Mathematics of Computation最新文献

{"title":"Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping","authors":"Katherine Baker, Lehel Banjai, Mariya Ptashnyk","doi":"10.1090/mcom/3945","DOIUrl":"https://doi.org/10.1090/mcom/3945","url":null,"abstract":"<p>We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of nonlocal in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"t equals 0\"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">t = 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"147 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931313","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}

{"title":"A posteriori error estimates for the Richards equation","authors":"K. Mitra, M. Vohralík","doi":"10.1090/mcom/3932","DOIUrl":"https://doi.org/10.1090/mcom/3932","url":null,"abstract":"&lt;p&gt;The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1 Baseline left-parenthesis upper H Superscript negative 1 Baseline right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;H&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;H&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;H^1(H^{-1})&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;, &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared left-parenthesis upper L squared right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;L^2(L^2)&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;, and the &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L squared left-parenthesis upper H Superscript 1 Baseline right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;H&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;L^2(H^1)&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Superscript 1 Baseline left-parenthesis upper H Superscript negative 1 Baseline right-parenthesis intersection upper L squared left-parenthesis upper H Superscript 1 Baseline right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;H&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msup&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;H&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;−&lt;!-- − --&gt;&lt;/mml:mo&gt; &lt;mml","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"47 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942574","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}

{"title":"Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations","authors":"Constantin Carle, Marlis Hochbruck","doi":"10.1090/mcom/3952","DOIUrl":"https://doi.org/10.1090/mcom/3952","url":null,"abstract":"<p>This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"38 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931379","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}

{"title":"Quinary forms and paramodular forms","authors":"N. Dummigan, A. Pacetti, G. Rama, G. Tornaría","doi":"10.1090/mcom/3815","DOIUrl":"https://doi.org/10.1090/mcom/3815","url":null,"abstract":"<p>We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"43 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931393","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}

{"title":"Energy diminishing implicit-explicit Runge–Kutta methods for gradient flows","authors":"Zhaohui Fu, Tao Tang, Jiang Yang","doi":"10.1090/mcom/3950","DOIUrl":"https://doi.org/10.1090/mcom/3950","url":null,"abstract":"<p>This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge–Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"38 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931340","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}

{"title":"Construction of diagonal quintic threefolds with infinitely many rational points","authors":"Maciej Ulas","doi":"10.1090/mcom/3953","DOIUrl":"https://doi.org/10.1090/mcom/3953","url":null,"abstract":"&lt;p&gt;In this note we present a construction of an infinite family of diagonal quintic threefolds defined over &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;Q&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;mathbb {Q}&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; each containing infinitely many rational points. As an application, we prove that there are infinitely many quadruples &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B equals left-parenthesis upper B 0 comma upper B 1 comma upper B 2 comma upper B 3 right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;B=(B_{0}, B_{1}, B_{2}, B_{3})&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; of co-prime integers such that for a suitable chosen integer &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;b&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;b&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; (depending on &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;B&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;), the equation &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 0 upper X 0 Superscript 5 Baseline plus upper B 1 upper X 1 Superscript 5 Baseline plus upper B 2 upper X 2 Superscript 5 Baseline plus upper B 3 upper X 3 Superscript 5 Baseline equals b\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:msubsup&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;mml:mn&gt;5&lt;/mml:mn&gt; &lt;/mml:msubsup&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:msubsup&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;mml:mn&gt;5&lt;/mml:mn&gt; &lt;/mml:msubsup&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;B&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:msubsup&gt; &lt;mml:mi&gt;X&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:mro","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"147 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931203","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}

{"title":"High-order splitting finite element methods for the subdiffusion equation with limited smoothing property","authors":"Buyang Li, Zongze Yang, Zhi Zhou","doi":"10.1090/mcom/3944","DOIUrl":"https://doi.org/10.1090/mcom/3944","url":null,"abstract":"<p>In contrast with the diffusion equation which smoothens the initial data to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C Superscript normal infinity\"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mi mathvariant=\"normal\">∞<!-- ∞ --></mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">C^infty</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=\"t greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">t&gt;0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data. In this paper, a new splitting of the solution is constructed for high-order finite element approximations to the subdiffusion equation with nonsmooth initial data. The method is constructed by splitting the solution into two parts, i.e., a time-dependent smooth part and a time-independent nonsmooth part, and then approximating the two parts via different strategies. The time-dependent smooth part is approximated by using high-order finite element method in space and convolution quadrature in time, while the steady nonsmooth part could be approximated by using smaller mesh size or other methods that could yield high-order accuracy. Several examples are presented to show how to accurately approximate the steady nonsmooth part, including piecewise smooth initial data, Dirac–Delta point initial data, and Dirac measure concentrated on an interface. The argument could be directly extended to subdiffusion equations with nonsmooth source data. Extensive numerical experiments are presented to support the theoretical analysis and to illustrate the performance of the proposed high-order splitting finite element methods.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"23 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931235","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}

{"title":"Effective homology and periods of complex projective hypersurfaces","authors":"Pierre Lairez, Eric Pichon-Pharabod, Pierre Vanhove","doi":"10.1090/mcom/3947","DOIUrl":"https://doi.org/10.1090/mcom/3947","url":null,"abstract":"<p>We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard–Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section.</p> <p>We provide a SageMath implementation. For example, on a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"64 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931320","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}

{"title":"Convergence, finiteness and periodicity of several new algorithms of 𝑝-adic continued fractions","authors":"Zhaonan Wang, Yingpu Deng","doi":"10.1090/mcom/3948","DOIUrl":"https://doi.org/10.1090/mcom/3948","url":null,"abstract":"&lt;p&gt;Classical continued fractions can be introduced in the field of &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;-adic numbers, where &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;-adic continued fractions offer novel perspectives on number representation and approximation. While numerous &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;-adic continued fraction expansion algorithms have been proposed by the researchers, the establishment of several excellent properties, such as the Lagrange’s Theorem for classic continued fractions, which indicates that every quadratic irrationals can be expanded periodically, remains elusive. In this paper, we introduce several new algorithms designed for expanding algebraic numbers in &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper Q Subscript p\"&gt; &lt;mml:semantics&gt; &lt;mml:msub&gt; &lt;mml:mrow&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;Q&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;mathbb {Q}_p&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; for a given prime &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;. We give an upper bound of the number of partial quotients for the expansion of rational numbers, and prove that for small primes &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;, our algorithm generates periodic continued fraction expansions for all quadratic irrationals. Experimental data demonstrates that our algorithms exhibit better performance in the periodicity of expansions for quadratic irrationals compared to the existing algorithms. Furthermore, for bigger primes &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;p&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;p&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;, we propos","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"64 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931310","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}

{"title":"Few hamiltonian cycles in graphs with one or two vertex degrees","authors":"Jan Goedgebeur, Jorik Jooken, On-Hei Solomon Lo, Ben Seamone, Carol Zamfirescu","doi":"10.1090/mcom/3943","DOIUrl":"https://doi.org/10.1090/mcom/3943","url":null,"abstract":"&lt;p&gt;Inspired by Sheehan’s conjecture that no &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"4\"&gt; &lt;mml:semantics&gt; &lt;mml:mn&gt;4&lt;/mml:mn&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;4&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;-regular graph contains exactly one hamiltonian cycle, we prove results on hamiltonian cycles in regular graphs and nearly regular graphs. We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe’s computational enumerative results from [Exp. Math. &lt;bold&gt;27&lt;/bold&gt; (2018), no. 4, 426–430]. Thereafter, we use the Lovász Local Lemma to extend Thomassen’s independent dominating set method. This extension allows us to find a second hamiltonian cycle that inherits linearly many edges from the first hamiltonian cycle. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen by showing that for &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k element-of StartSet 5 comma 6 EndSet\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;mml:mo&gt;∈&lt;!-- ∈ --&gt;&lt;/mml:mo&gt; &lt;mml:mo fence=\"false\" stretchy=\"false\"&gt;{&lt;/mml:mo&gt; &lt;mml:mn&gt;5&lt;/mml:mn&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mn&gt;6&lt;/mml:mn&gt; &lt;mml:mo fence=\"false\" stretchy=\"false\"&gt;}&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;k in {5, 6}&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; there exist infinitely many &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;k&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;-regular hamiltonian graphs having no independent dominating set with respect to a prescribed hamiltonian cycle. Motivated by an observation of Aldred and Thomassen, we prove that for every &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"kappa element-of StartSet 2 comma 3 EndSet\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;κ&lt;!-- κ --&gt;&lt;/mml:mi&gt; &lt;mml:mo&gt;∈&lt;!-- ∈ --&gt;&lt;/mml:mo&gt; &lt;mml:mo fence=\"false\" stretchy=\"false\"&gt;{&lt;/mml:mo&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mn&gt;3&lt;/mml:mn&gt; &lt;mml:mo fence=\"false\" stretchy=\"false\"&gt;}&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;kappa in { 2, 3 }&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; and any positive integer &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;k&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;k&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt;, there are infinitely many non-regular gra","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"26 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931236","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}