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39-759: Berlekamp invented an algorithm to factor polynomials and the Berlekamp switching game , and was one of the inventors of the Berlekamp–Welch algorithm and the Berlekamp–Massey algorithms , which are used to implement Reed–Solomon error correction . He also co-invented the Berlekamp–Rabin algorithm , Berlekamp–Zassenhaus algorithm , and the Berlekamp–Van Lint–Seidel graph . Berlekamp had also been active in investing , and ran Axcom, which became

78-619: A Golden Jubilee Award for Technological Innovation from the IEEE Information Theory Society . Along with Tom M. Rodgers he was one of the founders of Gathering 4 Gardner and was on its board for many years. In the mid-1980s, he was president of Cyclotomics, Inc. , a corporation that developed error-correcting code technology. He studied various games, including dots and boxes , fox and geese , and, especially, Go . Berlekamp and co-author David Wolfe describe methods for analyzing certain classes of Go endgames in

117-467: A finite field F q {\displaystyle \mathbb {F} _{q}} and gives as output a polynomial g ( x ) {\displaystyle g(x)} with coefficients in the same field such that g ( x ) {\displaystyle g(x)} divides f ( x ) {\displaystyle f(x)} . The algorithm may then be applied recursively to these and subsequent divisors, until we find

156-544: A gathering of people who shared Gardner's interests, especially puzzles, magic, and mathematics. Rodgers invited the world's foremost puzzle composers and collectors, and enlisted magician Mark Setteducati and mathematician Elwyn Berlekamp to recruit leading magicians and recreational mathematicians, respectively. Gardner agreed to attend. Thus was born the first Gathering (G4G1), held in Atlanta, GA, in January 1993. For

195-830: A non-trivial factor. Since the ring of polynomials over a field is a Euclidean domain , we may compute these GCDs using the Euclidean algorithm . With some abstract algebra, the idea behind Berlekamp's algorithm becomes conceptually clear. We represent a finite field F q {\textstyle \mathbb {F} _{q}} , where q = p m {\textstyle q=p^{m}} for some prime p {\textstyle p} , as F p [ y ] / ( g ( y ) ) {\textstyle \mathbb {F} _{p}[y]/(g(y))} . We can assume that f ( x ) ∈ F q [ x ] {\textstyle f(x)\in \mathbb {F} _{q}[x]}

234-402: A playful and fun approach to learning, G4Gn events explore ideas in fields of interest to Gardner. The term G4G is also used to denote the community of people who participate in these events. With the "n" denoting the number in the series, G4Gn is an invitation-only bi-annual conference that started with G4G1 in January 1993. A second gathering (G4G2) was held in Atlanta in 1996, and from then on

273-566: A related series of events called Celebration of Mind (CoM). Martin Gardner's prolific output as a columnist and writer—he authored over 100 books between 1951 and 2010—put him in contact with a large number of people on a wide range of subjects from magic, mathematics, puzzles, physics, philosophy, logic and rationality, to G. K. Chesterton , Alice in Wonderland , and the Wizard of Oz . As

312-414: A result, he had a large following of amateurs and professionals eager to pay tribute to him, but many of them had only infrequent contact with each other. Moreover, Gardner was famously shy, and generally declined to appear at any events honoring him. In the early 1990s, Atlanta-based entrepreneur and puzzle collector Thomas M. Rodgers (1943–2012), a friend of Martin Gardner's, conceived a plan to create

351-599: A son with his wife Jennifer. He lived in Piedmont, California and died in April 2019 at the age of 78 from complications of pulmonary fibrosis . Berlekamp was a professor of electrical engineering at the University of California, Berkeley from 1964 until 1966, when he became a mathematics researcher at Bell Labs . In 1971, Berlekamp returned to Berkeley as professor of mathematics and computer science, where he served as

390-423: Is always the prime subfield of that field extension. Thus, Fix p ( F q [ x ] / ( f ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f(x)))} has p {\textstyle p} elements if and only if f ( x ) {\textstyle f(x)} is irreducible. Moreover, we can use

429-513: Is fixed by Frobenius. We note that at this point we have an efficiently computable irreducibility criterion, and the remaining analysis shows how to use this to find factors. The algorithm now breaks down into two cases: For further details one can consult. One important application of Berlekamp's algorithm is in computing discrete logarithms over finite fields F p n {\displaystyle \mathbb {F} _{p^{n}}} , where p {\displaystyle p}

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468-543: Is in fact the kernel of a certain n × n {\displaystyle n\times n} matrix over F q {\displaystyle \mathbb {F} _{q}} , which is derived from the so-called Berlekamp matrix of the polynomial, denoted Q {\displaystyle {\mathcal {Q}}} . If Q = [ q i , j ] {\displaystyle {\mathcal {Q}}=[q_{i,j}]} then q i , j {\displaystyle q_{i,j}}

507-521: Is in the Berlekamp subalgebra if and only if g ( Q − I ) = 0 {\displaystyle g({\mathcal {Q}}-I)=0} (where I {\displaystyle I} is the n × n {\displaystyle n\times n} identity matrix ), i.e. if and only if it is in the null space of Q − I {\displaystyle {\mathcal {Q}}-I} . By computing

546-450: Is prime and n ≥ 2 {\displaystyle n\geq 2} . Computing discrete logarithms is an important problem in public key cryptography and error-control coding . For a finite field, the fastest known method is the index calculus method , which involves the factorisation of field elements. If we represent the field F p n {\displaystyle \mathbb {F} _{p^{n}}} in

585-486: Is relatively straightforward to see that the row vector g Q {\displaystyle g{\mathcal {Q}}} corresponds, in the same way, to the reduction of g ( x ) q {\displaystyle g(x)^{q}} modulo f ( x ) {\displaystyle f(x)} . Consequently, a polynomial g ( x ) ∈ R {\displaystyle g(x)\in R}

624-705: Is square free, by taking all possible pth roots and then computing the gcd with its derivative. Now, suppose that f ( x ) = f 1 ( x ) … f n ( x ) {\textstyle f(x)=f_{1}(x)\ldots f_{n}(x)} is the factorization into irreducibles. Then we have a ring isomorphism, σ : F q [ x ] / ( f ( x ) ) → ∏ i F q [ x ] / ( f i ( x ) ) {\textstyle \sigma :\mathbb {F} _{q}[x]/(f(x))\to \prod _{i}\mathbb {F} _{q}[x]/(f_{i}(x))} , given by

663-413: Is the coefficient of the j {\displaystyle j} -th power term in the reduction of x i q {\displaystyle x^{iq}} modulo f ( x ) {\displaystyle f(x)} , i.e.: With a certain polynomial g ( x ) ∈ R {\displaystyle g(x)\in R} , say: we may associate the row vector: It

702-898: The Renaissance Technologies ' Medallion Fund. Berlekamp was born in Dover, Ohio . His family moved to Northern Kentucky, where Berlekamp graduated from Ft. Thomas Highlands high school in Ft. Thomas, Campbell county, Kentucky. While an undergraduate at the Massachusetts Institute of Technology (MIT), he was a Putnam Fellow in 1961. He completed his bachelor's and master's degrees in electrical engineering in 1962. Continuing his studies at MIT, he finished his Ph.D. in electrical engineering in 1964; his advisors were Robert G. Gallager , Peter Elias , Claude Shannon , and John Wozencraft . Berlekamp had two daughters and

741-500: The COVID-19 pandemic ) and was a hybrid event, with some participants joining remotely as speakers and audience members. The logo of Gathering for Gardner, as well as the logo for the first CoM event, employs ambigrams designed by long-time Gardner associate Scott Kim . Packed with wonderful treats and startling surprises, it provided one "aha!" moment after another. – Ivars Peterson Continuing Martin Gardner's pursuit of

780-702: The WolframAlpha [1] website. Gathering 4 Gardner Gathering 4 Gardner (G4G) is an educational foundation and non-profit corporation (Gathering 4 Gardner, Inc.) devoted to preserving the legacy and spirit of prolific writer Martin Gardner . G4G organizes conferences where people who have been inspired by or have a strong personal connection to Martin Gardner can meet and celebrate his influence. These events explore ideas and developments in recreational mathematics, magic, illusion, puzzles, philosophy, and rationality, and foster creative work in all of these areas by enthusiasts of all ages. G4G also facilitates

819-1376: The Chinese remainder theorem. The crucial observation is that the Frobenius automorphism x → x p {\textstyle x\to x^{p}} commutes with σ {\textstyle \sigma } , so that if we denote Fix p ( R ) = { f ∈ R : f p = f } {\textstyle {\text{Fix}}_{p}(R)=\{f\in R:f^{p}=f\}} , then σ {\textstyle \sigma } restricts to an isomorphism Fix p ( F q [ x ] / ( f ( x ) ) ) → ∏ i = 1 n Fix p ( F q [ x ] / ( f i ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f(x)))\to \prod _{i=1}^{n}{\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f_{i}(x)))} . By finite field theory, Fix p ( F q [ x ] / ( f i ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f_{i}(x)))}

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858-421: The Gardner legacy. Berlekamp was one of the founders of G4G and was on its board of directors for many years. Berlekamp%27s algorithm In mathematics , particularly computational algebra , Berlekamp's algorithm is a well-known method for factoring polynomials over finite fields (also known as Galois fields ). The algorithm consists mainly of matrix reduction and polynomial GCD computations. It

897-508: The Scientific American: Lewis Carroll’s Appearances in ‘Mathematical Games”. Ingrid Daubechies was also a featured speaker with a talk titled, "Reunited: An Art Historical and Digital Adventure". To date, all G4Gn conferences have been held in Atlanta, GA. Gardner (and his wife), who disliked travel in addition to wanting to avoid the limelight, attended only the first two G4Gs. Notable presenters over

936-419: The above product will be a non-trivial factor of f ( x ) {\displaystyle f(x)} , but some are, providing the factors we seek. Berlekamp's algorithm finds polynomials g ( x ) {\displaystyle g(x)} suitable for use with the above result by computing a basis for the Berlekamp subalgebra. This is achieved via the observation that Berlekamp subalgebra

975-758: The advisor for over twenty doctoral students. He was a member of the National Academy of Engineering (1977) and the National Academy of Sciences (1999). He was elected a Fellow of the American Academy of Arts and Sciences in 1996, and became a fellow of the American Mathematical Society in 2012. In 1991, he received the IEEE Richard W. Hamming Medal , and in 1993, the Claude E. Shannon Award . In 1998, he received

1014-538: The basis for a 46-minute episode of The Nature of Things made by David Suzuki . Titled, Mystery and Magic of Mathematics: Martin Gardner and Friends , it showcases Martin's numerous passions, and reminds us of the amazing panoply of people that he informally assembled and mentored over the decades. Developed in 2010 after Martin Gardner's death that spring, Celebration of Mind (CoM) is a worldwide series of events held on or around Gardner's birthday: October 21. Anyone, anywhere, can host one, and most of them are open to

1053-448: The book Winning Ways for your Mathematical Plays , leading to his recognition as one of the founders of combinatorial game theory . The dedication of their book says, "To Martin Gardner, who has brought more mathematics to more millions than anyone else." Berlekamp and Gardner both had great love for and were strong advocates of recreational mathematics. Conferences called Gathering 4 Gardner (G4G) are held every two years to celebrate

1092-418: The book Mathematical Go. Berlekamp was a close friend of Scientific American columnist Martin Gardner and was an important member of the gifted and diverse group of people that Gardner nurtured and acted as a conduit for; people who inspired Gardner and who were in turn inspired by him. Berlekamp teamed up with John Horton Conway and Richard K. Guy , two other close associates of Gardner, to co-author

1131-462: The congruence: These polynomials form a subalgebra of R (which can be considered as an n {\displaystyle n} -dimensional vector space over F q {\displaystyle \mathbb {F} _{q}} ), called the Berlekamp subalgebra . The Berlekamp subalgebra is of interest because the polynomials g ( x ) {\displaystyle g(x)} it contains satisfy In general, not every GCD in

1170-493: The decomposition of f ( x ) {\displaystyle f(x)} into powers of irreducible polynomials (recalling that the ring of polynomials over a finite field is a unique factorization domain ). All possible factors of f ( x ) {\displaystyle f(x)} are contained within the factor ring The algorithm focuses on polynomials g ( x ) ∈ R {\displaystyle g(x)\in R} which satisfy

1209-407: The events have been held bi-annually, in the springs of even-numbered years, up until G4G13 which took place in April 2018 and featured Fields medallist Manjul Bhargava and inventor of the eponymous cube Ernő Rubik . After a delay because of Covid, G4G14 finally took place in April 2022. Lewis Carroll expert Mark Burstein was a featured speaker with a talk titled, “A Literary Englishman and

Elwyn Berlekamp - Misplaced Pages Continue

1248-506: The fact that the Frobenius automorphism is F p {\textstyle \mathbb {F} _{p}} -linear to calculate the fixed set. That is, we note that Fix p ( F q [ x ] / ( f ( x ) ) ) {\textstyle {\text{Fix}}_{p}(\mathbb {F} _{q}[x]/(f(x)))} is a F p {\textstyle \mathbb {F} _{p}} -subspace, and an explicit basis for it can be calculated in

1287-426: The matrix Q − I {\displaystyle {\mathcal {Q}}-I} and reducing it to reduced row echelon form and then easily reading off a basis for the null space, we may find a basis for the Berlekamp subalgebra and hence construct polynomials g ( x ) {\displaystyle g(x)} in it. We then need to successively compute GCDs of the form above until we find

1326-558: The next two decades G4G was supported mainly by Rodgers with "seemingly unfettered access to his personal time and resources". In 2007 board members Rodgers, Berlekamp, Setteducati, Thane Plambeck, and Scott Hudson de Tarnowsky decided that G4G should broaden its reach and expand the scope of its educational programs. To that end they formed the 501(c)(3) non-profit corporation Gathering 4 Gardner, Inc. So far there have been 14 Gatherings, all held in downtown Atlanta. The 14th Gathering took place in April 2022 (delayed for 2 years because of

1365-443: The polynomial ring F p [ x , y ] / ( f , g ) {\textstyle \mathbb {F} _{p}[x,y]/(f,g)} by computing ( x i y j ) p {\textstyle (x^{i}y^{j})^{p}} and establishing the linear equations on the coefficients of x , y {\textstyle x,y} polynomials that are satisfied iff it

1404-498: The public. These CoMs vary in size from three people meeting to perform rubber-band magic for each other, to crowds of hundreds of students in highly organized exploratory paper folding activities. The goal is to celebrate the boundless creativity and curiosity of the human mind, and they have been held in locations from Boston to Beijing, and from Riga to Rio. There have been Celebration of Mind events hosted on all seven continents, and G4G encourages organizers to register their events at

1443-474: The usual way - that is, as polynomials over the base field F p {\displaystyle \mathbb {F} _{p}} , reduced modulo an irreducible polynomial of degree n {\displaystyle n} - then this is simply polynomial factorisation, as provided by Berlekamp's algorithm. Berlekamp's algorithm may be accessed in the PARI/GP package using the factormod command, and

1482-574: The years have included Activities typically include lectures, performance art, puzzle and book displays, close-up and stage magic acts. Another tradition held at these off-site events was the directed community building of original mathematical sculptures under the leadership of George Hart , Chaim Goodman-Strauss and others. Each conference has a Gift Exchange, in which attendees swap puzzles, magic tricks, artwork, mathematical papers, novelty items, books, and CDs/DVDs. Gardner and many of his admirers were filmed at G4G2 in 1996, and this footage formed

1521-525: Was invented by Elwyn Berlekamp in 1967. It was the dominant algorithm for solving the problem until the Cantor–Zassenhaus algorithm of 1981. It is currently implemented in many well-known computer algebra systems . Berlekamp's algorithm takes as input a square-free polynomial f ( x ) {\displaystyle f(x)} (i.e. one with no repeated factors) of degree n {\displaystyle n} with coefficients in

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