The euclidean algorithm for computing the greatest common divisor of two integers is, as D. E. Knuth has remarked, ''the oldest nontrivial algorithm that has survived to the present day.'' Credit for t...
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The euclidean algorithm for computing the greatest common divisor of two integers is, as D. E. Knuth has remarked, ''the oldest nontrivial algorithm that has survived to the present day.'' Credit for the first analysis of the running time of the algorithm is traditionally assigned to Gabriel Lame, for his 1844 paper. This article explores the historical origins of the analysis of the euclidean algorithm. A weak bound on the running time of this algorithm was given as early as 1811 by Antoine-Andre-Louis Reynaud. Furthermore, Lame's basic result was known to Emile Leger in 1837, and a complete, valid proof along different lines was given by Pierre-Joseph-Etienne Finck in 1841. (C) 1994 Academic Press, Inc.
We introduce a generalization of the euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an o...
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We introduce a generalization of the euclidean algorithm for rings equipped with an involution, and completely enumerate all isomorphism classes of orders over definite, rational quaternion algebras equipped with an orthogonal involution that admit such an algorithm. We give two applications: first, any order that admits such an algorithm has class number 1;second, we show how the existence of such an algorithm relates to the problem of constructing explicit Dirichlet domains for Kleinian subgroups of the isometry group of hyperbolic 4-space. (C) 2020 Elsevier Inc. All rights reserved.
We prove a quadratic expression for the Bezoutian of two univariate polynomials in terms of the remainders for the euclidean algorithm. In case of two polynomials of the same degree, or of consecutive degrees, this al...
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We prove a quadratic expression for the Bezoutian of two univariate polynomials in terms of the remainders for the euclidean algorithm. In case of two polynomials of the same degree, or of consecutive degrees, this allows us to interpret their Bezoutian as the Christoffel-Darboux kernel for a finite family of orthogonal polynomials, arising from the euclidean algorithm. We give orthogonality properties of remainders, and reproducing properties of Bezoutians.
The binary euclidean algorithm is a modification of the classical euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The...
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The binary euclidean algorithm is a modification of the classical euclidean algorithm for computation of greatest common divisors which avoids ordinary integer division in favour of division by powers of two only. The expectation of the number of steps taken by the binary euclidean algorithm when applied to pairs of integers of bounded size was first investigated by R.P. Brent in 1976 via a heuristic model of the algorithm as a random dynamical system. Based on numerical investigations of the expectation of the associated Ruelle transfer operator, Brent obtained a conjectural asymptotic expression for the mean number of steps performed by the algorithm when processing pairs of odd integers whose size is bounded by a large integer. In 1998 B. Vallee modified Brent's model via an induction scheme to rigorously prove an asymptotic formula for the average number of steps performed by the algorithm;however, the relationship of this result with Brent's heuristics remains conjectural. In this article we establish previously conjectural properties of Brent's transfer operator, showing directly that it possesses a spectral gap and preserves a unique continuous density. This density is shown to extend holomorphically to the complex right half-plane and to have a logarithmic singularity at zero. By combining these results with methods from classical analytic number theory we prove the correctness of three conjectured formulae for the expected number of steps, resolving several open questions promoted by D.E. Knuth in The Art of Computer Programming. (C) 2016 The Author. Published by Elsevier Inc.
Let K be a number field with unit rank at least four, containing a subfield M such that K/M is Galois of degree at least four. We show that the ring of integers of K is a euclidean domain if and only if it is a princi...
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Let K be a number field with unit rank at least four, containing a subfield M such that K/M is Galois of degree at least four. We show that the ring of integers of K is a euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann Hypothesis for Dedekind zeta functions. We prove this unconditionally.
As early as the 16th century. Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the euclidean algorithm occurs if these numbers are equimult...
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As early as the 16th century. Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the euclidean algorithm occurs if these numbers are equimultiples of two consecutive members of the Fibonacci sequence. (C) 1995 Academic Press, Inc.
As with euclidean rings and rings admitting a restricted Nagata's pairwise algorithm, we will give an internal characterization of 2-stage euclidean rings. Applying this characterization we are capable of providin...
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As with euclidean rings and rings admitting a restricted Nagata's pairwise algorithm, we will give an internal characterization of 2-stage euclidean rings. Applying this characterization we are capable of providing infinitely many integral domains which are omega-stage euclidean but not 2-stage euclidean. Our examples solve finally a fundamental question related to the notion of k-stage euclidean rings raised by G.E. Cooke [G.E. Cooke, A weakening of the euclidean property for integral domains and applications to algebraic number theory I, J. Reine Angew, Math. 282 (1976) 133-156]. The question was stated as follows: "I do not know of an example of an omega-stage euclidean ring which is not 2-stage euclidean." Also, in this article we will give a method to construct the smallest restricted Nagata's pairwise algorithm theta on a unique factorization domain which admits a restricted Nagata's pairwise algorithm. It is of interest to point out that in a euclidean domain the shortest length d(a, b) of all terminating division chains starting from a pair (a, b) and the value theta(a, b) with g.c.d.(a, b) not equal 1 can be determined by each other. (C) 2011 Elsevier Inc. All rights reserved.
The need to reduce a periodic structure given in terms of a large supercell and associated lattice generators arises frequently in different fields of application of crystallography, in particular in the ab initio the...
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The need to reduce a periodic structure given in terms of a large supercell and associated lattice generators arises frequently in different fields of application of crystallography, in particular in the ab initio theoretical modelling of materials at the atomic scale. This paper considers the reduction of crystals and addresses the reduction associated with the existence of a commensurate translation that leaves the crystal invariant, providing a practical scheme for it. The reduction procedure hinges on a convenient integer factorization of the full period of the cycle (or grid) generated by the repeated applications of the invariant translation, and its iterative reduction into sub-cycles, each of which corresponds to a factor in the decomposition of the period. This is done in successive steps, each time solving a Diophantine linear equation by means of a euclidean reduction algorithm in order to provide the generators of the reduced lattice.
We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Eucli...
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ISBN:
(纸本)9784885522925
We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the euclidean algorithm with respect to composition of polynomials. For a given received word, our decoding algorithm computes a list of all codewords that are closest to the received word with respect to the rank metric.
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