Stirling numbers of the first kind are common in number theory and combinatorics; through Ewen's sampling formula, these numbers enter into the calculation of several population genetics statistics, such as Fu's Fs. In previous papers we have considered an asymptotic estimator for a finite sum of Stirling numbers, which enables rapid and accurate calculation of Fu's Fs. These sums can also be viewed as a cumulative distribution function; this formulation leads directly to an inversion problem, where, given a value for Fu's Fs, the goal is to solve for one of the input parameters. We solve this inversion using Newton iteration for small parameters. For large parameters we need to extend the earlier obtained asymptotic results to handle the inversion problem asymptotically. Numerical experiments are given to show the efficiency of both solving the inversion problem and the expanded estimator for the statistical quantities.
{"title":"A distribution function from population genetics statistics using Stirling numbers of the first kind: Asymptotics, inversion and numerical evaluation","authors":"S. Chen, N. Temme","doi":"10.1090/mcom/3711","DOIUrl":"https://doi.org/10.1090/mcom/3711","url":null,"abstract":"Stirling numbers of the first kind are common in number theory and combinatorics; through Ewen's sampling formula, these numbers enter into the calculation of several population genetics statistics, such as Fu's Fs. In previous papers we have considered an asymptotic estimator for a finite sum of Stirling numbers, which enables rapid and accurate calculation of Fu's Fs. These sums can also be viewed as a cumulative distribution function; this formulation leads directly to an inversion problem, where, given a value for Fu's Fs, the goal is to solve for one of the input parameters. We solve this inversion using Newton iteration for small parameters. For large parameters we need to extend the earlier obtained asymptotic results to handle the inversion problem asymptotically. Numerical experiments are given to show the efficiency of both solving the inversion problem and the expanded estimator for the statistical quantities.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79460116","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}
Let $mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $mathcal{A}_n$'s stabilizes for the real fields $F=mathbb{Q}(sqrt{f})$ for any integer $0
设$mathcal{A}_n$是实二次元域$F$的环切$mathbb{Z}_2$扩展的$n$第n$层理想类群的$2$部分。$mathcal{A}_n$的基数与整组环切单位的索引有关。本文提出了一种研究后一指标的方法。作为一个应用，我们证明了$mathcal{A}_n$的序列对于实域$F=mathbb{Q}(sqrt{F})$对于任意整数$0< F <10000$是稳定的。格林伯格的猜想同样适用于这些领域。
{"title":"Greenberg's conjecture for real quadratic fields and the cyclotomic Z2-extensions","authors":"L. Pagani","doi":"10.1090/mcom/3712","DOIUrl":"https://doi.org/10.1090/mcom/3712","url":null,"abstract":"Let $mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th layer of the cyclotomic $mathbb{Z}_2$-extension of a real quadratic number field $F$. The cardinality of $mathcal{A}_n$ is related to the index of cyclotomic units in the full group of units. We present a method to study the latter index. As an application we show that the sequence of the $mathcal{A}_n$'s stabilizes for the real fields $F=mathbb{Q}(sqrt{f})$ for any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those fields.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73093813","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}
. A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.
{"title":"Improved computation of fundamental domains for arithmetic Fuchsian groups","authors":"James Rickards","doi":"10.1090/mcom/3777","DOIUrl":"https://doi.org/10.1090/mcom/3777","url":null,"abstract":". A practical algorithm to compute the fundamental domain of an arithmetic Fuchsian group was given by Voight, and implemented in Magma. It was later expanded by Page to the case of arithmetic Kleinian groups. We combine and improve on parts of both algorithms to produce a more efficient algorithm for arithmetic Fuchsian groups. This algorithm is implemented in PARI/GP, and we demonstrate the improvements by comparing running times versus the live Magma implementation.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85355990","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}
Carlos Beltrán, Laurent Bétermin, P. Grabner, S. Steinerberger
{"title":"How well-conditioned can the eigenvector problem be?","authors":"Carlos Beltrán, Laurent Bétermin, P. Grabner, S. Steinerberger","doi":"10.1090/mcom/3706","DOIUrl":"https://doi.org/10.1090/mcom/3706","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91452714","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}
{"title":"Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation","authors":"Dong Li, Chaoyu Quan, T. Tang","doi":"10.1090/mcom/3704","DOIUrl":"https://doi.org/10.1090/mcom/3704","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74781507","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}
A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24%$ larger than the width of the regular octagon: $cos(pi/8)$. In addition, the paper proposes a family of equilateral small $n$-gons, for $n=2^s$ with $sge 4$, whose widths are within $O(1/n^4)$ of the maximal width.
小多边形是单位直径的多边形。具有$n=2^s$顶点的等边小多边形的最大宽度不知道，当$s ge 3$。本文解决了第一种开放情况，找到了最优的等边小八边形。它的宽度大约$3.24%$大于正八边形的宽度$cos(pi/8)$。此外，对于$n=2^s$和$sge 4$，本文提出了一类宽度在最大宽度$O(1/n^4)$以内的等边小的$n$ -gons。
{"title":"The equilateral small octagon of maximal width","authors":"Christian Bingane, Charles Audet","doi":"10.1090/mcom/3733","DOIUrl":"https://doi.org/10.1090/mcom/3733","url":null,"abstract":"A small polygon is a polygon of unit diameter. The maximal width of an equilateral small polygon with $n=2^s$ vertices is not known when $s ge 3$. This paper solves the first open case and finds the optimal equilateral small octagon. Its width is approximately $3.24%$ larger than the width of the regular octagon: $cos(pi/8)$. In addition, the paper proposes a family of equilateral small $n$-gons, for $n=2^s$ with $sge 4$, whose widths are within $O(1/n^4)$ of the maximal width.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74012378","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}
We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters, $gamma_j$, satisfy the summability condition $sum_{j=1}^{infty}gamma_j^{1/alpha}
{"title":"Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate","authors":"J. Dick, T. Goda, Kosuke Suzuki","doi":"10.1090/mcom/3769","DOIUrl":"https://doi.org/10.1090/mcom/3769","url":null,"abstract":"We study a randomized quadrature algorithm to approximate the integral of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich (2019) shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters, $gamma_j$, satisfy the summability condition $sum_{j=1}^{infty}gamma_j^{1/alpha}<infty$, where $alpha$ is a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces. In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition $sum_{j=1}^{infty}gamma_j^{1/alpha}<infty$ holds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88362704","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}
We devise 3-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology. We prove that the sequence of numerical approximations converges to a weak solution of the problem. We develop a block preconditioner based on augmented Lagrangian stabilisation for a discretisation based on the Scott-Vogelius finite element pair for the velocity and pressure. The preconditioner involves a specialised multigrid algorithm that makes use of a space-decomposition that captures the kernel of the divergence and non-standard intergrid transfer operators. The preconditioner exhibits robust convergence behaviour when applied to the Navier-Stokes and power-law systems, including temperature-dependent viscosity, heat conductivity and viscous dissipation.
{"title":"Finite element approximation and preconditioning for anisothermal flow of implicitly-constituted non-Newtonian fluids","authors":"P. Farrell, P. A. Gazca-Orozco, E. Süli","doi":"10.1090/mcom/3703","DOIUrl":"https://doi.org/10.1090/mcom/3703","url":null,"abstract":"We devise 3-field and 4-field finite element approximations of a system describing the steady state of an incompressible heat-conducting fluid with implicit non-Newtonian rheology. We prove that the sequence of numerical approximations converges to a weak solution of the problem. We develop a block preconditioner based on augmented Lagrangian stabilisation for a discretisation based on the Scott-Vogelius finite element pair for the velocity and pressure. The preconditioner involves a specialised multigrid algorithm that makes use of a space-decomposition that captures the kernel of the divergence and non-standard intergrid transfer operators. The preconditioner exhibits robust convergence behaviour when applied to the Navier-Stokes and power-law systems, including temperature-dependent viscosity, heat conductivity and viscous dissipation.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88746456","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}
The numerical simulation of the incompressible Navier-Stokes equations presents a challenging computational task primarily because of two reasons: (a) the coupling of the velocity and pressure by the incompressibility constraint and (b) the nonlinearity of the convection term [14, 18]. The development of splitting schemes aims to overcome these difficulties by decoupling the nonlinearity in the convection term from the pressure term. For an overview of such methods, we refer to the works of Glowinski [15] and of Guermond, Minev, and Shen [18]. In this paper, we will focus on pressure correction schemes. The basic idea of a non-incremental pressure correction scheme in time was first proposed by Chorin and Temam [5, 28]. This scheme was subsequently modified by several mathematicians leading to two major variations: (1) the incremental scheme where a previous value of the pressure gradient is added [16,30] and (2) the rotational scheme where the non-physical boundary condition for the pressure is corrected by using the rotational form of the Laplacian [29]. The main contribution of our work is the theoretical analysis of a discontinuous Galerkin (dG) discretization of the pressure correction approach. We derive stability and a priori error bounds on a family of regular meshes. The discrete velocities are approximated by discontinuous piecewise polynomials of degree k1 and the discrete potential and pressure by polynomials of degree k2. Stability of the solutions is obtained under the constraint k1−1 ≤ k2 ≤ k1+1 whereas the convergence of the scheme is obtained for the case k2 = k1 − 1 because of approximation properties. The proofs are technical and rely on several tools including special lift operators. The semi-discrete error analysis of pressure correction schemes has been extensively studied, see for example the work by Shen and Guermond [21, 27]. The use
不可压缩Navier-Stokes方程的数值模拟是一项具有挑战性的计算任务，主要有两个原因:(a)不可压缩约束对速度和压力的耦合;(b)对流项的非线性[14,18]。分裂格式的发展旨在通过将对流项和压力项的非线性解耦来克服这些困难。对于这些方法的概述，我们参考了Glowinski[15]和Guermond, Minev, and Shen[15]的作品。在本文中，我们将重点讨论压力校正方案。非增量压力及时校正方案的基本思想最早由Chorin和Temam提出[5,28]。随后，几位数学家对该格式进行了修改，导致了两个主要的变化:(1)增量格式，其中添加了先前的压力梯度值[16,30];(2)旋转格式，其中使用拉普拉斯[29]的旋转形式修正了压力的非物理边界条件。本文的主要贡献是对压力校正方法的不连续伽辽金离散化进行了理论分析。我们推导了一类规则网格的稳定性和先验误差界。离散速度近似为k1阶的不连续分段多项式，离散势和压力近似为k2阶的多项式。在k1−1≤k2≤k1+1的约束下，得到了解的稳定性，而在k2 = k1−1的近似条件下，得到了解的收敛性。证明是技术性的，依赖于几种工具，包括特殊的升降机操作员。压力校正方案的半离散误差分析已经得到了广泛的研究，例如参见Shen和Guermond[21,27]的工作。使用
{"title":"A discontinuous Galerkin pressure correction scheme for the incompressible Navier-Stokes equations: stability and convergence","authors":"R. Masri, Chen Liu, B. Rivière","doi":"10.1090/mcom/3731","DOIUrl":"https://doi.org/10.1090/mcom/3731","url":null,"abstract":"The numerical simulation of the incompressible Navier-Stokes equations presents a challenging computational task primarily because of two reasons: (a) the coupling of the velocity and pressure by the incompressibility constraint and (b) the nonlinearity of the convection term [14, 18]. The development of splitting schemes aims to overcome these difficulties by decoupling the nonlinearity in the convection term from the pressure term. For an overview of such methods, we refer to the works of Glowinski [15] and of Guermond, Minev, and Shen [18]. In this paper, we will focus on pressure correction schemes. The basic idea of a non-incremental pressure correction scheme in time was first proposed by Chorin and Temam [5, 28]. This scheme was subsequently modified by several mathematicians leading to two major variations: (1) the incremental scheme where a previous value of the pressure gradient is added [16,30] and (2) the rotational scheme where the non-physical boundary condition for the pressure is corrected by using the rotational form of the Laplacian [29]. The main contribution of our work is the theoretical analysis of a discontinuous Galerkin (dG) discretization of the pressure correction approach. We derive stability and a priori error bounds on a family of regular meshes. The discrete velocities are approximated by discontinuous piecewise polynomials of degree k1 and the discrete potential and pressure by polynomials of degree k2. Stability of the solutions is obtained under the constraint k1−1 ≤ k2 ≤ k1+1 whereas the convergence of the scheme is obtained for the case k2 = k1 − 1 because of approximation properties. The proofs are technical and rely on several tools including special lift operators. The semi-discrete error analysis of pressure correction schemes has been extensively studied, see for example the work by Shen and Guermond [21, 27]. The use","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80458661","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}
M. Kirschmer, Fabien Narbonne, C. Ritzenthaler, Damien Robert
Let E E be an ordinary elliptic curve over a finite field and g g be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of E g E^g . The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to E 3 E^3 and of the Igusa modular form in dimension 4 4 . We illustrate our algorithms with examples of curves with many rational points over finite fields.
设E E为有限域上的普通椭圆曲线，g g为正整数。在一定的技术假设下，给出了一种跨出等同构类E g E^g中的主极化阿贝尔变体同构类的算法。这些变化首先被描述为(不一定是最大的)二次阶的厄米格，然后在几何上根据它们的代数零点。我们还展示了如何通过仔细选择零点的仿射升力，在常数中以多项式形式给出偶权的西格尔模形式的代数计算。然后，我们利用这些结果给出了在4维4上的Igusa模形式的主要极化阿贝尔三倍等齐e ^3的Serre障碍的代数计算。我们用有限域上有许多有理点的曲线的例子来说明我们的算法。
{"title":"Spanning the isogeny class of a power of an elliptic curve","authors":"M. Kirschmer, Fabien Narbonne, C. Ritzenthaler, Damien Robert","doi":"10.1090/MCOM/3672","DOIUrl":"https://doi.org/10.1090/MCOM/3672","url":null,"abstract":"Let \u0000\u0000 \u0000 E\u0000 E\u0000 \u0000\u0000 be an ordinary elliptic curve over a finite field and \u0000\u0000 \u0000 g\u0000 g\u0000 \u0000\u0000 be a positive integer. Under some technical assumptions, we give an algorithm to span the isomorphism classes of principally polarized abelian varieties in the isogeny class of \u0000\u0000 \u0000 \u0000 E\u0000 g\u0000 \u0000 E^g\u0000 \u0000\u0000. The varieties are first described as hermitian lattices over (not necessarily maximal) quadratic orders and then geometrically in terms of their algebraic theta null point. We also show how to algebraically compute Siegel modular forms of even weight given as polynomials in the theta constants by a careful choice of an affine lift of the theta null point. We then use these results to give an algebraic computation of Serre’s obstruction for principally polarized abelian threefolds isogenous to \u0000\u0000 \u0000 \u0000 E\u0000 3\u0000 \u0000 E^3\u0000 \u0000\u0000 and of the Igusa modular form in dimension \u0000\u0000 \u0000 4\u0000 4\u0000 \u0000\u0000. We illustrate our algorithms with examples of curves with many rational points over finite fields.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79735849","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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