We present new algorithms for computing the low nn bits or the high nn bits of the product of two nn-bit integers. We show that these problems may be solved in asymptotically 7575% of the time required to compute the full 2n2n-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers.
我们提出了计算两个 n n 位整数乘积的低 n n 位或高 n n 位的新算法。我们的研究表明，假设底层整数乘法算法依赖于计算实数序列的循环卷积，那么这些问题可以在计算全部 2 n 2n 位乘积所需时间的 75% 左右的时间内得到解决。
{"title":"Faster truncated integer multiplication","authors":"David Harvey","doi":"10.1090/mcom/3939","DOIUrl":"https://doi.org/10.1090/mcom/3939","url":null,"abstract":"<p>We present new algorithms for computing the low <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bits or the high <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bits of the product of two <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=\"application/x-tex\">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bit integers. We show that these problems may be solved in asymptotically <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"75\"> <mml:semantics> <mml:mn>75</mml:mn> <mml:annotation encoding=\"application/x-tex\">75%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the time required to compute the full <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2 n\"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">2n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bit product, assuming that the underlying integer multiplication algorithm relies on computing cyclic convolutions of sequences of real numbers.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra QGmathbb {Q}G for GG a finite generalized strongly monomial group. For the same groups with no exceptional simple components in QGmathbb {Q}G, we describe a subgroup of finite index in the group of units U(ZG)mathcal {U}(mathbb {Z}G) of the integral group ring ZGmathbb {Z}G that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement are a class of generalized strongly monomial groups where the theory developed in this paper is applicable.
对于有限广义强单项式群 G G，我们提出了一种明确计算有理群代数 Q G mathbb {Q}G 中舒尔指数为 1 的简单分量中完整的正交原始幂级数的方法。对于 Q G mathbb {Q}G 中没有特殊简单成分的相同群，我们描述了积分群环 Z G mathbb {Z}G 的单位群 U ( Z G ) mathcal {U}(mathbb {Z}G) 中的有限指数子群，该子群由三个零能群生成，我们给出了它们的生成子的明确描述。我们用一个详细的具体例子来举例说明理论构造。我们还证明了具有循环补集的奇阶弗罗贝纽斯群是一类广义强单项式群，本文所建立的理论适用于这类群。
{"title":"Rational group algebras of generalized strongly monomial groups: Primitive idempotents and units","authors":"Gurmeet Bakshi, Jyoti Garg, Gabriela Olteanu","doi":"10.1090/mcom/3937","DOIUrl":"https://doi.org/10.1090/mcom/3937","url":null,"abstract":"<p>We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q upper G\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a finite generalized strongly monomial group. For the same groups with no exceptional simple components in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q upper G\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Q}G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we describe a subgroup of finite index in the group of units <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper U left-parenthesis double-struck upper Z upper G right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"script\">U</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathcal {U}(mathbb {Z}G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the integral group ring <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Z upper G\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {Z}G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement are a class of generalized strongly monomial groups where the theory developed in this paper is applicable.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance (W2W_{2}). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on S1mathbb {S}^{1} is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to O(N)O(N) from O(N3)O(N^{3}) of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.
{"title":"Optimal transportation for electrical impedance tomography","authors":"Gang Bao, Yixuan Zhang","doi":"10.1090/mcom/3919","DOIUrl":"https://doi.org/10.1090/mcom/3919","url":null,"abstract":"This work establishes a framework for solving inverse boundary problems with the geodesic-based quadratic Wasserstein distance (<inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper W 2\"> <mml:semantics> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=\"application/x-tex\">W_{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). A general form of the Fréchet gradient is systematically derived from the optimal transportation (OT) theory. In addition, a fast algorithm based on the new formulation of OT on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper S Superscript 1\"> <mml:semantics> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">S</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {S}^{1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is developed to solve the corresponding optimal transport problem. The computational complexity of the algorithm is reduced to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N cubed right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(N^{3})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the traditional method. Combining with the adjoint-state method, this framework provides a new computational approach for solving the challenging electrical impedance tomography problem. Numerical examples are presented to illustrate the effectiveness of our method.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134992705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Geir Bogfjellmo, Elena Celledoni, Robert McLachlan, Brynjulf Owren, G. Quispel
The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures and integrals. Computerized methods based on discrete Darboux polynomials have recently been used for finding these measures and integrals. However, if the differential system contains many parameters, this approach can lead to highly complex results that can be difficult to interpret and analyse. But this complexity can in some cases be substantially reduced by using aromatic series. These are a mathematical tool introduced independently by Chartier and Murua and by Iserles, Quispel and Tse. We develop an algorithm for this purpose and derive some necessary conditions for the Kahan map to have preserved measures and integrals expressible in terms of aromatic functions. An important reason for the success of this method lies in the equivariance of the map from vector fields to their aromatic functions. We demonstrate the algorithm on a number of examples showing a great reduction in complexity compared to what had been obtained by a fixed basis such as monomials.
{"title":"Using aromas to search for preserved measures and integrals in Kahan’s method","authors":"Geir Bogfjellmo, Elena Celledoni, Robert McLachlan, Brynjulf Owren, G. Quispel","doi":"10.1090/mcom/3921","DOIUrl":"https://doi.org/10.1090/mcom/3921","url":null,"abstract":"The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures and integrals. Computerized methods based on discrete Darboux polynomials have recently been used for finding these measures and integrals. However, if the differential system contains many parameters, this approach can lead to highly complex results that can be difficult to interpret and analyse. But this complexity can in some cases be substantially reduced by using aromatic series. These are a mathematical tool introduced independently by Chartier and Murua and by Iserles, Quispel and Tse. We develop an algorithm for this purpose and derive some necessary conditions for the Kahan map to have preserved measures and integrals expressible in terms of aromatic functions. An important reason for the success of this method lies in the equivariance of the map from vector fields to their aromatic functions. We demonstrate the algorithm on a number of examples showing a great reduction in complexity compared to what had been obtained by a fixed basis such as monomials.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Don Coppersmith, Michael Mossinghoff, Danny Scheinerman, Jeffrey VanderKam
For given positive integers mm and nn with m>nm>n, the Prouhet–Tarry–Escott problem asks if there exist two disjoint multisets of integers of size nn having identical kkth moments for 1≤k≤m1leq kleq m; in the ideal case one requires m=n−1m=n-1, which is maximal. We describe some searches for ideal solutions to the Prouhet–Tarry–Escott problem, especially solutions possessing a particular symmetry, both over Zmathbb {Z} and over the ring of intege
In this work, we develop a multifactor approximation for dd-dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L2L^2-estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in (0,1/2)(0,1/2), we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.
{"title":"Approximation of stochastic Volterra equations with kernels of completely monotone type","authors":"Aurélien Alfonsi, Ahmed Kebaier","doi":"10.1090/mcom/3911","DOIUrl":"https://doi.org/10.1090/mcom/3911","url":null,"abstract":"In this work, we develop a multifactor approximation for <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=\"application/x-tex\">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared\"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">L^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 0 comma 1 slash 2 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(0,1/2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135874750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present two approaches that can be used to compute modular forms on noncongruence subgroups. The first approach uses Hejhal’s method for which we improve the arbitrary precision solving techniques so that the algorithm becomes about up to two orders of magnitude faster in practical computations. This allows us to obtain high precision numerical estimates of the Fourier coefficients from which the algebraic expressions can be identified using the LLL algorithm. The second approach is restricted to genus zero subgroups and uses efficient methods to compute the Belyi map from which the modular forms can be constructed.
{"title":"On the computation of modular forms on noncongruence subgroups","authors":"David Berghaus, Hartmut Monien, Danylo Radchenko","doi":"10.1090/mcom/3903","DOIUrl":"https://doi.org/10.1090/mcom/3903","url":null,"abstract":"We present two approaches that can be used to compute modular forms on noncongruence subgroups. The first approach uses Hejhal’s method for which we improve the arbitrary precision solving techniques so that the algorithm becomes about up to two orders of magnitude faster in practical computations. This allows us to obtain high precision numerical estimates of the Fourier coefficients from which the algebraic expressions can be identified using the LLL algorithm. The second approach is restricted to genus zero subgroups and uses efficient methods to compute the Belyi map from which the modular forms can be constructed.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136018097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a matrix A∈GLd(Z)Ain mathrm {GL}_d(mathbb {Z}). We study the pseudorandomness of vectors unmathbf {u}_n generated by a linear recurrence relation of the form un+1≡Aun(modpt),n=0,1,…,begin{equation*} mathbf {u}_{n+1} equiv A mathbf {u}_n pmod {p^t}, qquad n = 0, 1, ldots , end{equation*} modulo ptp^t w
给定矩阵a∈gl d(Z) a inmathrm GL＿d{(}mathbb Z{)。我们研究向量u n }mathbf u＿n{的伪随机性，由形式为u n + 1≡a u n (mod p t)， n = 0,1，…，}begin{equation*} mathbf {u}_{n+1} equiv A mathbf {u}_n pmod {p^t}, qquad n = 0, 1, ldots , end{equation*}模p t p^t与固定素数p p和足够大的整数t大于或等于1 t geqslant 1的线性递归关系生成。我们研究这样的序列在非常短的片段长度，这是无法通过以前使用的方法访问。我们的技术是基于N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972)， pp. 654-670, 672]估计双Weyl和的方法和K. Ford的Vinogradov中值定理的完全显式形式[Proc. London mathematics]。Soc。(3) 85 (2002)， pp. 565-633。这与I. E. Shparlinski [Proc. Voronezh State Pedagogical institute .， 197 (1978)， 74-85 (in Russian)]的一些想法相结合，它允许我们构建u n mathbf u＿n{坐标的多项式表示，并在多项式表示中控制其系数的p p进阶。}
{"title":"Distribution of recursive matrix pseudorandom number generator modulo prime powers","authors":"László Mérai, Igor Shparlinski","doi":"10.1090/mcom/3895","DOIUrl":"https://doi.org/10.1090/mcom/3895","url":null,"abstract":"Given a matrix <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A element-of normal upper G normal upper L Subscript d Baseline left-parenthesis double-struck upper Z right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">G</mml:mi> <mml:mi mathvariant=\"normal\">L</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Ain mathrm {GL}_d(mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We study the pseudorandomness of vectors <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold u Subscript n\"> <mml:semantics> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"bold\">u</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">mathbf {u}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generated by a linear recurrence relation of the form <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold u Subscript n plus 1 Baseline identical-to upper A bold u Subscript n Baseline left-parenthesis mod p Superscript t Baseline right-parenthesis comma n equals 0 comma 1 comma ellipsis comma\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"bold\">u</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mi>A</mml:mi> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"bold\">u</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mspace width=\"0.667em\" /> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=\"0.333em\" /> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>t</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width=\"2em\" /> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…<!-- … --></mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">begin{equation*} mathbf {u}_{n+1} equiv A mathbf {u}_n pmod {p^t}, qquad n = 0, 1, ldots , end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> modulo <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Superscript t\"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>t</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">p^t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> w","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135112108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Alex Best, L. Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, Yujie Xu
Kim gave a new proof of Siegel’s Theorem that there are only finitely many SS-integral points on PZ1∖{0,1,∞}mathbb {P}^1_mathbb {Z}setminus {0,1,infty }. One advantage of Kim’s method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of SS increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where SS has size 22 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.
Kim给出了西格尔定理的一个新的证明，证明在P Z 1∈{0,1，∞}mathbb P{^1＿ }mathbb Z{}setminus ｛0,1, infty｝上只有有限多个S -积分点。Kim的方法的一个优点是，它原则上允许人们实际找到这些点，但随着S的大小增加，计算变得非常复杂。在本文中，我们实现了Kim的方法的改进，以显式地计算由Betts和Dogra引入的S的大小为22的各种示例。在这样做的过程中，我们展示了Kim猜想的自然推广的新例子。
{"title":"Refined Selmer equations for the thrice-punctured line in depth two","authors":"Alex Best, L. Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, Yujie Xu","doi":"10.1090/mcom/3898","DOIUrl":"https://doi.org/10.1090/mcom/3898","url":null,"abstract":"Kim gave a new proof of Siegel’s Theorem that there are only finitely many <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-integral points on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P Subscript double-struck upper Z Superscript 1 Baseline minus StartSet 0 comma 1 comma normal infinity EndSet\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo class=\"MJX-variant\">∖<!-- ∖ --></mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {P}^1_mathbb {Z}setminus {0,1,infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One advantage of Kim’s method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has size <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\"application/x-tex\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135219379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.
{"title":"Minimal residual methods in negative or fractional Sobolev norms","authors":"Harald Monsuur, Rob Stevenson, Johannes Storn","doi":"10.1090/mcom/3904","DOIUrl":"https://doi.org/10.1090/mcom/3904","url":null,"abstract":"For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135923277","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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