Sebastian Heintze, Robert Tichy, Ingrid Vukusic, Volker Ziegler
Let (Un)n∈N(U_n)_{nin mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants BB and N0N_0 such that for any b,c∈Zb,cin mathbb {Z} with b>Bb> B the equation Un−bm=cU_n - b^m = c has at most two distinct solutions
让U (n) n∈{n的n (U_n)在 mathbb {n}}成为一个固定recurrence线性序列):通过和一些技术restrictions integers杂志》()。我们证明,以至于有存在effectively computable constants B B和N 0 N_0如此那车上为B、c∈Z B、c和B在 mathbb {Z} >B b>B《equation U n−B = c U_n - B ^ m = c已经在大多数二distinct解决方案2 (n, m)∈n (n, m)在 mathbb {n ^ 2的n和n≥0 geq N_0和m≥1 geq 1。而且,我们专心论点特别Tribonacci数字赐予的凯斯》由T = T = 2 = 1 T_1 = T_2 = 1 , 3 = 2 T_3 = 2 T T T和n = n−1 + T + n−2 T n−3 T_ {} = T_ {n-1} T_{已经开始}+ T_ {n-3}为n≥4 geq 4。我们可以证明N =2 N_0=2和B=e = B=e。corresponding算法正在以Sage的方式实现。
{"title":"On the Diophantine equation 𝑈_{𝑛}-𝑏^{𝑚}=𝑐","authors":"Sebastian Heintze, Robert Tichy, Ingrid Vukusic, Volker Ziegler","doi":"10.1090/mcom/3854","DOIUrl":"https://doi.org/10.1090/mcom/3854","url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper U Subscript n Baseline right-parenthesis Subscript n element-of double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:msub> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(U_n)_{nin mathbb {N}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B\"> <mml:semantics> <mml:mi>B</mml:mi> <mml:annotation encoding=\"application/x-tex\">B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N 0\"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">N_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b comma c element-of double-struck upper Z\"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>c</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">b,cin mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"b greater-than upper B\"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>></mml:mo> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">b> B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the equation <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper U Subscript n Baseline minus b Superscript m Baseline equals c\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>−<!-- − --></mml:mo> <mml:msup> <mml:mi>b</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">U_n - b^m = c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has at most two distinct solutions <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134959022","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}
In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the L2L^2 norm of the numerical solution does not increase in time, under the time step condition τ≤F(h/c,d/c2)tau le mathcal {F}(h/c, d/c^2), with the convection coefficient cc, the diffusion coefficient dd, and the mesh size hh. The function
{"title":"Uniform stability for local discontinuous Galerkin methods with implicit-explicit Runge-Kutta time discretizations for linear convection-diffusion equation","authors":"Haijin Wang, Fengyan Li, Chi-Wang Shu, Qiang Zhang","doi":"10.1090/mcom/3842","DOIUrl":"https://doi.org/10.1090/mcom/3842","url":null,"abstract":"In this paper, we consider the linear convection-diffusion equation in one dimension with periodic boundary conditions, and analyze the stability of fully discrete methods that are defined with local discontinuous Galerkin (LDG) methods in space and several implicit-explicit (IMEX) Runge-Kutta methods in time. By using the forward temporal differences and backward temporal differences, respectively, we establish two general frameworks of the energy-method based stability analysis. From here, the fully discrete schemes being considered are shown to have monotonicity stability, i.e. the <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> norm of the numerical solution does not increase in time, under the time step condition <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"tau less-than-or-equal-to script upper F left-parenthesis h slash c comma d slash c squared right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">F</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>h</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msup> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">tau le mathcal {F}(h/c, d/c^2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with the convection coefficient <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"c\"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding=\"application/x-tex\">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the diffusion coefficient <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>, and the mesh size <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=\"application/x-tex\">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The function <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper F\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" ","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134959096","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}
In a recent paper an Inexact Restoration method for solving continuous constrained optimization problems was analyzed from the point of view of worst-case functional complexity and convergence. On the other hand, the Inexact Restoration methodology was employed, in a different research, to handle minimization problems with inexact evaluation and simple constraints. These two methodologies are combined in the present report, for constrained minimization problems in which both the objective function and the constraints, as well as their derivatives, are subject to evaluation errors. Together with a complete description of the method, complexity and convergence results will be proved.
{"title":"Inexact restoration for minimization with inexact evaluation both of the objective function and the constraints","authors":"L. Bueno, F. Larreal, J. Martínez","doi":"10.1090/mcom/3855","DOIUrl":"https://doi.org/10.1090/mcom/3855","url":null,"abstract":"In a recent paper an Inexact Restoration method for solving continuous constrained optimization problems was analyzed from the point of view of worst-case functional complexity and convergence. On the other hand, the Inexact Restoration methodology was employed, in a different research, to handle minimization problems with inexact evaluation and simple constraints. These two methodologies are combined in the present report, for constrained minimization problems in which both the objective function and the constraints, as well as their derivatives, are subject to evaluation errors. Together with a complete description of the method, complexity and convergence results will be proved.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135421550","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 prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.
{"title":"Computing eigenvalues of the Laplacian on rough domains","authors":"Frank Rösler, Alexei Stepanenko","doi":"10.1090/mcom/3827","DOIUrl":"https://doi.org/10.1090/mcom/3827","url":null,"abstract":"We prove a general Mosco convergence theorem for bounded Euclidean domains satisfying a set of mild geometric hypotheses. For bounded domains, this notion implies norm-resolvent convergence for the Dirichlet Laplacian which in turn ensures spectral convergence. A key element of the proof is the development of a novel, explicit Poincaré-type inequality. These results allow us to construct a universal algorithm capable of computing the eigenvalues of the Dirichlet Laplacian on a wide class of rough domains. Many domains with fractal boundaries, such as the Koch snowflake and certain filled Julia sets, are included among this class. Conversely, we construct a counterexample showing that there does not exist a universal algorithm of the same type capable of computing the eigenvalues of the Dirichlet Laplacian on an arbitrary bounded domain.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135572689","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}
Danica Basarić, Mária Lukáčova-Medvidova, Hana Mizerová, Bangwei She, Yuhuan Yuan
In this paper we study the convergence rate of a finite volume approximation of the compressible Navier–Stokes–Fourier system. To this end we first show the local existence of a regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution.
{"title":"Error estimates of a finite volume method for the compressible Navier–Stokes–Fourier system","authors":"Danica Basarić, Mária Lukáčova-Medvidova, Hana Mizerová, Bangwei She, Yuhuan Yuan","doi":"10.1090/mcom/3852","DOIUrl":"https://doi.org/10.1090/mcom/3852","url":null,"abstract":"In this paper we study the convergence rate of a finite volume approximation of the compressible Navier–Stokes–Fourier system. To this end we first show the local existence of a regular unique strong solution and analyse its global extension in time as far as the density and temperature remain bounded. We make a physically reasonable assumption that the numerical density and temperature are uniformly bounded from above and below. The relative energy provides us an elegant way to derive a priori error estimates between finite volume solutions and the strong solution.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135807200","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 establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These algorithms extend the classical trace test in numerical algebraic geometry. Our results rely on both the analysis of the structure of sparse resultants as well as an extension of Esterov’s results on monodromy groups of sparse systems.
{"title":"Sparse trace tests","authors":"Taylor Brysiewicz, Michael Burr","doi":"10.1090/mcom/3849","DOIUrl":"https://doi.org/10.1090/mcom/3849","url":null,"abstract":"We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These algorithms extend the classical trace test in numerical algebraic geometry. Our results rely on both the analysis of the structure of sparse resultants as well as an extension of Esterov’s results on monodromy groups of sparse systems.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135806629","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 is concerned with coupling conditions for linear hyperbolic relaxation systems with multiple relaxation times. In the region with a small relaxation time, an equilibrium system can be used for computational efficiency. The key assumption is that the relaxation system satisfies Yong’s structural stability condition [J. Differential Equations, 155 (1999), pp. 89–132]. For the non-characteristic case, we derive a coupling condition at the interface to couple two systems in a domain decomposition setting. We prove the validity by the energy estimate and Laplace transform, which shows how the error of the domain decomposition method depends on the smaller relaxation time and the boundary-layer effects. In addition, we propose a discontinuous Galerkin (DG) numerical scheme for solving the interface problem with the derived coupling condition and prove the L2L^2 stability. We validate our analysis on the linearized Carleman model and the linearized Grad’s moment system and show the effectiveness of the DG scheme.
{"title":"Coupling conditions for linear hyperbolic relaxation systems in two-scale problems","authors":"Juntao Huang, Ruo Li, Yizhou Zhou","doi":"10.1090/mcom/3845","DOIUrl":"https://doi.org/10.1090/mcom/3845","url":null,"abstract":"This work is concerned with coupling conditions for linear hyperbolic relaxation systems with multiple relaxation times. In the region with a small relaxation time, an equilibrium system can be used for computational efficiency. The key assumption is that the relaxation system satisfies Yong’s structural stability condition [J. Differential Equations, 155 (1999), pp. 89–132]. For the non-characteristic case, we derive a coupling condition at the interface to couple two systems in a domain decomposition setting. We prove the validity by the energy estimate and Laplace transform, which shows how the error of the domain decomposition method depends on the smaller relaxation time and the boundary-layer effects. In addition, we propose a discontinuous Galerkin (DG) numerical scheme for solving the interface problem with the derived coupling condition and prove the <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> stability. We validate our analysis on the linearized Carleman model and the linearized Grad’s moment system and show the effectiveness of the DG scheme.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135845488","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 introduce a hybridizable discontinuous Galerkin (HDG) method for the convected Helmholtz equation based on the total flux formulation, in which the vector unknown represents both diffusive and convective phenomena. This HDG method is constricted with the same interpolation degree for all the unknowns and a physically informed value for the penalization parameter is computed. A detailed analysis including local and global well-posedness as well as a super-convergence result is carried out. We then provide numerical experiments to illustrate the theoretical results.
{"title":"Construction and analysis of a HDG solution for the total-flux formulation of the convected Helmholtz equation","authors":"Hélène Barucq, Nathan Rouxelin, Sébastien Tordeux","doi":"10.1090/mcom/3850","DOIUrl":"https://doi.org/10.1090/mcom/3850","url":null,"abstract":"We introduce a hybridizable discontinuous Galerkin (HDG) method for the convected Helmholtz equation based on the total flux formulation, in which the vector unknown represents both diffusive and convective phenomena. This HDG method is constricted with the same interpolation degree for all the unknowns and a physically informed value for the penalization parameter is computed. A detailed analysis including local and global well-posedness as well as a super-convergence result is carried out. We then provide numerical experiments to illustrate the theoretical results.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136265188","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}
A singularly perturbed convection-diffusion problem posed on the unit square in R2mathbb {R}^2, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most k>0k>0 on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than O(N−(k+1/2))O(N^{-(k+1/2)}) accurate in the energy norm induced by the bilinear form of the weak formulation, where NN mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish O(N
{"title":"Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem","authors":"Yao Cheng, Shan Jiang, Martin Stynes","doi":"10.1090/mcom/3844","DOIUrl":"https://doi.org/10.1090/mcom/3844","url":null,"abstract":"A singularly perturbed convection-diffusion problem posed on the unit square in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R squared\"> <mml:semantics> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 slash 2 right-parenthesis Baseline 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:mo>−<!-- − --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <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:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(N^{-(k+1/2)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> accurate in the energy norm induced by the bilinear form of the weak formulation, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\"application/x-tex\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 right-parenthesis Baseline 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:mro","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136231610","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}
Let KK be a number field, and let GG be a finitely generated subgroup of K×K^times. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes pmathfrak p of KK such that the order of (Gmodp)(Gbmod mathfrak p) is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes pmathfrak p for which the order is kk
{"title":"Divisibility conditions on the order of the reductions of algebraic numbers","authors":"Pietro Sgobba","doi":"10.1090/mcom/3848","DOIUrl":"https://doi.org/10.1090/mcom/3848","url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a number field, and let <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> be a finitely generated subgroup of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Superscript times\"> <mml:semantics> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>×<!-- × --></mml:mo> </mml:msup> <mml:annotation encoding=\"application/x-tex\">K^times</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German p\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">p</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathfrak p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the order of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper G mod German p right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo lspace=\"thickmathspace\" rspace=\"thickmathspace\">mod</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">p</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(Gbmod mathfrak p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German p\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">p</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathfrak p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the order is <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134922467","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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