{"title":"Stability of finite difference schemes approximation for hyperbolic boundary value problems in an interval","authors":"Antoine Benoit","doi":"10.1090/mcom/3698","DOIUrl":"https://doi.org/10.1090/mcom/3698","url":null,"abstract":",","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73196812","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}
Summary: In this paper we present lower and upper bounds for Kummer’s function ratios of the form M ( a,b,z ) ′ M ( a,b,z ) when 0 < a < b . The derived bounds are asymptotically precise, theoretically well-defined, numerically accurate, and easy to compute. Moreover, we show how the bounds can be used as starting values for monotonically convergent sequences to approximate the ratio with even higher precision while avoiding the anomalous convergence discussed by Gautschi [Math. Comp. 31 (1977), pp. 994-999]. This allows to apply the results in multiple areas, as for example the estimation of Watson distributions in statistical modelling. Furthermore, we extend the convergence results provided by Gautschi and the list of known bounds for the inverse of Kummer’s function ratio given by Sra and Karp [J. Multivariate Anal. 114 (2013), pp. 256-269]. In addition, the derived starting bounds are compared and connected to other results from the literature.
摘要:本文给出了当0 < a < b时，M (a,b,z) ' M (a,b,z)的Kummer函数比的下界和上界。所导出的界是渐近精确的，理论上定义良好的，数值精确的，并且易于计算。此外，我们展示了如何使用边界作为单调收敛序列的起始值，以更高的精度近似比率，同时避免了Gautschi [Math]讨论的异常收敛。汇编31(1977)，第994-999页]。这允许将结果应用于多个领域，例如统计建模中沃森分布的估计。进一步推广了Gautschi给出的收敛性结果以及由Sra和Karp给出的Kummer函数比逆的已知界列表[J]。多元肛门。114 (2013)，pp. 256-269]。此外，还将推导出的起始界与文献中的其他结果进行了比较和联系。
{"title":"On bounds for Kummer's function ratio","authors":"Lukas Sablica, K. Hornik","doi":"10.1090/mcom/3690","DOIUrl":"https://doi.org/10.1090/mcom/3690","url":null,"abstract":"Summary: In this paper we present lower and upper bounds for Kummer’s function ratios of the form M ( a,b,z ) ′ M ( a,b,z ) when 0 < a < b . The derived bounds are asymptotically precise, theoretically well-defined, numerically accurate, and easy to compute. Moreover, we show how the bounds can be used as starting values for monotonically convergent sequences to approximate the ratio with even higher precision while avoiding the anomalous convergence discussed by Gautschi [Math. Comp. 31 (1977), pp. 994-999]. This allows to apply the results in multiple areas, as for example the estimation of Watson distributions in statistical modelling. Furthermore, we extend the convergence results provided by Gautschi and the list of known bounds for the inverse of Kummer’s function ratio given by Sra and Karp [J. Multivariate Anal. 114 (2013), pp. 256-269]. In addition, the derived starting bounds are compared and connected to other results from the literature.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85518546","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 error analysis for a second-order approximation of 1D nonlocal Schrödinger equation under DtN-type boundary conditions","authors":"Jihong Wang, Jiwei Zhang, Chunxiong Zheng","doi":"10.1090/mcom/3685","DOIUrl":"https://doi.org/10.1090/mcom/3685","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86345707","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 the $q$-analogue of the average of Montgomery's function $F(alpha, T)$ over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain upper and lower bounds for this average over an interval that are quite close to the pointwise conjectured value of 1. To compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee, Chirre, and Milinovich, and apply computational methods of non-smooth programming.
{"title":"On the q-analogue of the Pair Correlation Conjecture via Fourier optimization","authors":"Oscar E. Quesada-Herrera","doi":"10.1090/mcom/3747","DOIUrl":"https://doi.org/10.1090/mcom/3747","url":null,"abstract":"We study the $q$-analogue of the average of Montgomery's function $F(alpha, T)$ over bounded intervals. Assuming the Generalized Riemann Hypothesis for Dirichlet $L$-functions, we obtain upper and lower bounds for this average over an interval that are quite close to the pointwise conjectured value of 1. To compute our bounds, we extend a Fourier analysis approach by Carneiro, Chandee, Chirre, and Milinovich, and apply computational methods of non-smooth programming.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75177861","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":"Nitsche's method for Navier-Stokes equations with slip boundary conditions","authors":"I. Gjerde, L. Scott","doi":"10.1090/mcom/3682","DOIUrl":"https://doi.org/10.1090/mcom/3682","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73736900","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 Lq-approximation and integration for functions from the Sobolev space W s p (Ω) and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use iid sample points, uniformly distributed on the domain. The main result is that we obtain the same optimal rate of convergence if we restrict to iid sampling, a common assumption in learning and uncertainty quantification. The only exception is when p = q = ∞, where a logarithmic loss cannot be avoided.
{"title":"Recovery of Sobolev functions restricted to iid sampling","authors":"David Krieg, E. Novak, Mathias Sonnleitner","doi":"10.1090/mcom/3763","DOIUrl":"https://doi.org/10.1090/mcom/3763","url":null,"abstract":"We study Lq-approximation and integration for functions from the Sobolev space W s p (Ω) and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use iid sample points, uniformly distributed on the domain. The main result is that we obtain the same optimal rate of convergence if we restrict to iid sampling, a common assumption in learning and uncertainty quantification. The only exception is when p = q = ∞, where a logarithmic loss cannot be avoided.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75933849","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 describe an algorithm for computing p-adic L-functions of characters of totally real elds. Such p-adic L-functions were constructed in the 1970’s independently by Barsky and CassouNoguès [Bar78, CN79] based on the explicit formula for zeta values of Shintani [Shi76] and by Serre and Deligne–Ribet [Ser73, DR80] using Hilbert modular forms and an idea of Siegel [Sie68] going back to Hecke [Hec24, Satz 3]. An algorithm for computing via the approach of Cassou-Noguès was developed by Roblot1 [Rob15]. Our algorithm follows the approach of Serre and Siegel, and its computational e ciency rests upon a method for computing with p-adic spaces of modular forms developed in previous work by the authors. The idea of our method is simple. In Serre’s approach, the value of the p-adic L-function of a totally real eld of degree d at a non-positive integer 1 − k is interpreted as the constant term of a classical modular form of weight dk obtained by diagonally restricting a Hilbert Eisenstein series. For small values of k these constants can be computed easily using an idea of Siegel, which goes back to Hecke. To compute the p-adic L-function at arbitrary points in its domain, to some nite p-adic precision, we use a method for computing p-adically with modular forms in larger weight developed in [Lau11, Von15]. We compute the required constant term in very large weight indirectly, by nding su ciently many of its higher Fourier coe cients and using linear algebra to deduce the unknown constant term. Thus our approach is an algorithmic incarnation of Serre’s approach to p-adic L-functions of totally real elds [Ser73], obtaining p-adic congruences between the constant terms of modular forms by studying their higher Fourier coe cients.
{"title":"Computing p-adic L-functions of totally real fields","authors":"Jan Vonk, Contents","doi":"10.1090/mcom/3678","DOIUrl":"https://doi.org/10.1090/mcom/3678","url":null,"abstract":"We describe an algorithm for computing p-adic L-functions of characters of totally real elds. Such p-adic L-functions were constructed in the 1970’s independently by Barsky and CassouNoguès [Bar78, CN79] based on the explicit formula for zeta values of Shintani [Shi76] and by Serre and Deligne–Ribet [Ser73, DR80] using Hilbert modular forms and an idea of Siegel [Sie68] going back to Hecke [Hec24, Satz 3]. An algorithm for computing via the approach of Cassou-Noguès was developed by Roblot1 [Rob15]. Our algorithm follows the approach of Serre and Siegel, and its computational e ciency rests upon a method for computing with p-adic spaces of modular forms developed in previous work by the authors. The idea of our method is simple. In Serre’s approach, the value of the p-adic L-function of a totally real eld of degree d at a non-positive integer 1 − k is interpreted as the constant term of a classical modular form of weight dk obtained by diagonally restricting a Hilbert Eisenstein series. For small values of k these constants can be computed easily using an idea of Siegel, which goes back to Hecke. To compute the p-adic L-function at arbitrary points in its domain, to some nite p-adic precision, we use a method for computing p-adically with modular forms in larger weight developed in [Lau11, Von15]. We compute the required constant term in very large weight indirectly, by nding su ciently many of its higher Fourier coe cients and using linear algebra to deduce the unknown constant term. Thus our approach is an algorithmic incarnation of Serre’s approach to p-adic L-functions of totally real elds [Ser73], obtaining p-adic congruences between the constant terms of modular forms by studying their higher Fourier coe cients.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74501495","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}
In this short note, we identify an error made in an earlier paper [Math. Comp. 68 (1999), no. 225, pp. 385–388] on simultaneous Pell equations, provide a revised statement of the main results contained therein, and show how this modification lends itself to the correctness of the proofs given in the earlier paper.
{"title":"Corrigendum to \"On two classes of simultaneous Pell equations with no solutions\"","authors":"P. Walsh","doi":"10.1090/MCOM/3677","DOIUrl":"https://doi.org/10.1090/MCOM/3677","url":null,"abstract":"In this short note, we identify an error made in an earlier paper [Math. Comp. 68 (1999), no. 225, pp. 385–388] on simultaneous Pell equations, provide a revised statement of the main results contained therein, and show how this modification lends itself to the correctness of the proofs given in the earlier paper.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86855895","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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