, and Lizhen Ji.
Arithmetic Groups and Their Generalizations: What, Why, and How; by Lizhen Ji.
Geometry, Analysis and Topology of Discrete Groups; co-edited by Lizhen Ji, Kefeng Liu, Yang Lo, and Shing-Tung Yau.
Compactifications of Symmetric Spaces (Progress in Mathematics, Vol 156); by Y. Guivarc'H, Lizhen Ji, and J. C. Taylor.
Geometry Analysis and Topology of Discrete Groups; by Lizhen Ji, Kefeng Liu, and Yang Lo.
Handbook of Geometric Analysis; by Lizhen Ji.
Mathematics and Mathematical People; Chief-editor Lizhen Ji.
Advanced Lectures in Mathematics; Chief-editor Lizhen Ji.
Editorial Work
An editor of the Asian Journal of Mathematics
An editor of Science in China, Series A: Mathematics
A founding editor and chief editor of Pure and Applied Mathematics Quarterly
One of the chief-editors of a book series on Popular Mathematics Mathematics and Humanities
References
External links
The homepage of Lizhen Ji
The Mathematics Genealogy Project - Lizhen Ji
Lizhen Ji's homepage at Zhejiang University
ScienceNet: 季理真：追求真爱的数学人 (Lizhen Ji: Who Loves Math)
1964 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Chinese emigrants to the United States
Hangzhou University alumni
Massachusetts Institute of Technology School of Science faculty
Northeastern University alumni
Educators from Wenzhou
Sloan Research Fellows
University of California, San Diego alumni
University of Michigan faculty
Zhejiang Un |
https://en.wikipedia.org/wiki/Valenzuela%20City%20School%20of%20Mathematics%20and%20Science | The Valenzuela City School of Mathematics and Science (VCSMS; ), also referred to as ValMaSci, is a specialized public high school in Valenzuela City, Philippines.
Established in 2003 as the Valenzuela City Science High School (), it offers a special advanced curriculum with emphasis in the fields of mathematics and science to residents of the city. The school also has a specialized range of subjects in technology, engineering, and language.
History
Foundation
The Valenzuela City School of Mathematics and Science traces its roots to the Science Oriented Experimental Class (SOEC), an education program organized by former Valenzuela City mayor Jose Emmanuel Carlos. SOEC was initiated in school year 1996–97 "to shape and produce globally competitive and morally upright individuals who will serve as foundation of [the] nation's great success." The class, which was composed of 40 academically outstanding students, was housed at the city's national high school, the Valenzuela National High School (VNHS), in Barangay Marulas.
On May 14, 2003, seven years after SOEC was created, an ordinance establishing a science high school from the original SOEC was approved. The school, under the name Valenzuela City Science High School, was to be funded by the local School Board of Valenzuela.
Valenzuela City Science High School
On February 8, 2003, a memorandum was sent to all principals of public and private elementary schools of the city, requesting them to inform their graduating students to apply for the qualifying examination for the upcoming admission of first year high school students in VCSHS. With 200 applicants administered, 108 students passed the admission tests (consisting of an intelligence quotient test, proficiency test, and interview). They enrolled at the new school with Brian E. Ilan as the first principal. On June 9, 2003, the first VCSHS Parents, Teachers, and Community Association (PTCA) was formed.
From the original six teachers, the institution added eight teachers. Additional students made the VCSHS community larger as the years passed. At the beginning of the school year 2005–06, VCSHS moved to its newly constructed building at A. Marcelo Street in Barangay Dalandanan. City mayor Sherwin Gatchalian inaugurated the new house and 4th year students entered VCSHS. During the year, Ilan was replaced by Lagrimas B. Bayle as the new school principal. Through her time, a recorded shortest graduation (as observed by teachers) happened for the 15 students of SOEC passing the 4th year curriculum of ValSci.
By the school year 2006–07, Genindina M. Sumbillo entered ValSci from Valenzuela National High School–Mapulang Lupa Annex to replace Lagrimas B. Bayle. Before the end of school year 2007–08, Arneil D. Aro replaced Sumbillo who had been promoted as principal of Caruhatan National High School. And at the midst of 2011–12, Edelina I. Golloso took the principal seat replacing Aro. Ilan resumed his duties as principal, relieving Golloso of her |
https://en.wikipedia.org/wiki/Wilkie%27s%20theorem | In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
Formulations
In terms of model theory, Wilkie's theorem deals with the language Lexp = (+, −, ·, <, 0, 1, ex), the language of ordered rings with an exponential function ex. Suppose φ(x1, ..., xm) is a formula in this language. Then Wilkie's theorem states that there is an integer n ≥ m and polynomials f1, ..., fr ∈ Z[x1, ..., xn, ex1, ..., exn] such that φ(x1, ..., xm) is equivalent to the existential formula
Thus, while this theory does not have full quantifier elimination, formulae can be put in a particularly simple form. This result proves that the theory of the structure Rexp, that is the real ordered field with the exponential function, is model complete.
In terms of analytic geometry, the theorem states that any definable set in the above language — in particular the complement of an exponential variety — is in fact a projection of an exponential variety. An exponential variety over a field K is the set of points in Kn where a finite collection of exponential polynomials simultaneously vanish. Wilkie's theorem states that if we have any definable set in an Lexp structure K = (K, +, −, ·, 0, 1, ex), say X ⊂ Km, then there will be an exponential variety in some higher dimension Kn such that the projection of this variety down onto Km will be precisely X.
Gabrielov's theorem
The result can be considered as a variation of Gabrielov's theorem. This earlier theorem of Andrei Gabrielov dealt with sub-analytic sets, or the language Lan of ordered rings with a function symbol for each proper analytic function on Rm restricted to the closed unit cube [0, 1]m. Gabrielov's theorem states that any formula in this language is equivalent to an existential one, as above. Hence the theory of the real ordered field with restricted analytic functions is model complete.
Intermediate results
Gabrielov's theorem applies to the real field with all restricted analytic functions adjoined, whereas Wilkie's theorem removes the need to restrict the function, but only allows one to add the exponential function. As an intermediate result Wilkie asked when the complement of a sub-analytic set could be defined using the same analytic functions that described the original set. It turns out the required functions are the Pfaffian functions. In particular the theory of the real ordered field with restricted, totally defined Pfaffian functions is model complete. Wilkie's approach for this latter result is somewhat different from his proof of Wilkie's theorem, and the result that allowed him to show that the Pfaffian structure is model complete is sometimes known as Wilkie's theorem of the complement. See also.
References
Model theory
Theorems in the foundations of mathematics |
https://en.wikipedia.org/wiki/Howell%20Tong | Howell Tong (; born in 1944 in Hong Kong) is a statistician who has made fundamental contributions to nonlinear time series analysis, semi-parametric statistics, non-parametric statistics, dimension reduction, model selection, likelihood-free statistics and other areas. In the words of Professor Peter Whittle (FRS): "The striking feature of Howell Tong's … is the
continuing freshness, boldness and spirit of enquiry which inform them-indeed, proper qualities for
an explorer. He stands as the recognised innovator and authority in his subject, while remaining
disarmingly direct and enthusiastic." His work, in the words of Sir David Cox, "links two fascinating fields, nonlinear time series and deterministic dynamical systems."
He is the father of the threshold time series models, which have extensive applications in ecology, economics, epidemiology and finance. (See external links for detail.) Besides nonlinear time series analysis, he was the co-author of a seminal paper, which he read to the Royal Statistical Society, on dimension reduction in semi-parametric statistics by pioneering the approach based on minimum average variance estimation. He has also made numerous novel contributions to nonparametric statistics (obtaining the surprising result that cross-validation does not suffer from the curse of dimensionality for consistent estimation of the embedding dimension of a dynamical system), Markov chain modelling (with application to weather data), reliability, non-stationary time series analysis (in both the frequency domain and the time domain) and wavelets.
Life
Since October 1, 2009, he has been an emeritus professor at the London School of Economics and was twice (2009, 2010) holder of the Saw Swee Hock Professorship of Statistics at the National University of Singapore. He was a guest professor, Academy of Mathematics and System Sciences, the Chinese Academy of Sciences from 2000 to 2004, a distinguished visiting professor of statistics at the University of Hong Kong from 2005 to 2013, a distinguished professor-at-large, University of Electronic Science & Technology of China from 2016–2021 and is a distinguished visiting professor, Tsinghua University, China, since 2019.
Tong, a scholarship boy, left Wah Yan College 香港華仁書院 (founded by the Irish Jesuits in 1919) in Hong Kong in 1961, and was sent by his father to complete his matriculation at Barnsbury Boys' School in North London (one of the earliest comprehensive schools in England, now no longer in existence). He got his Bachelor of Science in 1966 (with first class honours in Mathematics), Master of Science in 1969 and Doctor of Philosophy in 1972, all from the University of Manchester Institute of Science and Technology (UMIST, now merged into the University of Manchester), where he studied under Maurice Priestley. Tong remained at UMIST first as a lecturer and then as a senior lecturer. While in Manchester, he started his married life with Ann Mary Leong. In 1982, he moved to the |
https://en.wikipedia.org/wiki/Jan%20Mandel | Jan Mandel is a Czech-American mathematician. He received his PhD from the faculty of mathematics and physics, Charles University in Prague and was a senior research scientist there. Since 1986, he is professor of mathematics at the University of Colorado Denver. Since 2013, he is senior scientist at the Institute of Computer Science of the Czech Academy of Sciences.
He has worked in the field of multigrid methods and domain decomposition methods. He developed the balancing domain decomposition method and, with coauthors, published the convergence proofs of the FETI, FETI-DP, and BDDC methods, and the proof of the equivalence of the FETI-DP and the BDDC methods. He has been involved in the field of dynamic data driven application systems and data assimilation with applications in wildfire and epidemic modeling. He has contributed to the WRF-Fire software.
References
External links
Home page at University of Colorado Denver
20th-century American mathematicians
21st-century American mathematicians
Czech mathematicians
Living people
University of Colorado Denver faculty
Charles University alumni
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Julian%20Hails | Julian Hails (born 20 November 1967) is an English former professional footballer who played in the Football League as a midfielder for Fulham and Southend United. He is a maths teacher at the St Albans High School For Girls.
Biography
Football career
Hails was studying for a maths degree and playing part-time at Hemel Hempstead Town, before being offered a trial at Fulham. Part of the deal that took Hails to Fulham was that he could stay on and finish his degree. He joined Fulham permanently in 1990. He played as a right-winger for Fulham, a position his father used to play for Lincoln City, Peterborough United, Luton Town and Northampton Town. Hails made 126 appearances in all competitions scoring 13 goals, being voted as the "Player of the Season" by Fulham fans during his spell with the London club.
Peter Taylor signed Hails for Southend United in early December 1994. He made 182 appearances for Southend in all competitions scoring seven goals. Hails was moved into right-back position in September 1997, when Alvin Martin took control as the manager. He won the "Player of the Season" award that season after a number of impressive performances. He was forced to retire in 2000, after a two-year struggle with knee injuries.
Life after football
Hails has a BSc honours in Mathematical Studies. He is now a maths teacher at The Haberdashers' Aske's Boys' School, when he joined in May 2006, and has had various football and tennis coaching roles at the school.
Personal life
Hails was born in Lincoln and is married. He now lives with his wife and 3 sons in Hertfordshire. His father, William, was also a professional footballer in the 1950s and 1960s, who played for Lincoln City, Peterborough United, Luton Town and Northampton Town.
References
1967 births
Living people
Footballers from Lincoln, England
English men's footballers
Schoolteachers from Lincolnshire
Hemel Hempstead Town F.C. players
Fulham F.C. players
Southend United F.C. players
English Football League players
Men's association football midfielders |
https://en.wikipedia.org/wiki/J%C3%A1nos%20Koll%C3%A1r | János Kollár (born 7 June 1956) is a Hungarian mathematician, specializing in algebraic geometry.
Professional career
Kollár began his studies at the Eötvös University in Budapest and later received his PhD at Brandeis University in 1984 under the direction of Teruhisa Matsusaka with a thesis on canonical threefolds. He was Junior Fellow at Harvard University from 1984 to 1987 and professor at the University of Utah from 1987 until 1999. Currently, he is professor at Princeton University.
Contributions
Kollár is known for his contributions to the minimal model program for threefolds and hence the compactification of moduli of algebraic surfaces, for pioneering the notion of rational connectedness (i.e. extending the theory of rationally connected varieties for varieties over the complex field to varieties over local fields), and finding counterexamples to a conjecture of John Nash. (In 1952 Nash conjectured a converse to a famous theorem he proved, and Kollár was able to provide many 3-dimensional counterexamples from an important new structure theory for a class of 3-dimensional algebraic varieties.)
Kollár also gave the first algebraic proof of effective Nullstellensatz: let be polynomials of degree at most in variables; if they have no common zero, then the equation has a solution such that each polynomial has degree at most .
Awards and honors
Kollár is a member of the National Academy of Sciences since 2005 and received the Cole Prize in 2006. He is an external member of the Hungarian Academy of Sciences since 1995. In 2012 he became a fellow of the American Mathematical Society. In 2016 he became a fellow of the American Academy of Arts and Sciences. In 2017 he received the Shaw Prize in Mathematical Sciences.
In 1990 he was an invited speaker at the International Congress of Mathematicians (ICM) in Kyōto. In 1996 he gave one of the plenary addresses at the European Mathematical Congress in Budapest (Low degree polynomial equations: arithmetic, geometry and topology). He was also selected as a plenary speaker at the ICM held in 2014 in Seoul.
As a high school student, Kollár represented Hungary and won Gold medals at both the 1973 and 1974 International Mathematical Olympiads.
Works
(Japanese by Iwanami Shoten).
References
External links
Homepage in Princeton
1956 births
Living people
Algebraic geometers
20th-century American mathematicians
20th-century Hungarian mathematicians
21st-century American mathematicians
21st-century Hungarian mathematicians
Institute for Advanced Study visiting scholars
Harvard Fellows
University of Utah faculty
Princeton University faculty
Brandeis University alumni
Eötvös Loránd University alumni
Members of the Hungarian Academy of Sciences
Members of the United States National Academy of Sciences
Fellows of the American Mathematical Society
International Mathematical Olympiad participants
Fellows of the American Academy of Arts and Sciences |
https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted%E2%80%93Knuth%20correspondence | In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices with non-negative integer entries and pairs of semistandard Young tableaux of equal shape, whose size equals the sum of the entries of . More precisely the weight of is given by the column sums of , and the weight of by its row sums.
It is a generalization of the Robinson–Schensted correspondence, in the sense that taking to be a permutation matrix, the pair will be the pair of standard tableaux associated to the permutation under the Robinson–Schensted correspondence.
The Robinson–Schensted–Knuth correspondence extends many of the remarkable properties of the Robinson–Schensted correspondence, notably its symmetry: transposition of the matrix results in interchange of the tableaux .
The Robinson–Schensted–Knuth correspondence
Introduction
The Robinson–Schensted correspondence is a bijective mapping between permutations and pairs of standard Young tableaux, both having the same shape. This bijection can be constructed using an algorithm called Schensted insertion, starting with an empty tableau and successively inserting the values σ1,…,σn of the permutation σ at the numbers 1,2,…n; these form the second line when σ is given in two-line notation:
.
The first standard tableau is the result of successive insertions; the other standard tableau records the successive shapes of the intermediate tableaux during the construction of .
The Schensted insertion easily generalizes to the case where σ has repeated entries; in that case the correspondence will produce a semistandard tableau rather than a standard tableau, but will still be a standard tableau. The definition of the RSK correspondence reestablishes symmetry between the P and Q tableaux by producing a semistandard tableau for as well.
Two-line arrays
The two-line array (or generalized permutation) corresponding to a matrix is defined as
in which for any pair that indexes an entry of , there are columns equal to , and all columns are in lexicographic order, which means that
, and
if and then .
Example
The two-line array corresponding to
is
Definition of the correspondence
By applying the Schensted insertion algorithm to the bottom line of this two-line array, one obtains a pair consisting of a semistandard tableau and a standard tableau , where the latter can be turned into a semistandard tableau by replacing each entry of by the -th entry of the top line of . One thus obtains a bijection from matrices to ordered pairs, of semistandard Young tableaux of the same shape, in which the set of entries of is that of the second line of , and the set of entries of is that of the first line of . The number of entries in is therefore equal to the sum of the entries in column of , and the number of entries in is equal to the sum of the entries in row of .
Example
In the above example |
https://en.wikipedia.org/wiki/Daniel%20Ocone | Daniel Leonard Ocone (born 1953) is a Professor in the Mathematics Department at Rutgers University, where he specializes in probability theory and stochastic processes. He obtained his Ph.D. at MIT in 1980 under the supervision of Sanjoy K. Mitter. He is known for the Clark–Ocone theorem in stochastic analysis. The continuous Ocone martingale is also named after him; it is a continuous martingale that is conditionally Gaussian, given its quadratic variation process.
References
External links
Home Page at Rutgers (includes Photo)
D. Ocone: Topics in Nonlinear Filtering Theory (Ph.D. thesis, 1980)
1953 births
Living people
Massachusetts Institute of Technology alumni
Rutgers University faculty
20th-century American mathematicians
21st-century American mathematicians |
https://en.wikipedia.org/wiki/List%20of%20cities%20and%20towns%20in%20Denmark | This article shows a list of cities in Denmark by population. The population is measured by Statistics Denmark for urban areas (Danish: Byområder), defined as a contiguous built-up area with a maximum distance of 200 meters between houses, unless further distance is caused by public areas, cemeteries or similar. Furthermore, to obtain the status of being a city (byområde), the area must have at least 200 inhabitants. Smaller settlements are by Danmarks Statistik included in numbers for rural areas (landdistrikter).
The largest urban area is the Hovedstadsområdet, the metropolitan area of Copenhagen.
See also
List of urban areas in Sweden by population
List of towns and cities in Norway
List of urban areas in the Nordic countries
World's largest cities
List of municipalities of Denmark
References and notes
External links
Cities
Denmark
Denmark |
https://en.wikipedia.org/wiki/Journal%20of%20Formalized%20Reasoning | The Journal of Formalized Reasoning is a peer-reviewed open access academic journal established in 2009. It publishes formalization efforts in any area, including classical mathematics, constructive mathematics, formal algorithms, and program verifications. It is maintained by AlmaDL, the digital library of the University of Bologna.
Abstracting and indexing
The journal is abstracted and indexed in Scopus, MathSciNet, and Zentralblatt MATH.
External links
Computer science journals
Open access journals
Reasoning
Academic journals established in 2008
Biannual journals
University of Bologna
English-language journals |
https://en.wikipedia.org/wiki/Paul%20Rabinowitz | Paul H. Rabinowitz (born 1939) is the Edward Burr Van Vleck Professor of Mathematics and a Vilas Research Professor at the University of Wisconsin, Madison. He received a Ph.D. from New York University in 1966 under the direction of Jürgen Moser. From 1966 to 1969 he held a position as assistant professor at Stanford University. He has visited many mathematical institutions all over the world (among them universities at Aarhus, Pisa, Paris and ETH in Zurich). In 1978 Paul Rabinowitz became a fellow of the John Simon Guggenheim Memorial Foundation.
He works in the fields of partial differential equations and nonlinear analysis. He is best known for his global bifurcation theorem and the mountain pass theorem, the latter done jointly with Antonio Ambrosetti. However also the linking and saddle point theorems, results concerning the existence of periodic solutions to hamiltonian systems, variational methods in the theory of critical points of strongly indefinite functional under the absence of compactness conditions of the Palais–Smale type and other achievements of Paul Rabinowitz have found their place in the history of mathematics.
He is the recipient of numerous honors and awards, including the George David Birkhoff Prize in 1998. He was elected as a member of the United States National Academy of Sciences in 1998. In 2012 he became a fellow of the American Mathematical Society. In 2014 Paul Rabinowitz was awarded with the Juliusz Schauder Medal, the prize established by the Juliusz Schauder Center for Nonlinear Studies at the Nicolaus Copernicus University in Toruń, Poland, in recognition of his important contribution in the field of topological methods in nonlinear analysis.
References and notes
External links
Paul Rabinowitz at the Mathematics Genealogy Project
Announcement of Election to National Academy
Juliusz P. Schauder Center for Nonlinear Studies
20th-century American mathematicians
21st-century American mathematicians
University of Wisconsin–Madison faculty
1939 births
Living people
Fellows of the American Mathematical Society
Fellows of the Society for Industrial and Applied Mathematics
Members of the United States National Academy of Sciences
Foreign Members of the Russian Academy of Sciences |
https://en.wikipedia.org/wiki/Polyominoid | In geometry, a polyominoid (or minoid for short) is a set of equal squares in 3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include the polyominoes, which are just the planar polyominoids. The surface of a cube is an example of a hexominoid, or 6-cell polyominoid, and many other polycubes have polyominoids as their boundaries. Polyominoids appear to have been first proposed by Richard A. Epstein.
Classification
90-degree connections are called hard; 180-degree connections are called soft. This is because, in manufacturing a model of the polyominoid, a hard connection would be easier to realize than a soft one. Polyominoids may be classified as hard if every junction includes a 90° connection, soft if every connection is 180°, and mixed otherwise, except in the unique case of the monominoid, which has no connections of either kind. The set of soft polyominoids is equal to the set of polyominoes.
As with other polyforms, two polyominoids that are mirror images may be distinguished. One-sided polyominoids distinguish mirror images; free polyominoids do not.
Enumeration
The table below enumerates free and one-sided polyominoids of up to 6 cells.
Generalization to higher dimensions
In general one can define an n,k-polyominoid as a polyform made by joining k-dimensional hypercubes at 90° or 180° angles in n-dimensional space, where 1≤k≤n.
Polysticks are 2,1-polyominoids.
Polyominoes are 2,2-polyominoids.
The polyforms described above are 3,2-polyominoids.
Polycubes are 3,3-polyominoids.
References
Polyforms |
https://en.wikipedia.org/wiki/Levan%20Kakubava | Levan Kakubava (; born 15 October 1990) is Georgian football player, currently playing for FC Gagra as a centre back.
Career statistics (Dinamo Tbilisi)
External links
UEFA profile
1990 births
Living people
Men's footballers from Georgia (country)
Georgia (country) men's under-21 international footballers
Georgia (country) men's international footballers
Men's association football defenders
Erovnuli Liga players
FC Dinamo Tbilisi players
FC Metalurgi Rustavi players
FC Tskhinvali players
AC Omonia players
FC Samtredia players
FC Chikhura Sachkhere players
FC Saburtalo Tbilisi players
Footballers from Tbilisi |
https://en.wikipedia.org/wiki/Tsallis%20distribution | In statistics, a Tsallis distribution is a probability distribution derived from the maximization of the Tsallis entropy under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the Tsallis distribution". The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. Similarly, if the domain of the variable is constrained to be positive in the maximum entropy procedure, the q-exponential distribution is derived.
The Tsallis distributions have been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distributions are often used for their heavy tails.
Note that Tsallis distributions are obtained as Box–Cox transformation over usual distributions, with deformation parameter . This deformation transforms exponentials into q-exponentials.
Procedure
In a similar procedure to how the normal distribution can be derived using the standard Boltzmann–Gibbs entropy or Shannon entropy, the q-Gaussian can be derived from a maximization of the Tsallis entropy subject to the appropriate constraints.
Common Tsallis distributions
q-Gaussian
See q-Gaussian.
q-exponential distribution
See q-exponential distribution
q-Weibull distribution
See q-Weibull distribution
See also
Constantino Tsallis
Tsallis statistics
Tsallis entropy
Notes
Further reading
Juniper, J. (2007) "The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty", Centre of Full Employment and Equity, The University of Newcastle, Australia
Shigeru Furuichi, Flavia-Corina Mitroi-Symeonidis, Eleutherius Symeonidis, On some properties of Tsallis hypoentropies and hypodivergences, Entropy, 16(10) (2014), 5377-5399;
Shigeru Furuichi, Flavia-Corina Mitroi, Mathematical inequalities for some divergences, Physica A 391 (2012), pp. 388-400, ;
Shigeru Furuichi, Nicușor Minculete, Flavia-Corina Mitroi, Some inequalities on generalized entropies, J. Inequal. Appl., 2012, 2012:226.
External links
Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions
Statistical mechanics
Types of probability distributions
Probability distributions with non-finite variance |
https://en.wikipedia.org/wiki/Lasso%20%28statistics%29 | In statistics and machine learning, lasso (least absolute shrinkage and selection operator; also Lasso or LASSO) is a regression analysis method that performs both variable selection and regularization in order to enhance the prediction accuracy and interpretability of the resulting statistical model. It was originally introduced in geophysics, and later by Robert Tibshirani, who coined the term.
Lasso was originally formulated for linear regression models. This simple case reveals a substantial amount about the estimator. These include its relationship to ridge regression and best subset selection and the connections between lasso coefficient estimates and so-called soft thresholding. It also reveals that (like standard linear regression) the coefficient estimates do not need to be unique if covariates are collinear.
Though originally defined for linear regression, lasso regularization is easily extended to other statistical models including generalized linear models, generalized estimating equations, proportional hazards models, and M-estimators. Lasso's ability to perform subset selection relies on the form of the constraint and has a variety of interpretations including in terms of geometry, Bayesian statistics and convex analysis.
The LASSO is closely related to basis pursuit denoising.
History
Lasso was introduced in order to improve the prediction accuracy and interpretability of regression models. It selects a reduced set of the known covariates for use in a model.
Lasso was developed independently in geophysics literature in 1986, based on prior work that used the penalty for both fitting and penalization of the coefficients. Statistician Robert Tibshirani independently rediscovered and popularized it in 1996, based on Breiman's nonnegative garrote.
Prior to lasso, the most widely used method for choosing covariates was stepwise selection. That approach only improves prediction accuracy in certain cases, such as when only a few covariates have a strong relationship with the outcome. However, in other cases, it can increase prediction error.
At the time, ridge regression was the most popular technique for improving prediction accuracy. Ridge regression improves prediction error by shrinking the sum of the squares of the regression coefficients to be less than a fixed value in order to reduce overfitting, but it does not perform covariate selection and therefore does not help to make the model more interpretable.
Lasso achieves both of these goals by forcing the sum of the absolute value of the regression coefficients to be less than a fixed value, which forces certain coefficients to zero, excluding them from impacting prediction. This idea is similar to ridge regression, which also shrinks the size of the coefficients; however, ridge regression does not set coefficients to zero (and, thus, does not perform variable selection).
Basic form
Least squares
Consider a sample consisting of N cases, each of which consists of p covar |
https://en.wikipedia.org/wiki/Invertible%20module | In mathematics, particularly commutative algebra, an invertible module is intuitively a module that has an inverse with respect to the tensor product. Invertible modules form the foundation for the definition of invertible sheaves in algebraic geometry.
Formally, a finitely generated module M over a ring R is said to be invertible if it is locally a free module of rank 1. In other words, for all primes P of R. Now, if M is an invertible R-module, then its dual is its inverse with respect to the tensor product, i.e. .
The theory of invertible modules is closely related to the theory of codimension one varieties including the theory of divisors.
See also
Picard group
References
Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Springer,
Mathematical structures
Algebra |
https://en.wikipedia.org/wiki/Long%20code%20%28mathematics%29 | In theoretical computer science and coding theory, the long code is an error-correcting code that is locally decodable. Long codes have an extremely poor rate, but play a fundamental role in the theory of hardness of approximation.
Definition
Let for be the list of all functions from .
Then the long code encoding of a message is the string where denotes concatenation of strings.
This string has length .
The Walsh-Hadamard code is a subcode of the long code, and can be obtained by only using functions that are linear functions when interpreted as functions on the finite field with two elements. Since there are only such functions, the block length of the Walsh-Hadamard code is .
An equivalent definition of the long code is as follows:
The Long code encoding of is defined to be the truth table of the Boolean dictatorship function on the th coordinate, i.e., the truth table of with .
Thus, the Long code encodes a -bit string as a -bit string.
Properties
The long code does not contain repetitions, in the sense that the function computing the th bit of the output is different from any function computing the th bit of the output for .
Among all codes that do not contain repetitions, the long code has the longest possible output.
Moreover, it contains all non-repeating codes as a subcode.
References
Coding theory
Error detection and correction |
https://en.wikipedia.org/wiki/George%20Sinclair%20%28mathematician%29 | George Sinclair (Sinclar) (ca.1630–1696) was a Scottish mathematician, engineer and demonologist. The first Professor of Mathematics at the University of Glasgow, he is known for Satan's Invisible World Discovered, (c. 1685), a work on witchcraft. He wrote in all three areas of his interests, including an account of the "Glenluce Devil", a poltergeist case from , in a 1672 book mainly on hydrostatics but also a pioneering study of geological structures through his experience in coal mines.
Life
He was probably from the East Lothian area. He became a professor of the University of Glasgow, 18 April 1654, initially in a philosophy chair, then in a chair founded for mathematics. In 1655 he made descents in a diving bell off the Isle of Mull, to look at the wreck of a ship from the Spanish Armada there.
He was deprived of his university post in 1666, as a Presbyterian. He then worked as a mineral surveyor and engineer, and was employed in particular by Sir James Hope. He was brought in by the magistrates of Edinburgh, about 1670, to oversee piping of water from Comiston into the city.
On 3 March 1691, University of Edinburgh appointed him again to the professorship of mathematics, which had been vacant.
Sinclair invented an early example of a perpetual motion machine based on the principle of the siphon. He first proposed this in a Latin work on pneumatics in 1669.
Glenluce Devil
In his book Satan's Invisible World Discovered (1685), Sinclair described an alleged poltergeist incident known as the Devil of Glenluce. Sinclair described the incident as having a "usefulness for refuting atheism."
The incident is described as having taken place at the house of weaver Gilbert Campbell in Glenluce during October, 1654. A beggar named Alexander Agnew was refused a handout by Campbell. Agnew had promised to cause the family harm and over the next two years strange phenomena were alleged to have occurred at the house. This included the mysterious cutting of warp thread, demonic voices, strange whistling noises and stones being thrown. The poltergeist claims have been dismissed by researchers as a hoax. Magic historian Thomas Frost suggested that the phenomena was the result of conjuring trickery. The story was given to Sinclair by Campbell's son Thomas, a philosophy student from a college in Glasgow who was living at the household. Folklorist Andrew Lang suggested that Thomas had produced the phenomena fraudulently.
Historian David Damrosch has noted that Alexander Agnew commonly called the "Jock of Broad Scotland" was the first person in Scottish history to publicly deny the existence of God. He was hanged at Dumfries for blasphemy on 21 May 1656.
Geological pioneer
A long-neglected aspect deriving from Sinclair's work as a mineral surveyor is that the last part of Hydrostaticks - a short History of Coal - includes the first published geological cross section in which he treats the strata in purely geometric terms.
Controversy
James Gregory, then a |
https://en.wikipedia.org/wiki/List%20of%20Luton%20Town%20F.C.%20records%20and%20statistics | Luton Town Football Club is an English professional football club based in Luton, Bedfordshire. The club was founded in 1885 and became the first professional club in southern England in 1891. Luton Town have played at all professional levels of English football and are currently contesting the 2023–24 season in the first tier, Premier League. Luton Town have been Football League members for 104 seasons: from 1897 to 1900; from 1920 to 2009, and from 2014 to 2023.
The record for most games played for the club is held by Bob Morton, who made 562 appearances between 1946 and 1964. Gordon Turner is the club's record goalscorer, with 276 goals across his 450 appearances for Luton. Mal Donaghy made 58 appearances for Northern Ireland and so is the Luton Town player who has gained the most caps while with the club. The highest transfer fee paid by the club is the £2 million paid to Barnsley for striker Carlton Morris in 2022, and the highest fees received is the £8 million fee paid by Leicester City for Luton-born James Justin, in 2019. The highest attendance recorded at Kenilworth Road was 30,069 for the visit of Blackpool in 1959. One Football League record is held by a Luton Town player—the 10 goals scored by forward Joe Payne in 1936 against Bristol Rovers is the most scored in any Football League match by a single player.
All records are correct as of the 2022–23 season.
Honours and achievements
Luton Town have won some major honours in English football. The club reached its first major final in 1959, when the team reached the FA Cup Final, and the 1988 Football League Cup Final was the side's first major cup victory. The team have also won a Football League Trophy (in 2008–09) and finished as runners-up in the Full Members Cup and Football League Cup (in 1987–88 and 1988–89 respectively).
Luton Town have won all three of the present Football League divisions, and have achieved promotion as runners-up on four other occasions. Outside of the League, the club have finished as runners-up in the Southern League twice in a row (starting in 1894–95), runners-up in the United League in 1896–97, and United League champions in 1897–98. More recently, the club were crowned as Conference Premier champions in the 2013–14 season.
Football pyramid
Football League First Division (tier 1)
Best finish: seventh, 1986–87
Football League Second Division (tier 2)
Champions: 1981–82
Runners-up: 1954–55, 1973–74
Play-off winners: 2022–23
Football League Third Division (tier 3)
Champions: 1936–37 (South), 2004–05, 2018–19
Runners-up: 1935–36 (South), 1969–70
Football League Fourth Division / Football League Third Division (tier 4)
Champions: 1967–68
Runners-up: 2001–02, 2017–18
Conference Premier (tier 5)
Champions: 2013–14
Luton Town were the first club to be relegated from the top division to the fourth (relegated from First Division in 1959–60, started playing in Fourth Division in 1965–66) and then subsequently win promotion back to the top flight (prom |
https://en.wikipedia.org/wiki/Picture%20%28mathematics%29 | In combinatorial mathematics, a picture is a bijection between skew diagrams satisfying certain properties, introduced by in a generalization of the Robinson–Schensted correspondence and the Littlewood–Richardson rule.
References
Algebraic combinatorics
Combinatorial algorithms |
https://en.wikipedia.org/wiki/Simplicial%20sphere | In geometry and combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained in this way.
One important open problem in the field was the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere. In December 2018, the g-conjecture was proven by Karim Adiprasito in the more general context of rational homology spheres.
Examples
For any n ≥ 3, the simple n-cycle Cn is a simplicial circle, i.e. a simplicial sphere of dimension 1. This construction produces all simplicial circles.
The boundary of a convex polyhedron in R3 with triangular faces, such as an octahedron or icosahedron, is a simplicial 2-sphere.
More generally, the boundary of any (d+1)-dimensional compact (or bounded) simplicial convex polytope in the Euclidean space is a simplicial d-sphere.
Properties
It follows from Euler's formula that any simplicial 2-sphere with n vertices has 3n − 6 edges and 2n − 4 faces. The case of n = 4 is realized by the tetrahedron. By repeatedly performing the barycentric subdivision, it is easy to construct a simplicial sphere for any n ≥ 4. Moreover, Ernst Steinitz gave a characterization of 1-skeleta (or edge graphs) of convex polytopes in R3 implying that any simplicial 2-sphere is a boundary of a convex polytope.
Branko Grünbaum constructed an example of a non-polytopal simplicial sphere (that is, a simplicial sphere that is not the boundary of a polytope). Gil Kalai proved that, in fact, "most" simplicial spheres are non-polytopal. The smallest example is of dimension d = 4 and has f0 = 8 vertices.
The upper bound theorem gives upper bounds for the numbers fi of i-faces of any simplicial d-sphere with f0 = n vertices. This conjecture was proved for simplicial convex polytopes by Peter McMullen in 1970 and by Richard Stanley for general simplicial spheres in 1975.
The ''g-conjecture, formulated by McMullen in 1970, asks for a complete characterization of f-vectors of simplicial d-spheres. In other words, what are the possible sequences of numbers of faces of each dimension for a simplicial d-sphere? In the case of polytopal spheres, the answer is given by the g''-theorem, proved in 1979 by Billera and Lee (existence) and Stanley (necessity). It has been conjectured that the same conditions are necessary for general simplicial spheres. The conjecture was proved by Karim Adiprasito in December 2018.
See also
Dehn–Sommerville equations
References
Algebraic combinatorics
Topology |
https://en.wikipedia.org/wiki/Lattice%20word | 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
Algebraic combinatorics
Combinatorics on words |
https://en.wikipedia.org/wiki/Akihiro%20Kanamori | is a Japanese-born American mathematician. He specializes in set theory and is the author of the monograph on large cardinals, The Higher Infinite. He has written several essays on the history of mathematics, especially set theory.
Kanamori graduated from California Institute of Technology and earned a Ph.D. from the University of Cambridge (King's College), and is a professor of mathematics at Boston University.
With Matthew Foreman, Kanamori is the editor of the Handbook of Set Theory (2010).
Selected publications
A. Kanamori, M. Magidor: The evolution of large cardinal axioms in set theory, in: Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977), Lecture Notes in Mathematics, 669, Springer, 99–275.
R. M. Solovay, W. N. Reinhardt, A. Kanamori: Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, 13(1978), 73–116.
A. Kanamori: The Higher Infinite. Large Cardinals in Set Theory from their Beginnings., Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp.
Honors
Marshall Scholarship
References
External links
Kanamori's homepage with selected publications
Short CV
1948 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Set theorists
Boston University faculty
Japanese emigrants to the United States
American academics of Japanese descent |
https://en.wikipedia.org/wiki/Mathematical%20Cranks | Mathematical Cranks is a book on pseudomathematics and the cranks who create it, written by Underwood Dudley. It was published by the Mathematical Association of America in their MAA Spectrum book series in 1992 ().
Topics
Previously, Augustus De Morgan wrote in A Budget of Paradoxes about cranks in multiple subjects, and Dudley wrote a book about angle trisection. However, this is the first book to focus on mathematical crankery as a whole.
The book consists of 57 essays, loosely organized by the most common topics in mathematics for cranks to focus their attention on. The "top ten" of these topics, as listed by reviewer Ian Stewart, are, in order:
squaring the circle,
angle trisection,
Fermat's Last Theorem,
non-Euclidean geometry and the parallel postulate,
the golden ratio,
perfect numbers,
the four color theorem,
advocacy for duodecimal and other non-standard number systems,
Cantor's diagonal argument for the uncountability of the real numbers, and
doubling the cube.
Other common topics for crankery, collected by Dudley, include calculations for the perimeter of an ellipse, roots of quintic equations, Fermat's little theorem, Gödel's incompleteness theorems, Goldbach's conjecture, magic squares, divisibility rules, constructible polygons, twin primes, set theory, statistics, and the Van der Pol oscillator.
As David Singmaster writes, many of these topics are the subject of mainstream mathematics "and only become crankery in extreme cases". The book omits or passes lightly over other topics that apply mathematics to crankery in other areas, such as numerology and pyramidology. Its attitude towards the cranks it covers is one of "sympathy and understanding", and in order to keep the focus on their crankery it names them only by initials. The book also attempts to analyze the motivation and psychology behind crankery, and to provide advice to professional mathematicians on how to respond to cranks.
Despite his work on the subject, which has "become enshrined in academic folklore", Dudley has stated "I've been at this for a decade and still can't pin down exactly what it is that makes a crank a crank", adding that "It's like obscenity – you can tell a crank when you see one."
Lawsuit
After the book was published, one of the cranks whose work was featured in the book, William Dilworth, sued Dudley for defamation in a federal court in Wisconsin. The court dismissed the Dilworth vs Dudley case on two grounds. First, it found that by publishing his work on Cantor's diagonal argument, Dilworth had made himself a public figure, creating a higher burden of proof for a defamation case. Second, it found that the word "crank" was "rhetorical hyperbole" rather than an actionably inaccurate description. The United States Court of Appeals for the Seventh Circuit concurred. After Dilworth repeated the lawsuit in a state court, he lost again and was forced to pay Dudley's legal expenses.
Reception and audience
Reviewer John N. Fujii calls the |
https://en.wikipedia.org/wiki/Michio%20Jimbo | is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist .
Career
After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at the Research Institute for Mathematical Sciences in Kyoto University. He has made important contributions to mathematical physics, including (independently of Vladimir Drinfeld) the initial development of the study of quantum groups, the development of the theory of -functions for the KP
(Kadomtsev–Petviashvili) integrable hierarchy, and other related integrable hierarchies
, and development of the theory of isomonodromic deformation systems for rational covariant derivative operators.
Awards
In 1993 he won the Japan Academy Prize for this work. In 2010 he received the Wigner Medal.
Selected books
with Tetsuji Miwa, Etsurō Date: Solitons – differential equations, symmetries and infinite dimensional algebras. Cambridge University Press 2000,
with Tetsuji Miwa: Algebraic analysis of solvable lattice models. American Mathematical Society 1993,
Editor: Yang-Baxter Equation in integrable systems. World Scientific 1990,
References
1951 births
Living people
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Academic staff of the University of Tokyo
Academic staff of Kyoto University
University of Tokyo alumni
Kyoto University alumni
Mathematical physicists |
https://en.wikipedia.org/wiki/K.%20S.%20Amur | Krishna Shyamacharya Amur (born 1931) was a professor emeritus of mathematics in differential geometry was head of the department of mathematics, Karnatak University, Dharwar.
Amur was vice-president of Karnatak Education Board, Dharwar. and a brother of G. S. Amur.
Born and raised in Suranagi village of Haveri taluka, he earned a M.Sc. and a Ph.D. (1964) in mathematics from the Karnataka University, Dharwar.
Amur was a postdoctoral fellow at the department of mathematics, University of North Carolina at Chapel Hill, from 1967 to 1968 and again in 1984, he went to the USA on a fellowship program for a year. He was also acting registrar of the Karnatak University, Dharwar from 1978-80.
Amur was the President of Sri Aurobindo society, Karnataka State.
References
1931 births
2018 deaths
Kannada people
People from Haveri district
Academic staff of Karnatak University
20th-century Indian mathematicians
Scientists from Karnataka
Indian expatriate academics in the United States
Karnatak University alumni |
https://en.wikipedia.org/wiki/Eduard%20Oscar%20Schmidt | Eduard Oscar Schmidt (21 February 1823, in Torgau – 17 January 1886, in Kappelrodeck) was a German zoologist and phycologist.
Biography
He initially studied mathematics and science at Halle, then continued his education in Berlin, where he came under the influence of Christian Gottfried Ehrenberg and Johannes Peter Müller. In 1847 he received his habilitation at the University of Jena, becoming an associate professor during the following year. In 1855 was he appointed professor of zoology at the University of Cracow. Later he taught classes at the Universities of Graz (from 1857) and Strasbourg (from 1872).
Schmidt was an early proponent of Darwinian evolutionary thought. He is remembered for his research of Porifera (sponges), particularly species from the Adriatic Sea. Schmidt also made contributions in the field of phycology.
As far back as 1862 Oscar Schmidt showed that "cuttings" of sponges will attach themselves and grow. This idea was followed through in the experiments of Croatian scientist Grgur Bučić on the island of Hvar, from 1863 to 1872, but these experiments were brought to a close by the hostility of the native fishermen.
Written works
Schmidt built a reputation based upon a handbook of comparative anatomy, the 9th edition of which, by Arnold Lang, was issued under the title Lehrbuch der vergleichenden Anatomie der wirbellosen Tiere (1888–1894). He made significant contributions to Brehms Tierleben, and was the author of several treatises on sponges. The following are some of his principal writings:
Bilder aus dem Norden - Images of the North, based on Schmidt's second expedition to the Faeroe Islands and the North Cape, 1851.
Goethes Verhältnis zu den organischen Naturwissenschaften, (1853).
Lehrbuch der Zoologie - Textbook of zoology, 1854.
Die Entwicklung der vergleichenden Anatomie - Development of comparative anatomy, 1855.
Die Spongien des adriatischen Meeres - Sponges of the Adriatic Sea, 1862.
Das Alter der Menschheit und das Paradies, (with Franz Unger, 1866).
Descendenzlehre und Darwinismus - The Doctrine of Descent and Darwinism, (1873, 3rd edition 1884).
Leitfaden der Zoologie (4th edition 1882).
Die Säugethiere in ihrem Verhältnis zur Vorwelt, (1884).
See also
:Category:Taxa named by Eduard Oscar Schmidt
Notes
References
Wikisource translated biography @ Allgemeine Deutsche Biographie
External links
1823 births
1886 deaths
19th-century German zoologists
Academic staff of Jagiellonian University
People from the Province of Saxony
People from Torgau
Spongiologists
Academic staff of the University of Graz
Academic staff of the University of Jena
Academic staff of the University of Strasbourg |
https://en.wikipedia.org/wiki/Overshoot%20%28signal%29 | In signal processing, control theory, electronics, and mathematics, overshoot is the occurrence of a signal or function exceeding its target. Undershoot is the same phenomenon in the opposite direction. It arises especially in the step response of bandlimited systems such as low-pass filters. It is often followed by ringing, and at times conflated with the latter.
Definition
Maximum overshoot is defined in Katsuhiko Ogata's Discrete-time control systems as "the maximum peak value of the response curve measured from the desired response of the system."
Control theory
In control theory, overshoot refers to an output exceeding its final, steady-state value. For a step input, the percentage overshoot (PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the overshoot is just the maximum value of the step response minus one. Also see the definition of overshoot in an electronics context.
For second-order systems, the percentage overshoot is a function of the damping ratio ζ and is given by
The damping ratio can also be found by
Electronics
In electronics, overshoot refers to the transitory values of any parameter that exceeds its final (steady state) value during its transition from one value to another. An important application of the term is to the output signal of an amplifier.
Usage: Overshoot occurs when the transitory values exceed final value. When they are lower than the final value, the phenomenon is called "undershoot".
A circuit is designed to minimize rise time while containing distortion of the signal within acceptable limits.
Overshoot represents a distortion of the signal.
In circuit design, the goals of minimizing overshoot and of decreasing circuit rise time can conflict.
The magnitude of overshoot depends on time through a phenomenon called "damping." See illustration under step response.
Overshoot often is associated with settling time, how long it takes for the output to reach steady state; see step response.
Also see the definition of overshoot in a control theory context.
Mathematics
In the approximation of functions, overshoot is one term describing quality of approximation. When a function such as a square wave is represented by a summation of terms, for example, a Fourier series or an expansion in orthogonal polynomials, the approximation of the function by a truncated number of terms in the series can exhibit overshoot, undershoot and ringing. The more terms retained in the series, the less pronounced the departure of the approximation from the function it represents. However, though the period of the oscillations decreases, their amplitude does not; this is known as the Gibbs phenomenon. For the Fourier transform, this can be modeled by approximating a step function by the integral up to a certain frequency, which yields the sine integral. This can be interpreted as convolution with the sinc function; in signal processing terms, this is a low-pass filt |
https://en.wikipedia.org/wiki/Louis%20Auslander | Louis Auslander (July 12, 1928 – February 25, 1997) was a Jewish American mathematician. He had wide-ranging interests both in pure and applied mathematics and worked on Finsler geometry, geometry of solvmanifolds and nilmanifolds, locally affine spaces, many aspects of harmonic analysis, representation theory of solvable Lie groups, and multidimensional Fourier transforms and the design of signal sets for communications and radar. He is the author of more than one hundred papers and ten books.
Education and career
Auslander received his Ph.D. at the University of Chicago in 1955 under Shiing-Shen Chern. He was a visiting scholar at the Institute for Advanced Study in 1955-57 and again in 1971-72.
After holding a variety of faculty positions at US universities, in 1965 Auslander joined the faculty at Graduate Center of the City University of New York and since 1971 he had been a Distinguished Professor of Mathematics and Computer Science there.
Personal life
Louis Auslander was married twice, first for over 25 years to Elinor Newstadt Auslander, with whom he had three children (Nathan, Rose, and Daniel), and later to Fernande Couturier Auslander. His brother Maurice Auslander was also a mathematician.
Selected publications
Articles
Books
with L. Markus: Flat Lorentz 3-Manifolds, AMS 1957
with Robert MacKenzie: Introduction to differentiable Manifolds, McGraw Hill 1963
with Leon W. Green and Frank J. Hahn: Flows on homogeneous spaces, Princeton University Press 1963 (with the assistance of Lawrence Markus and William S. Massey and an appendix by L. Greenberg)
with Calvin C. Moore: Unitary representations of solvable Lie groups, AMS 1966
Abelian Harmonic Analysis, Theta Functions and Function Algebras on a Nilmanifold, Springer, 1975
Lecture Notes on Nil-Theta Functions, CBMS lectures, American Mathematical Society, 1977
Minimal flows and their extensions, North-Holland 1988
as editor: Signal processing theory, 2 volumes, Springer 1990
References
Sources
Shiing-Shen Chern, Thomas Kailath, Bertram Kostant, Calvin C. Moore, Anna Tsao, Louis Auslander (1928–1997), Notices of the American Mathematical Society, volume 45, number 3, March 1998
External links
1928 births
1997 deaths
20th-century American mathematicians
University of Chicago alumni
CUNY Graduate Center faculty
Purdue University faculty
Institute for Advanced Study visiting scholars
People from Brooklyn
Mathematicians from New York (state) |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20NK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players for the football club Dinamo Zagreb. It also lists all matches that Dinamo Zagreb played in the 2006–07 season.
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Matches
Player seasonal records
Competitive matches only. Updated to games played 26 May 2007.
Goalscorers
Source: Competitive matches
References
External links
Dinamo Zagreb official website
GNK Dinamo Zagreb seasons
Dinamo Zagreb
Croatian football championship-winning seasons |
https://en.wikipedia.org/wiki/Cross-validation | Cross-validation may refer to:
Cross-validation (statistics), a technique for estimating the performance of a predictive model
Cross-validation (analytical chemistry), the practice of confirming an experimental finding by repeating the experiment using an independent assay technique
See also
Validation (disambiguation) |
https://en.wikipedia.org/wiki/Congruence%20of%20triangles | Congruence of triangles may refer to:
Congruence (geometry)#Congruence of triangles
Solution of triangles |
https://en.wikipedia.org/wiki/Clifford%20parallel | In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in spaces of at least three dimensions. Since parallel lines have the property of equidistance, the term "parallel" was appropriated from Euclidean geometry, although the "lines" of elliptic geometry are geodesic curves and, unlike the lines of Euclidean geometry, are of finite length.
The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.
Introduction
The lines on 1 in elliptic space are described by versors with a fixed axis r:
For an arbitrary point u in elliptic space, two Clifford parallels to this line pass through u.
The right Clifford parallel is
and the left Clifford parallel is
Generalized Clifford parallelism
Clifford's original definition was of curved parallel lines, but the concept generalizes to Clifford parallel objects of more than one dimension. In 4-dimensional Euclidean space Clifford parallel objects of 1, 2, 3 or 4 dimensions are related by isoclinic rotations. Clifford parallelism and isoclinic rotations are closely related aspects of the SO(4) symmetries which characterize the regular 4-polytopes.
Clifford surfaces
Rotating a line about another, to which it is Clifford parallel, creates a Clifford surface.
The Clifford parallels through points on the surface all lie in the surface. A Clifford surface is thus a ruled surface since every point is on two lines, each contained in the surface.
Given two square roots of minus one in the quaternions, written r and s, the Clifford surface through them is given by
History
Clifford parallels were first described in 1873 by the English mathematician William Kingdon Clifford.
In 1900 Guido Fubini wrote his doctoral thesis on Clifford's parallelism in elliptic spaces.
In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map.
In 2016 Hans Havlicek showed that there is a one-to-one correspondence between Clifford parallelisms and planes external to the Klein quadric.
See also
Clifford torus
Regular 4-polytopes
Citations
References
Laptev, B.L. & B.A. Rozenfel'd (1996) Mathematics of the 19th Century: Geometry, page 74, Birkhäuser Verlag .
Duncan Sommerville (1914) The Elements of Non-Euclidean Geometry, page 108 Paratactic lines, George Bell & Sons
Frederick S. Woods (1917) Higher Geometry, "Clifford parallels", page 255, via Internet Archive
Non-Euclidean geometry
Quaternions |
https://en.wikipedia.org/wiki/Quasi-analytic%20function | In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an interval [a,b] ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of [a,b]. Quasi-analytic classes are broader classes of functions for which this statement still holds true.
Definitions
Let be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions CM([a,b]) is defined to be those f ∈ C∞([a,b]) which satisfy
for all x ∈ [a,b], some constant A, and all non-negative integers k. If Mk = 1 this is exactly the class of real analytic functions on [a,b].
The class CM([a,b]) is said to be quasi-analytic if whenever f ∈ CM([a,b]) and
for some point x ∈ [a,b] and all k, then f is identically equal to zero.
A function f is called a quasi-analytic function if f is in some quasi-analytic class.
Quasi-analytic functions of several variables
For a function and multi-indexes , denote , and
and
Then is called quasi-analytic on the open set if for every compact there is a constant such that
for all multi-indexes and all points .
The Denjoy-Carleman class of functions of variables with respect to the sequence on the set can be denoted , although other notations abound.
The Denjoy-Carleman class is said to be quasi-analytic when the only function in it having all its partial derivatives equal to zero at a point is the function identically equal to zero.
A function of several variables is said to be quasi-analytic when it belongs to a quasi-analytic Denjoy-Carleman class.
Quasi-analytic classes with respect to logarithmically convex sequences
In the definitions above it is possible to assume that and that the sequence is non-decreasing.
The sequence is said to be logarithmically convex, if
is increasing.
When is logarithmically convex, then is increasing and
for all .
The quasi-analytic class with respect to a logarithmically convex sequence satisfies:
is a ring. In particular it is closed under multiplication.
is closed under composition. Specifically, if and , then .
The Denjoy–Carleman theorem
The Denjoy–Carleman theorem, proved by after gave some partial results, gives criteria on the sequence M under which CM([a,b]) is a quasi-analytic class. It states that the following conditions are equivalent:
CM([a,b]) is quasi-analytic.
where .
, where Mj* is the largest log convex sequence bounded above by Mj.
The proof that the last two conditions are equivalent to the second uses Carleman's inequality.
Example: pointed out that if Mn is given by one of the sequences
then the corresponding class is quasi-analytic. The first sequence gives analytic functions.
Additional properties
For a logarithmically convex sequence the following properties of the corresponding class of functions hold:
contains the analytic functions, and it is equal to it if and only i |
https://en.wikipedia.org/wiki/Real%20form%20%28Lie%20theory%29 | In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0:
The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan.
Real forms for Lie groups and algebraic groups
Using the Lie correspondence between Lie groups and Lie algebras, the notion of a real form can be defined for Lie groups. In the case of linear algebraic groups, the notions of complexification and real form have a natural description in the language of algebraic geometry.
Classification
Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified by Satake diagrams, which are obtained from the Dynkin diagram of the complex form by labeling some vertices black (filled), and connecting some other vertices in pairs by arrows, according to certain rules.
It is a basic fact in the structure theory of complex semisimple Lie algebras that every such algebra has two special real forms: one is the compact real form and corresponds to a compact Lie group under the Lie correspondence (its Satake diagram has all vertices blackened), and the other is the split real form and corresponds to a Lie group that is as far as possible from being compact (its Satake diagram has no vertices blackened and no arrows). In the case of the complex special linear group SL(n,C), the compact real form is the special unitary group SU(n) and the split real form is the real special linear group SL(n,R). The classification of real forms of semisimple Lie algebras was accomplished by Élie Cartan in the context of Riemannian symmetric spaces. In general, there may be more than two real forms.
Suppose that g0 is a semisimple Lie algebra over the field of real numbers. By Cartan's criterion, the Killing form is nondegenerate, and can be diagonalized in a suitable basis with the diagonal entries +1 or −1. By Sylvester's law of inertia, the number of positive entries, or the positive index of inertia, is an invariant of the bilinear form, i.e. it does not depend on the choice of the diagonalizing basis. This is a number between 0 and the dimension of g which is an important invariant of the real Lie algebra, called its index.
Split real form
A real form g0 of a finite-dimensional complex semisimple Lie algebra g is said to be split, or normal, if in each Cartan decomposition g0 = k0 ⊕ p0, the space p0 contains a maximal abelian subalgebra of g0, i.e. its Cartan subalgebra. Élie Cartan proved that every complex semisimple Lie algebra g has a split real form, which is unique up to isomorphism. It has maximal index among all real forms.
The split form corresponds to the Satake diagram with no vertices blackened and no arrows.
Compact real form
A real Lie algebra g0 |
https://en.wikipedia.org/wiki/Coarse%20space%20%28numerical%20analysis%29 | This article deals with a component of numerical methods. For coarse space in topology, see coarse structure.
In numerical analysis, coarse problem is an auxiliary system of equations used in an iterative method for the solution of a given larger system of equations. A coarse problem is basically a version of the same problem at a lower resolution, retaining its essential characteristics, but with fewer variables. The purpose of the coarse problem is to propagate information throughout the whole problem globally.
In multigrid methods for partial differential equations, the coarse problem is typically obtained as a discretization of the same equation on a coarser grid (usually, in finite difference methods) or by a Galerkin approximation on a subspace, called a coarse space. In finite element methods, the Galerkin approximation is typically used, with the coarse space generated by larger elements on the same domain. Typically, the coarse problem corresponds to a grid that is twice or three times coarser.
Coarse spaces (coarse model, surrogate model) are the backbone of algorithms and methodologies exploiting the space mapping concept for solving computationally intensive engineering modeling and design problems. In space mapping, a fine or high fidelity (high resolution, computationally intensive) model is used to calibrate or recalibrate—or update on the fly, as in aggressive space mapping—a suitable coarse model. An updated coarse model is often referred to as surrogate model or mapped coarse model. It permits fast, but more accurate, harnessing of the underlying coarse model in the exploration of designs or in design optimization.
In domain decomposition methods, the construction of a coarse problem follows the same principles as in multigrid methods, but the coarser problem has much fewer unknowns, generally only one or just a few unknowns per subdomain or substructure, and the coarse space can be of a quite different type that the original finite element space, e.g. piecewise constants with averaging in balancing domain decomposition or built from energy minimal functions in BDDC. The construction of the coarse problem in FETI is unusual in that it is not obtained as a Galerkin approximation of the original problem, however.
In Algebraic Multigrid Methods and in iterative aggregation methods in mathematical economics and Markov chains, the coarse problem is generally obtained by the Galerkin approximation on a subspace. In mathematical economics, the coarse problem may be obtained by the aggregation of products or industries into a coarse description with fewer variables. In Markov chains, a coarse Markov chain may be obtained by aggregating states.
The speed of convergence of multigrid and domain decomposition methods for elliptic partial differential equations without a coarse problem deteriorates with decreasing mesh step (or decreasing element size, or increasing number of subdomains or substructures), thus making a coarse problem |
https://en.wikipedia.org/wiki/Post-test%20odds | Post-test odds may refer to:
Bayes' theorem in terms of odds and likelihood ratio
Post test odds as related to pre- and post-test probability |
https://en.wikipedia.org/wiki/Monus | In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the − symbol because the natural numbers are a CMM under subtraction; it is also denoted with the symbol to distinguish it from the standard subtraction operator.
Notation
Definition
Let be a commutative monoid. Define a binary relation on this monoid as follows: for any two elements and , define if there exists an element such that . It is easy to check that is reflexive and that it is transitive. is called naturally ordered if the relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements and , a unique smallest element exists such that , then is called a commutative monoid with monus and the monus of any two elements and can be defined as this unique smallest element such that .
An example of a commutative monoid that is not naturally ordered is , the commutative monoid of the integers with usual addition, as for any there exists such that , so holds for any , so is not a partial order. There are also examples of monoids that are naturally ordered but are not semirings with monus.
Other structures
Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid) is a semiring where the commutative monoid induced by the addition operator is naturally ordered. When this monoid is a commutative monoid with monus, the semiring is called a semiring with monus, or m-semiring.
Examples
If is an ideal in a Boolean algebra, then is a commutative monoid with monus under and .
Natural numbers
The natural numbers including 0 form a commutative monoid with monus, with their ordering being the usual order of natural numbers and the monus operator being a saturating variant of standard subtraction, variously referred to as truncated subtraction, limited subtraction, proper subtraction, doz (difference or zero), and monus. Truncated subtraction is usually defined as
where − denotes standard subtraction. For example, 5 − 3 = 2 and 3 − 5 = −2 in regular subtraction, whereas in truncated subtraction 3 ∸ 5 = 0. Truncated subtraction may also be defined as
In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function (the inverse of the successor function):
A definition that does not need the predecessor function is:
Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers. Truncated subtraction is also used in the definition of the multiset difference operator.
Properties
The class of all commutative monoids with monus form a variety. The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:
Notes
Alg |
https://en.wikipedia.org/wiki/Symmetric%20Boolean%20function | In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones (or zeros) in the input. For this reason they are also known as Boolean counting functions.
There are 2n+1 symmetric n-ary Boolean functions. Instead of the truth table, traditionally used to represent Boolean functions, one may use a more compact representation for an n-variable symmetric Boolean function: the (n + 1)-vector, whose i-th entry (i = 0, ..., n) is the value of the function on an input vector with i ones. Mathematically, the symmetric Boolean functions correspond one-to-one with the functions that map n+1 elements to two elements, .
Symmetric Boolean functions are used to classify Boolean satisfiability problems.
Special cases
A number of special cases are recognized:
Majority function: their value is 1 on input vectors with more than n/2 ones
Threshold functions: their value is 1 on input vectors with k or more ones for a fixed k
All-equal and not-all-equal function: their values is 1 when the inputs do (not) all have the same value
Exact-count functions: their value is 1 on input vectors with k ones for a fixed k
One-hot or 1-in-n function: their value is 1 on input vectors with exactly one one
One-cold function: their value is 1 on input vectors with exactly one zero
Congruence functions: their value is 1 on input vectors with the number of ones congruent to k mod m for fixed k, m
Parity function: their value is 1 if the input vector has odd number of ones
The n-ary versions of AND, OR, XOR, NAND, NOR and XNOR are also symmetric Boolean functions.
Properties
In the following, denotes the value of the function when applied to an input vector of weight .
Weight
The weight of the function can be calculated from its value vector:
Algebraic normal form
The algebraic normal form either contains all monomials of certain order , or none of them; i.e. the Möbius transform of the function is also a symmetric function. It can thus also be described by a simple (n+1) bit vector, the ANF vector . The ANF and value vectors are related by a Möbius relation:where denotes all the weights k whose base-2 representation is covered by the base-2 representation of m (a consequence of Lucas’ theorem). Effectively, an n-variable symmetric Boolean function corresponds to a log(n)-variable ordinary Boolean function acting on the base-2 representation of the input weight.
For example, for three-variable functions:
So the three variable majority function with value vector (0, 0, 1, 1) has ANF vector (0, 0, 1, 0), i.e.:
Unit hypercube polynomial
The coefficients of the real polynomial agreeing with the function on are given by:For example, the three variable majority function polynomial has coefficients (0, 0, 1, -2):
Examples
See also
Symmetric function
References
Boolean algebra
Cryptography |
https://en.wikipedia.org/wiki/Parity%20function | In Boolean algebra, a parity function is a Boolean function whose value is one if and only if the input vector has an odd number of ones. The parity function of two inputs is also known as the XOR function.
The parity function is notable for its role in theoretical investigation of circuit complexity of Boolean functions.
The output of the parity function is the parity bit.
Definition
The -variable parity function is the Boolean function with the property that if and only if the number of ones in the vector is odd.
In other words, is defined as follows:
where denotes exclusive or.
Properties
Parity only depends on the number of ones and is therefore a symmetric Boolean function.
The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 n − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n.
Computational complexity
Some of the earliest work in computational complexity was 1961 bound of Bella Subbotovskaya showing the size of a Boolean formula computing parity must be at least . This work uses the method of random restrictions. This exponent of has been increased through careful analysis to by Paterson and Zwick (1993) and then to by Håstad (1998).
In the early 1980s, Merrick Furst, James Saxe and Michael Sipser and independently Miklós Ajtai established super-polynomial lower bounds on the size of constant-depth Boolean circuits for the parity function, i.e., they showed that polynomial-size constant-depth circuits cannot compute the parity function. Similar results were also established for the majority, multiplication and transitive closure functions, by reduction from the parity function.
established tight exponential lower bounds on the size of constant-depth Boolean circuits for the parity function. Håstad's Switching Lemma is the key technical tool used for these lower bounds and Johan Håstad was awarded the Gödel Prize for this work in 1994.
The precise result is that depth- circuits with AND, OR, and NOT gates require size to compute the parity function.
This is asymptotically almost optimal as there are depth- circuits computing parity which have size .
Infinite version
An infinite parity function is a function mapping every infinite binary string to 0 or 1, having the following property: if and are infinite binary strings differing only on finite number of coordinates then if and only if and differ on even number of coordinates.
Assuming axiom of choice it can be easily proved that parity functions exist and there are many of them - as many as the number of all functions from to . It is enough to take one representative per equivalence class of relation defined as follows: if and differ at finite number of coordinates. Having such representatives, we can map all of them to 0; the rest of values are deducted unambiguously.
Infinite parity functions ar |
https://en.wikipedia.org/wiki/1510%20%28number%29 | 1510 (one thousand five hundred [and] ten) is the natural number following 1509 and preceding 1511.
In mathematics
1510 is an even number.
1510 is a composite number.
1510 is a deficient number.
1510 is an odious number.
1510 is an apocalyptic power (21510 contains the consecutive digits 666).
1510 is a square-free integer.
1510 is an untouchable number.
1510 is a subprime number (1510 + 1 = 1511, which is a prime).
1510 is one less than an equidigital number.
If one leading zero is added to 1510, it becomes a palindromic number.
References to 1510
It is common knowledge that the character limit for messages sent using Pidgin is 1510.
1510 kHz was the broadcast frequency for WPGR, Pittsburgh's Gospel Radio station.
1510 Charlois is the name of an asteroid.
Tata 1510/1512 is the largest selling bus model seen regularly in India and neighboring countries and the Seychelles.
1510 is the ID of the USS Walter D. Munson
See also
1510 AD
Number
References
External links
The Positive Integer 1510
Number Gossip: 1510
Integers |
https://en.wikipedia.org/wiki/Commutant-associative%20algebra | In abstract algebra, a commutant-associative algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom:
,
where [A, B] = AB − BA is the commutator of A and B and
(A, B, C) = (AB)C – A(BC) is the associator of A, B and C.
In other words, an algebra M is commutant-associative if the commutant, i.e. the subalgebra of M generated by all commutators [A, B], is an associative algebra.
See also
Valya algebra
Malcev algebra
Alternative algebra
References
A. Elduque, H. C. Myung Mutations of alternative algebras, Kluwer Academic Publishers, Boston, 1994,
M.V. Karasev, V.P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization. American Mathematical Society, Providence, 1993.
A.G. Kurosh, Lectures on general algebra. Translated from the Russian edition (Moscow, 1960) by K. A. Hirsch. Chelsea, New York, 1963. 335 pp.
A.G. Kurosh, General algebra. Lectures for the academic year 1969/70. Nauka, Moscow,1974. (In Russian)
A.I. Mal'tsev, Algebraic systems. Springer, 1973. (Translated from Russian)
A.I. Mal'tsev, Analytic loops. Mat. Sb., 36 : 3 (1955) pp. 569–576 (In Russian)
V.E. Tarasov, "Quantum dissipative systems: IV. Analogues of Lie algebras and groups" Theoretical and Mathematical Physics. Vol.110. No.2. (1997) pp.168-178.
V.E. Tarasov Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Elsevier Science, Amsterdam, Boston, London, New York, 2008.
Non-associative algebras |
https://en.wikipedia.org/wiki/Valya%20algebra | In abstract algebra, a Valya algebra (or Valentina algebra) is a nonassociative algebra M over a field F whose multiplicative binary operation g satisfies the following axioms:
1. The skew-symmetry condition
for all .
2. The Valya identity
for all , where k=1,2,...,6, and
3. The bilinear condition
for all and .
We say that M is a Valya algebra if the commutant of this algebra is a Lie subalgebra. Each Lie algebra is a Valya algebra.
There is the following relationship between the commutant-associative algebra and Valentina algebra. The replacement of the multiplication g(A,B) in an algebra M by the operation of commutation [A,B]=g(A,B)-g(B,A), makes it into the algebra .
If M is a commutant-associative algebra, then is a Valya algebra. A Valya algebra is a generalization of a Lie algebra.
Examples
Let us give the following examples regarding Valya algebras.
(1) Every finite Valya algebra is the tangent algebra of an analytic local commutant-associative loop (Valya loop) as each finite Lie algebra is the tangent algebra of an analytic local group (Lie group). This is the analog of the classical correspondence between analytic local groups (Lie groups) and Lie algebras.
(2) A bilinear operation for the differential 1-forms
on a symplectic manifold can be introduced by the rule
where is 1-form. A set of all nonclosed 1-forms, together with this operation, is Lie algebra.
If and are closed 1-forms, then
and
A set of all closed 1-forms, together with this bracket, form a Lie algebra. A set of all nonclosed 1-forms together with the bilinear operation is a Valya algebra, and it is not a Lie algebra.
See also
Malcev algebra
Alternative algebra
Commutant-associative algebra
References
A. Elduque, H. C. Myung Mutations of alternative algebras, Kluwer Academic Publishers, Boston, 1994,
M.V. Karasev, V.P. Maslov, Nonlinear Poisson Brackets: Geometry and Quantization. American Mathematical Society, Providence, 1993.
A.G. Kurosh, Lectures on general algebra. Translated from the Russian edition (Moscow, 1960) by K. A. Hirsch. Chelsea, New York, 1963. 335 pp.
A.G. Kurosh, General algebra. Lectures for the academic year 1969/70. Nauka, Moscow,1974. (In Russian)
A.I. Mal'tsev, Algebraic systems. Springer, 1973. (Translated from Russian)
A.I. Mal'tsev, Analytic loops. Mat. Sb., 36 : 3 (1955) pp. 569–576 (In Russian)
V.E. Tarasov Quantum Mechanics of Non-Hamiltonian and Dissipative Systems. Elsevier Science, Amsterdam, Boston, London, New York, 2008.
V.E. Tarasov, "Quantum dissipative systems: IV. Analogues of Lie algebras and groups" Theoretical and Mathematical Physics. Vol.110. No.2. (1997) pp.168-178.
Non-associative algebras
Lie algebras |
https://en.wikipedia.org/wiki/Function%20field%20sieve | In mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic subexponential complexity. Leonard Adleman developed it in 1994 and then elaborated it together with M. D. Huang in 1999.
Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two.
The discrete logarithm problem in a finite field consists of solving the equation for , a prime number and an integer. The function for a fixed is a one-way function used in cryptography. Several cryptographic methods are based on the DLP such as the Diffie-Hellman key exchange, the El Gamal cryptosystem and the Digital Signature Algorithm.
Number theoretical background
Function Fields
Let be a polynomial defining an algebraic curve over a finite field . A function field may be viewed as the field of fractions of the affine coordinate ring , where denotes the ideal generated by . This is a special case of an algebraic function field. It is defined over the finite field and has transcendence degree one. The transcendent element will be denoted by .
There exist bijections between valuation rings in function fields and equivalence classes of places, as well as between valuation rings and equivalence classes of valuations. This correspondence is frequently used in the Function Field Sieve algorithm.
Divisors
A discrete valuation of the function field , namely a discrete valuation ring , has a unique maximal ideal called a prime of the function field. The degree of is and we also define .
A divisor is a -linear combination over all primes, so where and only finitely many elements of the sum are non-zero. The divisor of an element is defined as , where is the valuation corresponding to the prime .
The degree of a divisor is .
Method
The Function Field Sieve algorithm consists of a precomputation where the discrete logarithms of irreducible polynomials of small degree are found and a reduction step where they are combined to the logarithm of .
Functions that decompose into irreducible function of degree smaller than some bound are called -smooth. This is analogous to the definition of a smooth number and such functions are useful because their decomposition can be found relatively fast. The set of those functions is called the factor base.
A pair of functions is doubly-smooth if and are both smooth, where is the norm of an element of over , is some parameter and is viewed as an element of the function field of .
The sieving step of the algorithm consists of finding doubly-smooth pairs of functions. In the subsequent step we use them to find linear relations including the logarithms of the functions in the decompositions. By solving a linear system we then calculate the logarithms.
In the reduction step we express as a combination of the logarithm we found before and thus solve the DLP.
Precomputation
Parameter selection
The algorith |
https://en.wikipedia.org/wiki/Kemnitz%27s%20conjecture | In additive number theory, Kemnitz's conjecture states that every set of lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher, then an undergraduate student, and Carlos di Fiore, then a high school student.
The exact formulation of this conjecture is as follows:
Let be a natural number and a set of lattice points in plane. Then there exists a subset with points such that the centroid of all points from is also a lattice point.
Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that every integers have a subset of size whose average is an integer. In 2000, Lajos Rónyai proved a weakened form of Kemnitz's conjecture for sets with lattice points. Then, in 2003, Christian Reiher proved the full conjecture using the Chevalley–Warning theorem.
References
Further reading
Theorems in discrete mathematics
Lattice points
Combinatorics
Conjectures that have been proved |
https://en.wikipedia.org/wiki/Randall%20J.%20LeVeque | Randall J. LeVeque is a Professor of Applied Mathematics at University of Washington who works in many fields including numerical analysis, computational fluid dynamics, and mathematical theory of conservation laws. Among other contributions, he is lead developer of the open source software project Clawpack for solving hyperbolic partial differential equations using the finite volume method. With Zhilin Li, he has also devised a numerical technique called the immersed interface method for solving problems with elastic boundaries or surface tension.
He was an invited speaker at the 2006 International Congress of Mathematicians held in Madrid. He became a fellow of the Society for Industrial and Applied Mathematics in 2010, fellow of the American Mathematical Society in 2013, and a member of the National Academy of Sciences in 2021.
LeVeque is a son of the well-known mathematician William J. LeVeque.
Education and Career
LeVeque received his B.A. in mathematics from University of California, San Diego in 1977. He then continued to Stanford University to get his Ph.D. in computer science in 1982. Following a postdoctoral fellowship at the Courant Institute and the Hedrick Assistant Professorship at University of California, Los Angeles, he has been a faculty member at the University of Washington since 1985. He has advised twenty three PhD students.
Books
LeVeque has authored several textbooks and monographs:
Finite Volume Methods for Hyperbolic Problems, Cambridge University Press (2002).
Numerical Methods for Conservation Laws, 1st ed. (1992), 2nd ed., Birkhäuser Basel (2005).
Computational Methods for Astrophysical Fluid Flow, Springer (1998).
Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, SIAM (2007).
References
External links
20th-century American mathematicians
21st-century American mathematicians
Computational fluid dynamicists
University of Washington faculty
University of California, San Diego alumni
Stanford University alumni
Fellows of the American Mathematical Society
Living people
Fellows of the Society for Industrial and Applied Mathematics
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Abstract%20differential%20geometry | The adjective abstract has often been applied to differential geometry before, but the abstract differential geometry (ADG) of this article is a form of differential geometry without the calculus notion of smoothness, developed by Anastasios Mallios and Ioannis Raptis from 1998 onwards.
Instead of calculus, an axiomatic treatment of differential geometry is built via sheaf theory and sheaf cohomology using vector sheaves in place of bundles based on arbitrary topological spaces. Mallios says noncommutative geometry can be considered a special case of ADG, and that ADG is similar to synthetic differential geometry.
Applications
ADG Gravity
Mallios and Raptis use ADG to avoid the singularities in general relativity and propose this as a route to quantum gravity.
See also
Discrete differential geometry
Analysis on fractals
References
Further reading
Space-time foam dense singularities and de Rham cohomology, A Mallios, EE Rosinger, Acta Applicandae Mathematicae, 2001
Differential geometry
Sheaf theory
General relativity
Quantum gravity |
https://en.wikipedia.org/wiki/Clifford%20analysis | Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis and geometry, together with their applications. Examples of Dirac type operators include, but are not limited to, the Hodge–Dirac operator, on a Riemannian manifold, the Dirac operator in euclidean space and its inverse on and their conformal equivalents on the sphere, the Laplacian in euclidean n-space and the Atiyah–Singer–Dirac operator on a spin manifold, Rarita–Schwinger/Stein–Weiss type operators, conformal Laplacians, spinorial Laplacians and Dirac operators on SpinC manifolds, systems of Dirac operators, the Paneitz operator, Dirac operators on hyperbolic space, the hyperbolic Laplacian and Weinstein equations.
Euclidean space
In Euclidean space the Dirac operator has the form
where e1, ..., en is an orthonormal basis for Rn, and Rn is considered to be embedded in a complex Clifford algebra, Cln(C) so that .
This gives
where Δn is the Laplacian in n-euclidean space.
The fundamental solution to the euclidean Dirac operator is
where ωn is the surface area of the unit sphere Sn−1.
Note that
where
is the fundamental solution to Laplace's equation for .
The most basic example of a Dirac operator is the Cauchy–Riemann operator
in the complex plane. Indeed, many basic properties of one variable complex analysis follow through for many first order Dirac type operators. In euclidean space this includes a Cauchy Theorem, a Cauchy integral formula, Morera's theorem, Taylor series, Laurent series and Liouville Theorem. In this case the Cauchy kernel is G(x−y). The proof of the Cauchy integral formula is the same as in one complex variable and makes use of the fact that each non-zero vector x in euclidean space has a multiplicative inverse in the Clifford algebra, namely
Up to a sign this inverse is the Kelvin inverse of x. Solutions to the euclidean Dirac equation Df = 0 are called (left) monogenic functions. Monogenic functions are special cases of harmonic spinors on a spin manifold.
In 3 and 4 dimensions Clifford analysis is sometimes referred to as quaternionic analysis. When , the Dirac operator is sometimes referred to as the Cauchy–Riemann–Fueter operator. Further some aspects of Clifford analysis are referred to as hypercomplex analysis.
Clifford analysis has analogues of Cauchy transforms, Bergman kernels, Szegő kernels, Plemelj operators, Hardy spaces, a Kerzman–Stein formula and a Π, or Beurling–Ahlfors, transform. These have all found applications in solving boundary value problems, including moving boundary value problems, singular integrals and classic harmonic analysis. In particular Clifford analysis has been used to solve, in certain Sobolev spaces, the full water wave problem in 3D. This method works in all dimensions greater than 2.
Much of Clifford analysis works if we replace the complex Clifford algebra by a real Clifford algebra, Cln. This is not the |
https://en.wikipedia.org/wiki/Chris%20Tofts | Chris M. N. Tofts (born 1964) is an English computer scientist.
Education
Chris Tofts studied mathematics as an undergraduate at Clare College, Cambridge, followed by a Diploma in Computer Science from the same college. He went on to do a PhD supervised by Robin Milner in the Laboratory for Foundations of Computer Science at the University of Edinburgh, Scotland.
Career
Tofts' postdoctorate research saw some of the first applications of process algebra to the study of the behaviour of animals and disease processes, which led to his interest in the correctness of simulation models.
Tofts held lectureships at Swansea University (1992–94), the University of Manchester (1994–96), and the University of Leeds (1996–99). From 1999 to 2008 he was a scientist at Hewlett-Packard (HP) Research Laboratories in the UK. From 2008 to 2011 he was the Chief Mathematics Officer of Concinnitas Ltd before returning to HP.
Chris Tofts is a visiting Professor of Computer Science at Swansea University. He is a Fellow of the British Computer Society and Fellow of the Institute of Mathematics and its Applications, as well as a past President of the BCTCS.
Books
Chris Tofts, Concurrency, Complexity and Performance, Springer, 2007. .
References
External links
Chris Tofts — Publications and Patents
1964 births
Living people
Alumni of Clare College, Cambridge
Alumni of the University of Edinburgh
Academics of Swansea University
Academics of the University of Manchester
Academics of the University of Leeds
Hewlett-Packard people
English computer scientists
English science writers
Formal methods people
Fellows of the British Computer Society
Fellows of the Institute of Mathematics and its Applications |
https://en.wikipedia.org/wiki/Tur%C3%A1n%E2%80%93Kubilius%20inequality | The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic function. The theorem was proved in a special case in 1934 by Pál Turán and generalized in 1956 and 1964 by Jonas Kubilius.
Statement of the theorem
This formulation is from Tenenbaum. Other formulations are in Narkiewicz
and in Cojocaru & Murty.
Suppose f is an additive complex-valued arithmetic function, and write p for an arbitrary prime and for an arbitrary positive integer. Write
and
Then there is a function ε(x) that goes to zero when x goes to infinity, and such that for x ≥ 2 we have
Applications of the theorem
Turán developed the inequality to create a simpler proof of the Hardy–Ramanujan theorem about the normal order of the number ω(n) of distinct prime divisors of an integer n. There is an exposition of Turán's proof in Hardy & Wright, §22.11.
Tenenbaum gives a proof of the Hardy–Ramanujan theorem using the Turán–Kubilius inequality and states without proof several other applications.
Notes
Inequalities
Theorems in number theory |
https://en.wikipedia.org/wiki/Eli%20Upfal |
Eli Upfal is a computer science researcher, currently the Rush C. Hawkins Professor of Computer Science at Brown University. He completed his undergraduate studies in mathematics and statistics at the Hebrew University, Israel in 1978, received an M.Sc. in computer science from the Feinberg Graduate School of the Weizmann Institute of Science, Israel in 1980, and completed his PhD in computer science at the Hebrew University in 1983 under Eli Shamir. He has made contributions in a variety of areas. Most of his work involves randomized and/or online algorithms, stochastic processes, or the probabilistic analysis of deterministic algorithms. Particular applications include routing and communications networks, computational biology, and computational finance.
He is responsible for a large body of work, including, as of May 2012, more than 150 publications in journals and conferences as well as many patents. He has won several prizes, including the IBM Outstanding Innovation Award and the Levinson Prize in Mathematical Sciences. In 2002, Eli Upfal, was inducted as a Fellow of the Institute of Electrical and Electronics Engineers, and in 2005 he was inducted as a Fellow of the Association for Computing Machinery. He received, together with Yossi Azar, Andrei Broder, Anna Karlin, and Michael Mitzenmacher, the 2020 ACM Paris Kanellakis Award.
Eli is a coauthor of the book
References
External links
Eli Upfal's website
Living people
Theoretical computer scientists
Fellows of the Association for Computing Machinery
Fellow Members of the IEEE
Israeli computer scientists
Brown University faculty
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Group%20family | In probability theory, especially as that field is used in statistics, a group family of probability distributions is a family obtained by subjecting a random variable with a fixed distribution to a suitable family of transformations such as a location-scale family, or otherwise a family of probability distributions acted upon by a group.
Consideration of a particular family of distributions as a group family can, in statistical theory, lead to the identification of an ancillary statistic.
Types of group families
A group family can be generated by subjecting a random variable with a fixed distribution to some suitable transformations. Different types of group families are as follows :
Location Family
This family is obtained by adding a constant to a random variable. Let be a random variable and be a constant. Let . Then For a fixed distribution , as varies from to , the distributions that we obtain constitute the location family.
Scale Family
This family is obtained by multiplying a random variable with a constant. Let be a random variable and be a constant. Let . Then
Location - Scale Family
This family is obtained by multiplying a random variable with a constant and then adding some other constant to it. Let be a random variable , and be constants. Let . Then
Note that it is important that and in order to satisfy the properties mentioned in the following section.
Properties of the transformations
The transformation applied to the random variable must satisfy the following properties.
Closure under composition
Closure under inversion
References
Parametric statistics
Types of probability distributions |
https://en.wikipedia.org/wiki/Mean%20value%20analysis | In queueing theory, a discipline within the mathematical theory of probability, mean value analysis (MVA) is a recursive technique for computing expected queue lengths, waiting time at queueing nodes and throughput in equilibrium for a closed separable system of queues. The first approximate techniques were published independently by Schweitzer and Bard, followed later by an exact version by Lavenberg and Reiser published in 1980.
It is based on the arrival theorem, which states that when one customer in an M-customer closed system arrives at a service facility he/she observes the rest of the system to be in the equilibrium state for a system with M − 1 customers.
Problem setup
Consider a closed queueing network of K M/M/1 queues, with M customers circulating in the system. Suppose that the customers are indistinguishable from each other, so that the network has a single class of customers. To compute the mean queue length and waiting time at each of the nodes and throughput of the system we use an iterative algorithm starting with a network with 0 customers.
Write μi for the service rate at node i and P for the customer routing matrix where element pij denotes the probability that a customer finishing service at node i moves to node j for service. To use the algorithm we first compute the visit ratio row vector v, a vector such that v = v P.
Now write Li(n) for the mean number of customer at queue i when there are a total of n customers in the system (this includes the job currently being served at queue i) and Wj(n) for the mean time spent by a customer in queue i when there are a total of n customers in the system. Denote the throughput of a system with m customers by λm.
Algorithm
The algorithm starts with an empty network (zero customers), then increases the number of customers by 1 at each iteration until there are the required number (M) of customers in the system.
To initialise, set Lk(0) = 0 for k = 1,...,K. (This sets the average queue length in a system with no customers to zero at all nodes.)
Repeat for m = 1,...,M:
1. For k = 1, ..., K compute the waiting time at each node using the arrival theorem
2. Then compute the system throughput using Little's law
3. Finally, use Little's law applied to each queue to compute the mean queue lengths for k = 1, ..., K
End repeat.
Bard–Schweitzer method
The Bard–Schweitzer approximation estimates the average number of jobs at node to be
which is a linear interpolation. From the above formulas, this approximation yields fixed-point relationships which can be solved numerically. This iterative approach often goes under the name of approximate MVA (AMVA) and it is typically faster than the recursive approach of MVA.
Pseudocode
Multiclass networks
In the case of multiclass networks with R classes of customers, each queue k can feature different service rates μk,r for each job class r=1,...,R, although certain restrictions exist in the case of first-come first-served stations due to |
https://en.wikipedia.org/wiki/Invex%20function | In vector calculus, an invex function is a differentiable function from to for which there exists a vector valued function such that
for all x and u.
Invex functions were introduced by Hanson as a generalization of convex functions. Ben-Israel and Mond provided a simple proof that a function is invex if and only if every stationary point is a global minimum, a theorem first stated by Craven and Glover.
Hanson also showed that if the objective and the constraints of an optimization problem are invex with respect to the same function , then the Karush–Kuhn–Tucker conditions are sufficient for a global minimum.
Type I invex functions
A slight generalization of invex functions called Type I invex functions are the most general class of functions for which the Karush–Kuhn–Tucker conditions are necessary and sufficient for a global minimum. Consider a mathematical program of the form
where and are differentiable functions. Let denote the feasible region of this program. The function is a Type I objective function and the function is a Type I constraint function at with respect to if there exists a vector-valued function defined on such that
and
for all . Note that, unlike invexity, Type I invexity is defined relative to a point .
Theorem (Theorem 2.1 in): If and are Type I invex at a point with respect to , and the Karush–Kuhn–Tucker conditions are satisfied at , then is a global minimizer of over .
See also
Convex function
Pseudoconvex function
Quasiconvex function
References
Further reading
S. K. Mishra and G. Giorgi, Invexity and optimization, Nonconvex Optimization and Its Applications, Vol. 88, Springer-Verlag, Berlin, 2008.
S. K. Mishra, S.-Y. Wang and K. K. Lai, Generalized Convexity and Vector Optimization, Springer, New York, 2009.
Convex analysis
Generalized convexity
Real analysis
Types of functions |
https://en.wikipedia.org/wiki/Louis%20Billera | Louis Joseph Billera is a Professor of Mathematics at Cornell University.
Career
Billera completed his B.S. at the Rensselaer Polytechnic Institute in 1964. He earned his Ph.D. from the Graduate Center of the City University of New York in 1968, under the joint supervision of Moses Richardson and Michel Balinski.
Louis Billera served as the first Associate Director of the National Science Foundation Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) at Rutgers University.
In 2010 he gave the invited lecture, "Flag enumeration in polytopes, Eulerian partially ordered sets and Coxeter groups" at the International Congress of Mathematicians in Hyderabad.
Contributions
The common thread through much of his research is to study problems motivated by discrete and convex geometry. A sampling includes constructing polytopes to prove the sufficiency condition for the g-theorem (with Carl Lee), discovering fiber polytopes (with Bernd Sturmfels), and studying the space of phylogenetic trees (with Susan Holmes and Karen Vogtmann).
Awards and honors
In 1994 Billera won the Fulkerson Prize for his paper, Homology of smooth splines. This prize is given every three years to the best paper in discrete mathematics.
In 2012 he became a fellow of the American Mathematical Society.
Selected publications
Louis Billera, "Homology of smooth splines: Generic triangulations and a conjecture of Strang", Transactions of the American Mathematical Society 310 (1988). 325–340.
Louis Billera, Anders Björner, Curtis Greene, Rodica Simion, Richard P. Stanley (eds.): New Perspectives in Algebraic Combinatorics, MSRI Publications, Cambridge University Press 1999, http://library.msri.org/books/Book38/
See also
Simplicial complex
Combinatorial commutative algebra
Quasisymmetric function
References
External links
Billerafest 2008, Conference in Honor of Louis Billera's 65th Birthday. http://www.math.cornell.edu/event/conf/billera65/
Louis Billera's Homepage
profile at Cornell University archived
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
Cornell University faculty
Fellows of the American Mathematical Society
Place of birth missing (living people)
CUNY Graduate Center alumni
Combinatorialists
Rensselaer Polytechnic Institute alumni |
https://en.wikipedia.org/wiki/Monk%27s%20formula | In mathematics, Monk's formula, found by , is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold.
Write tij for the transposition (i j), and si = ti,i+1. Then 𝔖sr = x1 + ⋯ + xr, and Monk's formula states that for a permutation w,
where is the length of w. The pairs (i, j) appearing in the sum are exactly those such that i ≤ r < j, wi < wj, and there is no i < k < j with wi < wk < wj; each wtij is a cover of w in Bruhat order.
References
Symmetric functions |
https://en.wikipedia.org/wiki/Offered%20load | In the mathematical theory of probability, offered load is a concept in queuing theory. The offered load is a measure of traffic in a queue. The offered load is given by Little's law: the arrival rate into the queue (symbolized with λ) multiplied by the mean holding time (symbolized by τ), equals the average amount of time spent by items in the queue. Offered load is expressed in Erlang units or call-seconds per hour, a dimensionless measure.
References
Robert B. Cooper. Introduction to Queuing theory. North Holland, 1981, Second edition. . Chapter 1, page 4.
Queueing theory |
https://en.wikipedia.org/wiki/Isophote | In geometry, an isophote is a curve on an illuminated surface that connects points of equal brightness. One supposes that the illumination is done by parallel light and the brightness is measured by the following scalar product:
where is the unit normal vector of the surface at point and the unit vector of the light's direction. If , i.e. the light is perpendicular to the surface normal, then point is a point of the surface silhouette observed in direction Brightness 1 means that the light vector is perpendicular to the surface. A plane has no isophotes, because every point has the same brightness.
In astronomy, an isophote is a curve on a photo connecting points of equal brightness.
Application and example
In computer-aided design, isophotes are used for checking optically the smoothness of surface connections. For a surface (implicit or parametric), which is differentiable enough, the normal vector depends on the first derivatives. Hence, the differentiability of the isophotes and their geometric continuity is 1 less than that of the surface. If at a surface point only the tangent planes are continuous (i.e. G1-continuous), the isophotes have there a kink (i.e. is only G0-continuous).
In the following example (s. diagram), two intersecting Bezier surfaces are blended by a third surface patch. For the left picture, the blending surface has only G1-contact to the Bezier surfaces and for the right picture the surfaces have G2-contact. This difference can not be recognized from the picture. But the geometric continuity of the isophotes show: on the left side, they have kinks (i.e. G0-continuity), and on the right side, they are smooth (i.e. G1-continuity).
Determining points of an isophote
on an implicit surface
For an implicit surface with equation the isophote condition is
That means: points of an isophote with given parameter are solutions of the non linear system
which can be considered as the intersection curve of two implicit surfaces. Using the tracing algorithm of Bajaj et al. (see references) one can calculate a polygon of points.
on a parametric surface
In case of a parametric surface the isophote condition is
which is equivalent to
This equation describes an implicit curve in the s-t-plane, which can be traced by a suitable algorithm (see implicit curve) and transformed by into surface points.
See also
Contour line
References
J. Hoschek, D. Lasser: Grundlagen der geometrischen Datenverarbeitung, Teubner-Verlag, Stuttgart, 1989, , p. 31.
Z. Sun, S. Shan, H. Sang et al.: Biometric Recognition, Springer, 2014, , p. 158.
C.L. Bajaj, C.M. Hoffmann, R.E. Lynch, J.E.H. Hopcroft: Tracing Surface Intersections, (1988) Comp. Aided Geom. Design 5, pp. 285–307.
C. T. Leondes: Computer Aided and Integrated Manufacturing Systems: Optimization methods, Vol. 3, World Scientific, 2003, , p. 209.
External links
Patrikalakis-Maekawa-Cho: Isophotes (engl.)
A. Diatta, P. Giblin: Geometry of Isophote Curves
Jin Kim: Computing I |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20First%20League%20of%20the%20Republika%20Srpska | This page details the statistics of the First League of the Republika Srpska in the 2001–02 season.
At the end of the season, the top six clubs joined the Premier League of Bosnia and Herzegovina, to form the first nationwide football league of Bosnia and Herzegovina.
Clubs and stadiums
League standings
See also
2001–02 Premier League of Bosnia and Herzegovina
External links
FSRS Official website
Srpska 1
2001–02 in Bosnia and Herzegovina football
First League of the Republika Srpska seasons |
https://en.wikipedia.org/wiki/L%C3%B6vsta%2C%20Gotland | Lövsta (also known as Roma kyrkby) is a locality on the Swedish island of Gotland, with 261 inhabitants in 2010.
In 1995 the locality known as Roma was divided by Statistics Sweden into a part with the tentative name of "Roma kyrkby" (pop. 277) and the remaining part that was referred to as Roma (pop. 902). It was given the name "Romakloster" as a postal address to avoid confusion with the Italian capital.
Some confusion is caused by the fact that Lövsta/Roma kyrkby has been referred to as "Roma" in the statistical figures since 2000.
The medieval Roma Church is in Lövsta. , Roma Church belongs to Roma parish in Romaklosters pastorat.
Notes
References
Tätorter 1995 (Statistiska meddelanden. Serie Be, Stockholm: SCB, Programmet för regional planering och naturresurser, 1996. Beställningsnummer Be 16 SM 9601.
Tätorter 2000. Sveriges Officiella statistik. Statistiska meddelanden Serie MI. MI 38 SM 0101. Stockholm: SCB, 2002.
External links
Populated places in Gotland County |
https://en.wikipedia.org/wiki/Wesley%20%28footballer%2C%20born%201981%29 | Wesley Barbosa De Morais (born 10 November 1981, in São Paulo), known simply as Wesley, is a Brazilian footballer who plays for Figueirense.
Career statistics
(Correct )
External links
Wesley at ogol.com.br
1981 births
Brazilian men's footballers
Men's association football forwards
Living people
K League 1 players
Jeonnam Dragons players
Gangwon FC players
Brazilian expatriate sportspeople in South Korea
Brazilian expatriate men's footballers
Expatriate men's footballers in South Korea
Itumbiara Esporte Clube players
Mirassol Futebol Clube players
Grêmio Barueri Futebol players
Clube Atlético Mineiro players
Atlético Clube Goianiense players
Figueirense FC players
Campeonato Brasileiro Série A players
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Kim%20Sung-joon%20%28footballer%29 | Kim Sung-joon (; Hanja: 金聖埈; born 8 April 1988) is a South Korean football player who plays for Ulsan Hyundai as a midfielder.
Career statistics
Club
Honours
Club
Ulsan Hyundai
AFC Champions League: 2020
International
South Korea
EAFF East Asian Cup: 2017
External links
1988 births
Living people
South Korean men's footballers
Men's association football midfielders
South Korea men's youth international footballers
South Korea men's under-20 international footballers
South Korea men's international footballers
South Korean expatriate men's footballers
Daejeon Hana Citizen players
Seongnam FC players
Cerezo Osaka players
Gimcheon Sangmu FC players
FC Seoul players
Ulsan Hyundai FC players
K League 1 players
J1 League players
Expatriate men's footballers in Japan
South Korean expatriate sportspeople in Japan
People from Jinju
Footballers from South Gyeongsang Province |
https://en.wikipedia.org/wiki/Nilradical%20of%20a%20Lie%20algebra | In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible.
The nilradical of a finite-dimensional Lie algebra is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical of the Lie algebra . The quotient of a Lie algebra by its nilradical is a reductive Lie algebra . However, the corresponding short exact sequence
does not split in general (i.e., there isn't always a subalgebra complementary to in ). This is in contrast to the Levi decomposition: the short exact sequence
does split (essentially because the quotient is semisimple).
See also
Levi decomposition
Nilradical of a ring, a notion in ring theory.
References
.
Lie algebras |
https://en.wikipedia.org/wiki/Nilradical | Nilradical may refer to:
Nilradical of a ring
Nilradical of a Lie algebra |
https://en.wikipedia.org/wiki/Mathematics%20and%20art | Mathematics and art are related in a variety of ways. Mathematics has itself been described as an art motivated by beauty. Mathematics can be discerned in arts such as music, dance, painting, architecture, sculpture, and textiles. This article focuses, however, on mathematics in the visual arts.
Mathematics and art have a long historical relationship. Artists have used mathematics since the 4th century BC when the Greek sculptor Polykleitos wrote his Canon, prescribing proportions conjectured to have been based on the ratio 1: for the ideal male nude. Persistent popular claims have been made for the use of the golden ratio in ancient art and architecture, without reliable evidence. In the Italian Renaissance, Luca Pacioli wrote the influential treatise De divina proportione (1509), illustrated with woodcuts by Leonardo da Vinci, on the use of the golden ratio in art. Another Italian painter, Piero della Francesca, developed Euclid's ideas on perspective in treatises such as De Prospectiva Pingendi, and in his paintings. The engraver Albrecht Dürer made many references to mathematics in his work Melencolia I. In modern times, the graphic artist M. C. Escher made intensive use of tessellation and hyperbolic geometry, with the help of the mathematician H. S. M. Coxeter, while the De Stijl movement led by Theo van Doesburg and Piet Mondrian explicitly embraced geometrical forms. Mathematics has inspired textile arts such as quilting, knitting, cross-stitch, crochet, embroidery, weaving, Turkish and other carpet-making, as well as kilim. In Islamic art, symmetries are evident in forms as varied as Persian girih and Moroccan zellige tilework, Mughal jali pierced stone screens, and widespread muqarnas vaulting.
Mathematics has directly influenced art with conceptual tools such as linear perspective, the analysis of symmetry, and mathematical objects such as polyhedra and the Möbius strip. Magnus Wenninger creates colourful stellated polyhedra, originally as models for teaching. Mathematical concepts such as recursion and logical paradox can be seen in paintings by René Magritte and in engravings by M. C. Escher. Computer art often makes use of fractals including the Mandelbrot set, and sometimes explores other mathematical objects such as cellular automata. Controversially, the artist David Hockney has argued that artists from the Renaissance onwards made use of the camera lucida to draw precise representations of scenes; the architect Philip Steadman similarly argued that Vermeer used the camera obscura in his distinctively observed paintings.
Other relationships include the algorithmic analysis of artworks by X-ray fluorescence spectroscopy, the finding that traditional batiks from different regions of Java have distinct fractal dimensions, and stimuli to mathematics research, especially Filippo Brunelleschi's theory of perspective, which eventually led to Girard Desargues's projective geometry. A persistent view, based ultimately on the Pythagorea |
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