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Lms Journal of Computation and Mathematics最新文献

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Wieferich pairs and Barker sequences, II Wieferich对和Barker序列，2
Lms Journal of Computation and Mathematics Pub Date : 2013-05-31 DOI: 10.1112/S1461157013000223
P. Borwein, Michael J. Mossinghoff
{"title":"Wieferich pairs and Barker sequences, II","authors":"P. Borwein, Michael J. Mossinghoff","doi":"10.1112/S1461157013000223","DOIUrl":"https://doi.org/10.1112/S1461157013000223","url":null,"abstract":"We show that if a Barker sequence of length $n>13$\u0000 exists, then either n $=$\u0000 3 979 201 339 721 749 133 016 171 583 224 100, or $n > 4cdot 10^{33}$\u0000 . This improves the lower bound on the length of a long Barker sequence by a factor of nearly $2000$\u0000 . We also obtain eighteen additional integers $n<10^{50}$\u0000 that cannot be ruled out as the length of a Barker sequence, and find more than 237 000 additional candidates $n<10^{100}$\u0000 . These results are obtained by completing extensive searches for Wieferich prime pairs and using them, together with a number of arithmetic restrictions on $n$\u0000 , to construct qualifying integers below a given bound. We also report on some updated computations regarding open cases of the circulant Hadamard matrix problem.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"24-32"},"PeriodicalIF":0.0,"publicationDate":"2013-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000223","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 25
Bounds and algorithms for the K-Bessel function of imaginary order 虚阶k -贝塞尔函数的界和算法
Lms Journal of Computation and Mathematics Pub Date : 2013-04-10 DOI: 10.1112/S1461157013000028
A. Booker, Andreas Strömbergsson, H. Then
{"title":"Bounds and algorithms for the K-Bessel function of imaginary order","authors":"A. Booker, Andreas Strömbergsson, H. Then","doi":"10.1112/S1461157013000028","DOIUrl":"https://doi.org/10.1112/S1461157013000028","url":null,"abstract":"Using the paths of steepest descent, we prove precise bounds with numerical implied constants for the modified Bessel function ${K}_{ir} (x)$\u0000 of imaginary order and its first two derivatives with respect to the order. We also prove precise asymptotic bounds on more general (mixed) derivatives without working out numerical implied constants. Moreover, we present an absolutely and rapidly convergent series for the computation of ${K}_{ir} (x)$\u0000 and its derivatives, as well as a formula based on Fourier interpolation for computing with many values of $r$\u0000 . Finally, we have implemented a subset of these features in a software library for fast and rigorous computation of ${K}_{ir} (x)$\u0000 .","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"78-108"},"PeriodicalIF":0.0,"publicationDate":"2013-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000028","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 17
Rational functions with maximal radius of absolute monotonicity 具有绝对单调最大半径的有理函数
Lms Journal of Computation and Mathematics Pub Date : 2013-03-26 DOI: 10.1112/S1461157013000326
Lajos Lóczi, D. Ketcheson
{"title":"Rational functions with maximal radius of absolute monotonicity","authors":"Lajos Lóczi, D. Ketcheson","doi":"10.1112/S1461157013000326","DOIUrl":"https://doi.org/10.1112/S1461157013000326","url":null,"abstract":"We study the radius of absolute monotonicity $R$\u0000 of rational functions with numerator and denominator of degree $s$\u0000 that approximate the exponential function to order $p$\u0000 . Such functions arise in the application of implicit $s$\u0000 -stage, order $p$\u0000 Runge–Kutta methods for initial value problems, and the radius of absolute monotonicity governs the numerical preservation of properties like positivity and maximum-norm contractivity. We construct a function with $p=2$\u0000 and $R>2s$\u0000 , disproving a conjecture of van de Griend and Kraaijevanger. We determine the maximum attainable radius for functions in several one-parameter families of rational functions. Moreover, we prove earlier conjectured optimal radii in some families with two or three parameters via uniqueness arguments for systems of polynomial inequalities. Our results also prove the optimality of some strong stability preserving implicit and singly diagonally implicit Runge–Kutta methods. Whereas previous results in this area were primarily numerical, we give all constants as exact algebraic numbers.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"159-205"},"PeriodicalIF":0.0,"publicationDate":"2013-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000326","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411420","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven 计算6和7秩有限Chevalley群的Sylow -子群中的共轭类
Lms Journal of Computation and Mathematics Pub Date : 2013-02-27 DOI: 10.1112/S1461157013000284
Simon M. Goodwin, Peter Mosch, G. Röhrle
{"title":"Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven","authors":"Simon M. Goodwin, Peter Mosch, G. Röhrle","doi":"10.1112/S1461157013000284","DOIUrl":"https://doi.org/10.1112/S1461157013000284","url":null,"abstract":"Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In an earlier paper, the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of k(U(q)) for finite Chevalley groups of rank six and seven, except E_7. We observe that k(U(q)) is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write k(U(q)) as a polynomial in q-1, then the coefficients are non-negative. \u0000Under the assumption that k(U(q)) is a polynomial in q-1, we also give an explicit formula for the coefficients of k(U(q)) of degrees zero, one and two.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"109-122"},"PeriodicalIF":0.0,"publicationDate":"2013-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000284","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
Constructing supersingular elliptic curves with a given endomorphism ring 构造具有给定自同态环的超奇异椭圆曲线
Lms Journal of Computation and Mathematics Pub Date : 2013-01-29 DOI: 10.1112/S1461157014000254
I. Chevyrev, Steven D. Galbraith
{"title":"Constructing supersingular elliptic curves with a given endomorphism ring","authors":"I. Chevyrev, Steven D. Galbraith","doi":"10.1112/S1461157014000254","DOIUrl":"https://doi.org/10.1112/S1461157014000254","url":null,"abstract":"Let O be a maximal order in the quaternion algebra B_p over Q ramified at p and infinity. The paper is about the computational problem: Construct a supersingular elliptic curve E over F_p such that End(E) = O. We present an algorithm that solves this problem by taking gcds of the reductions modulo p of Hilbert class polynomials. New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice O^T of O, the order O is effectively characterized by the three successive minima and two other short vectors of O^T. The desired conditions turn out to hold whenever the j-invariant j(E), of the elliptic curve with End(E) = O, lies in F_p. We can then prove that our algorithm terminates with running time O(p^{1+epsilon}) under the aforementioned conditions. As a further application we present an algorithm to simultaneously match all maximal order types with their associated j-invariants. Our algorithm has running time O(p^{2.5+epsilon}) operations and is more efficient than Cervino's algorithm for the same problem.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"71-91"},"PeriodicalIF":0.0,"publicationDate":"2013-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000254","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
On a conjecture of Rudin on squares in Arithmetic Progressions 关于等差数列中平方的Rudin猜想
Lms Journal of Computation and Mathematics Pub Date : 2013-01-22 DOI: 10.1112/S1461157013000259
Enrique Gonz'alez-Jim'enez, X. Xarles
{"title":"On a conjecture of Rudin on squares in Arithmetic Progressions","authors":"Enrique Gonz'alez-Jim'enez, X. Xarles","doi":"10.1112/S1461157013000259","DOIUrl":"https://doi.org/10.1112/S1461157013000259","url":null,"abstract":"Let Q(N;q,a) denotes the number of squares in the arithmetic progression qn+a, for n=0, 1,...,N-1, and let Q(N) be the maximum of Q(N;q,a) over all non-trivial arithmetic progressions qn + a. Rudin's conjecture asserts that Q(N)=O(Sqrt(N)), and in its stronger form that Q(N)=Q(N;24,1) if N=> 6. We prove the conjecture above for 6 8 for some integer k, where GP_k is the k-th generalized pentagonal number, then Q(N)=Q(N;q,a) with gcd(q,a) squarefree and q> 0 if and only if (q,a)=(24,1).","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"58-76"},"PeriodicalIF":0.0,"publicationDate":"2013-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000259","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
A comprehensive perturbation theorem for estimating magnitudes of roots of polynomials 估计多项式根大小的一个综合摄动定理
Lms Journal of Computation and Mathematics Pub Date : 2013-01-01 DOI: 10.1112/S1461157012001192
M. Pakdemirli, Gözde Sari
{"title":"A comprehensive perturbation theorem for estimating magnitudes of roots of polynomials","authors":"M. Pakdemirli, Gözde Sari","doi":"10.1112/S1461157012001192","DOIUrl":"https://doi.org/10.1112/S1461157012001192","url":null,"abstract":"A comprehensive new perturbation theorem is posed and proven to estimate the magnitudes of roots of polynomials. The theorem successfully determines the magnitudes of roots for arbitrary degree of polynomial equations with no restrictions on the coefficients. In the previous papers ‘Pakdemirli and Elmas, Appl. Math. Comput. 216 (2010) 1645–1651’ and ‘Pakdemirli and Yurtsever, Appl. Math. Comput. 188 (2007) 2025–2028’, the given theorems were valid only for some restricted coefficients. The given theorem in this work is a generalization and unification of the past theorems and valid for arbitrary coefficients. Numerical applications of the theorem are presented as examples. It is shown that the theorem produces good estimates for the magnitudes of roots of polynomial equations of arbitrary order and unrestricted coefficients.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"1-8"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
The minimal generating sets of of size four 大小为4的最小发电集
Lms Journal of Computation and Mathematics Pub Date : 2013-01-01 DOI: 10.1112/S1461157013000193
Sebastian Jambor
{"title":"The minimal generating sets of of size four","authors":"Sebastian Jambor","doi":"10.1112/S1461157013000193","DOIUrl":"https://doi.org/10.1112/S1461157013000193","url":null,"abstract":"","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"419-423"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
Complex B-splines and Hurwitz zeta functions 复b样条和Hurwitz zeta函数
Lms Journal of Computation and Mathematics Pub Date : 2013-01-01 DOI: 10.1112/S146115701300003X
B. Forster, R. Garunk, P. Massopust, J. Steuding
{"title":"Complex B-splines and Hurwitz zeta functions","authors":"B. Forster, R. Garunk, P. Massopust, J. Steuding","doi":"10.1112/S146115701300003X","DOIUrl":"https://doi.org/10.1112/S146115701300003X","url":null,"abstract":"We characterize nonempty open subsets of the complex plane where the sum (s; ) + e is (s; 1 ) of Hurwitz zeta functions has no zeros in s for all 0 6 6 1. This problem is motivated by the construction of fundamental cardinal splines of complex order s.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"61-77"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S146115701300003X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
Václav Šimerka: quadratic forms and factorization Václav Šimerka:二次型和分解
Lms Journal of Computation and Mathematics Pub Date : 2013-01-01 DOI: 10.1112/S1461157013000065
F. Lemmermeyer
{"title":"Václav Šimerka: quadratic forms and factorization","authors":"F. Lemmermeyer","doi":"10.1112/S1461157013000065","DOIUrl":"https://doi.org/10.1112/S1461157013000065","url":null,"abstract":"In this article we show that the Czech mathematician V´aclav ˇSimerka discovered the factorization of 19 (10 17 − 1) using a method based on the class group of binary quadratic forms more than 120 years before Shanks and Schnorr developed similar algorithms. ˇSimerka also gave the first examples of what later became known as Carmichael numbers.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"118-129"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000065","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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