Pub Date : 2013-01-01DOI: 10.1112/S1461157013000090
Aydin Izgi
In this paper we deal with the operators $$begin{eqnarray*}{Z}_{n} (f; x)= frac{n}{{b}_{n} } { mathop{sum }nolimits}_{k= 0}^{n} {p}_{n, k} biggl(frac{x}{{b}_{n} } biggr)int nolimits nolimits_{0}^{infty } {s}_{n, k} biggl(frac{t}{{b}_{n} } biggr)f(t)hspace{0.167em} dt, quad 0leq xleq {b}_{n}end{eqnarray*}$$ and study some basic properties of these operators where ${p}_{n, k} (u)=bigl(hspace{-4pt}{scriptsize begin{array}{ l} displaystyle n displaystyle kend{array} } hspace{-4pt}bigr){u}^{k} mathop{(1- u)}nolimits ^{n- k} , (0leq kleq n), uin [0, 1] $ and ${s}_{n, k} (u)= {e}^{- nu} mathop{(nu)}nolimits ^{k} hspace{-3pt}/ k!, uin [0, infty )$ . Also, we establish the order of approximation by using weighted modulus of continuity.
{"title":"Approximation by a composition of Chlodowsky operators and Százs–Durrmeyer operators on weighted spaces","authors":"Aydin Izgi","doi":"10.1112/S1461157013000090","DOIUrl":"https://doi.org/10.1112/S1461157013000090","url":null,"abstract":"In this paper we deal with the operators $$begin{eqnarray*}{Z}_{n} (f; x)= frac{n}{{b}_{n} } { mathop{sum }nolimits}_{k= 0}^{n} {p}_{n, k} biggl(frac{x}{{b}_{n} } biggr)int nolimits nolimits_{0}^{infty } {s}_{n, k} biggl(frac{t}{{b}_{n} } biggr)f(t)hspace{0.167em} dt, quad 0leq xleq {b}_{n}end{eqnarray*}$$ and study some basic properties of these operators where ${p}_{n, k} (u)=bigl(hspace{-4pt}{scriptsize begin{array}{ l} displaystyle n displaystyle kend{array} } hspace{-4pt}bigr){u}^{k} mathop{(1- u)}nolimits ^{n- k} , (0leq kleq n), uin [0, 1] $ and ${s}_{n, k} (u)= {e}^{- nu} mathop{(nu)}nolimits ^{k} hspace{-3pt}/ k!, uin [0, infty )$ . Also, we establish the order of approximation by using weighted modulus of continuity.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"388-397"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000090","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410796","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}
Pub Date : 2013-01-01DOI: 10.1112/S1461157013000120
Hua Zhang
In this paper we study the weak approximation problem of E [ φ ( x ( T ))] by E [ φ ( y ( T ))], where x ( T ) is the solution of a stochastic differential delay equation and y ( T ) is defined by the Euler scheme. For φ ∈ C 3 b , Buckwar, Kuske, Mohammed and Shardlow (‘Weak convergence of the Euler scheme for stochastic differential delay equations’, LMS J. Comput. Math. 11 (2008) 60–69) have shown that the Euler scheme has weak order of convergence 1. Here we prove that the same results hold when φ is only assumed to be measurable and bounded under an additional non-degeneracy condition.
本文研究了E [φ (y (T))]对E [φ (x (T))]的弱逼近问题，其中x (T)是随机微分时滞方程的解，y (T)由欧拉格式定义。对于φ∈C 3b, Buckwar, Kuske, Mohammed and Shardlow(“随机微分时滞方程的欧拉格式的弱收敛性”，LMS J.计算机学报。数学。11(2008)60-69)已经证明欧拉格式具有弱收敛阶1。这里我们证明了当φ仅在附加的非退化条件下被假设为可测且有界时，同样的结果成立。
{"title":"Weak approximation of stochastic differential delay equations for bounded measurable function","authors":"Hua Zhang","doi":"10.1112/S1461157013000120","DOIUrl":"https://doi.org/10.1112/S1461157013000120","url":null,"abstract":"In this paper we study the weak approximation problem of E [ φ ( x ( T ))] by E [ φ ( y ( T ))], where x ( T ) is the solution of a stochastic differential delay equation and y ( T ) is defined by the Euler scheme. For φ ∈ C 3 b , Buckwar, Kuske, Mohammed and Shardlow (‘Weak convergence of the Euler scheme for stochastic differential delay equations’, LMS J. Comput. Math. 11 (2008) 60–69) have shown that the Euler scheme has weak order of convergence 1. Here we prove that the same results hold when φ is only assumed to be measurable and bounded under an additional non-degeneracy condition.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"16 1","pages":"319-343"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410849","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}
Pub Date : 2012-12-01DOI: 10.1112/S1461157012001039
B. Eick, A. Hulpke
We describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.
{"title":"Computing Hall subgroups of finite groups","authors":"B. Eick, A. Hulpke","doi":"10.1112/S1461157012001039","DOIUrl":"https://doi.org/10.1112/S1461157012001039","url":null,"abstract":"We describe an effective algorithm to compute a set of representatives for the conjugacy classes of Hall subgroups of a finite permutation or matrix group. Our algorithm uses the general approach of the so-called ‘trivial Fitting model’.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"205-218"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157012001039","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410369","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}
Pub Date : 2012-12-01DOI: 10.1112/S146115701200112X
Srinath Baba, H. Granath
We study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over ℚ of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.
{"title":"Differential equations and expansions for quaternionic modular forms in the discriminant 6 case","authors":"Srinath Baba, H. Granath","doi":"10.1112/S146115701200112X","DOIUrl":"https://doi.org/10.1112/S146115701200112X","url":null,"abstract":"We study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over ℚ of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"385-399"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S146115701200112X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410940","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}
Pub Date : 2012-12-01DOI: 10.1112/S1461157012000058
A. Islam
The existence of products of three pairwise coprime integers is investigated in short intervals of the form (x, x + x 1 2 ]. A general theorem is proved which shows that such integer products exist provided there is a bound on the product of any two of them. A particular case of relevance to elliptic curve cryptography, where all three integers are of order x 1 3 , is presented as a corollary to this result.
在(x, x + x 12]的短区间内研究了三个对素数整数积的存在性。证明了一个一般定理，证明了只要任意两个整数积有一个界，就存在这样的整数积。与椭圆曲线密码学相关的一个特殊情况，其中所有三个整数都是x 1 3阶，作为该结果的推论。
{"title":"Products of three pairwise coprime integers in short intervals","authors":"A. Islam","doi":"10.1112/S1461157012000058","DOIUrl":"https://doi.org/10.1112/S1461157012000058","url":null,"abstract":"The existence of products of three pairwise coprime integers is investigated in short intervals of the form (x, x + x 1 2 ]. A general theorem is proved which shows that such integer products exist provided there is a bound on the product of any two of them. A particular case of relevance to elliptic curve cryptography, where all three integers are of order x 1 3 , is presented as a corollary to this result.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"59-70"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157012000058","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410198","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}
Pub Date : 2012-12-01DOI: 10.1112/S1461157012001118
Felix Noeske
We revise the matching algorithm of Noeske ( LMS J. Comput. Math. 11 (2008) 213–222) and introduce a new approach via composition series to expedite the calculations. Furthermore, we show how the matching algorithm may be applied in the more general and frequently occurring setting that we are only given subalgebras of the condensed algebras which each contain the separable algebra of one of their Wedderburn–Malcev decompositions.
对Noeske (LMS J. Comput)的匹配算法进行了改进。数学。11(2008)213-222)，并介绍了一种新的方法，通过组成系列来加快计算。此外，我们展示了匹配算法如何应用于更一般的和频繁发生的设置，我们只给出了压缩代数的子代数，每个子代数都包含它们的一个Wedderburn-Malcev分解的可分离代数。
{"title":"Matching simple modules of condensation algebras","authors":"Felix Noeske","doi":"10.1112/S1461157012001118","DOIUrl":"https://doi.org/10.1112/S1461157012001118","url":null,"abstract":"We revise the matching algorithm of Noeske ( LMS J. Comput. Math. 11 (2008) 213–222) and introduce a new approach via composition series to expedite the calculations. Furthermore, we show how the matching algorithm may be applied in the more general and frequently occurring setting that we are only given subalgebras of the condensed algebras which each contain the separable algebra of one of their Wedderburn–Malcev decompositions.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"374-384"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410924","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}
Pub Date : 2012-12-01DOI: 10.1112/S1461157012000010
G. Berkolaiko, E. Buckwar, C. Kelly, A. Rodkina
In the original article [LMS J. Comput. Math. 15 (2012) 71–83], the authors use a discrete form of the Ito formula, developed by Appleby, Berkolaiko and Rodkina [Stochastics 81 (2009) no. 2, 99–127], to show that the almost sure asymptotic stability of a particular two-dimensional test system is preserved when the discretisation step size is small. In this Corrigendum, we identify an implicit assumption in the original proof of the discrete Ito formula that, left unaddressed, would preclude its application to the test system of interest. We resolve this problem by reproving the relevant part of the discrete Ito formula in such a way that confirms its applicability to our test equation. Thus, we reaffirm the main results and conclusions of the original article.
在原文中[LMS J. Comput]。数学。15(2012)71-83]，作者使用了离散形式的伊托公式，由Appleby, Berkolaiko和Rodkina[随机统计81 (2009)no。[2,99 - 127]，以证明当离散步长较小时，特定二维测试系统的几乎肯定渐近稳定性是保持的。在本勘误中，我们在离散伊藤公式的原始证明中确定了一个隐含的假设，如果不加以解决，将妨碍其应用于感兴趣的测试系统。我们通过重新证明离散伊藤公式的相关部分来解决这个问题，从而证实了它对我们的测试方程的适用性。因此，我们重申了原文章的主要结果和结论。
{"title":"Almost sure asymptotic stability analysis of the θ -Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations","authors":"G. Berkolaiko, E. Buckwar, C. Kelly, A. Rodkina","doi":"10.1112/S1461157012000010","DOIUrl":"https://doi.org/10.1112/S1461157012000010","url":null,"abstract":"In the original article [LMS J. Comput. Math. 15 (2012) 71–83], the authors use a discrete form of the Ito formula, developed by Appleby, Berkolaiko and Rodkina [Stochastics 81 (2009) no. 2, 99–127], to show that the almost sure asymptotic stability of a particular two-dimensional test system is preserved when the discretisation step size is small. In this Corrigendum, we identify an implicit assumption in the original proof of the discrete Ito formula that, left unaddressed, would preclude its application to the test system of interest. We resolve this problem by reproving the relevant part of the discrete Ito formula in such a way that confirms its applicability to our test equation. Thus, we reaffirm the main results and conclusions of the original article.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"71-83"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157012000010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410557","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}
Pub Date : 2012-12-01DOI: 10.1112/S1461157012001143
Jonas Siurys
We prove that for each positive integer k in the range 2≤ k ≤10 and for each positive integer k ≡79 ( mod 120) there is a k -step Fibonacci-like sequence of composite numbers and give some examples of such sequences. This is a natural extension of a result of Graham for the Fibonacci-like sequence.
{"title":"A linear recurrence sequence of composite numbers","authors":"Jonas Siurys","doi":"10.1112/S1461157012001143","DOIUrl":"https://doi.org/10.1112/S1461157012001143","url":null,"abstract":"We prove that for each positive integer k in the range 2≤ k ≤10 and for each positive integer k ≡79 ( mod 120) there is a k -step Fibonacci-like sequence of composite numbers and give some examples of such sequences. This is a natural extension of a result of Graham for the Fibonacci-like sequence.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"360-373"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157012001143","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411000","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}
Pub Date : 2012-12-01DOI: 10.1112/S146115701200109X
C. Diem
From power series expansions of functions on curves over finite fields, one can obtain sequences with perfect or almost perfect linear complexity profile. It has been suggested by various authors to use such sequences as key streams for stream ciphers. In this work, we show how long parts of such sequences can be computed efficiently from short ones. Such sequences should therefore considered to be cryptographically weak. Our attack leads in a natural way to a new measure of the complexity of sequences which we call expansion complexity.
{"title":"On the use of expansion series for stream ciphers","authors":"C. Diem","doi":"10.1112/S146115701200109X","DOIUrl":"https://doi.org/10.1112/S146115701200109X","url":null,"abstract":"From power series expansions of functions on curves over finite fields, one can obtain sequences with perfect or almost perfect linear complexity profile. It has been suggested by various authors to use such sequences as key streams for stream ciphers. In this work, we show how long parts of such sequences can be computed efficiently from short ones. Such sequences should therefore considered to be cryptographically weak. Our attack leads in a natural way to a new measure of the complexity of sequences which we call expansion complexity.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"326-340"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S146115701200109X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410865","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}
Pub Date : 2012-12-01DOI: 10.1112/S1461157012000071
Klaus Lux, Max Neunhöffer, Felix Noeske
We present an ecient algorithm for the condensation of homomorphism spaces. This provides an improvement over the known tensor condensation method which is essentially due to a better choice of bases. We explain the theory behind this approach and describe the implementation in detail. Finally, we give timings to compare with previous methods.
{"title":"Condensation of homomorphism spaces","authors":"Klaus Lux, Max Neunhöffer, Felix Noeske","doi":"10.1112/S1461157012000071","DOIUrl":"https://doi.org/10.1112/S1461157012000071","url":null,"abstract":"We present an ecient algorithm for the condensation of homomorphism spaces. This provides an improvement over the known tensor condensation method which is essentially due to a better choice of bases. We explain the theory behind this approach and describe the implementation in detail. Finally, we give timings to compare with previous methods.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"15 1","pages":"140-157"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157012000071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63410252","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}
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