Mathematics of Computation最新文献_第6页

Distribution of recursive matrix pseudorandom number generator modulo prime powers 递归矩阵伪随机数发生器模素数幂的分布

2区数学

Mathematics of Computation Pub Date : 2023-10-25 DOI: 10.1090/mcom/3895

László Mérai, Igor Shparlinski

{"title":"Distribution of recursive matrix pseudorandom number generator modulo prime powers","authors":"László Mérai, Igor Shparlinski","doi":"10.1090/mcom/3895","DOIUrl":"https://doi.org/10.1090/mcom/3895","url":null,"abstract":"Given a matrix <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A element-of normal upper G normal upper L Subscript d Baseline left-parenthesis double-struck upper Z right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>∈</mml:mo> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"normal\">G</mml:mi> <mml:mi mathvariant=\"normal\">L</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">Ain mathrm {GL}_d(mathbb {Z})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We study the pseudorandomness of vectors <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold u Subscript n\"> <mml:semantics> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"bold\">u</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=\"application/x-tex\">mathbf {u}_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> generated by a linear recurrence relation of the form <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold u Subscript n plus 1 Baseline identical-to upper A bold u Subscript n Baseline left-parenthesis mod p Superscript t Baseline right-parenthesis comma n equals 0 comma 1 comma ellipsis comma\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"bold\">u</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>≡</mml:mo> <mml:mi>A</mml:mi> <mml:msub> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"bold\">u</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mspace width=\"0.667em\" /> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width=\"0.333em\" /> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>t</mml:mi> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace width=\"2em\" /> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">begin{equation*} mathbf {u}_{n+1} equiv A mathbf {u}_n pmod {p^t}, qquad n = 0, 1, ldots , end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> modulo <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p Superscript t\"> <mml:semantics> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>t</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">p^t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> w","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135112108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Refined Selmer equations for the thrice-punctured line in depth two 深度二中三次穿刺线的改进Selmer方程

2区数学

Mathematics of Computation Pub Date : 2023-10-24 DOI: 10.1090/mcom/3898

Alex Best, L. Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, Yujie Xu

{"title":"Refined Selmer equations for the thrice-punctured line in depth two","authors":"Alex Best, L. Betts, Theresa Kumpitsch, Martin Lüdtke, Angus McAndrew, Lie Qian, Elie Studnia, Yujie Xu","doi":"10.1090/mcom/3898","DOIUrl":"https://doi.org/10.1090/mcom/3898","url":null,"abstract":"Kim gave a new proof of Siegel’s Theorem that there are only finitely many <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-integral points on <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P Subscript double-struck upper Z Superscript 1 Baseline minus StartSet 0 comma 1 comma normal infinity EndSet\"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">Z</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo class=\"MJX-variant\">∖</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant=\"normal\">∞</mml:mi> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathbb {P}^1_mathbb {Z}setminus {0,1,infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. One advantage of Kim’s method is that it in principle allows one to actually find these points, but the calculations grow vastly more complicated as the size of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S\"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=\"application/x-tex\">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has size <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding=\"application/x-tex\">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135219379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Minimal residual methods in negative or fractional Sobolev norms 负索博列夫范数或分数索博列夫范数的最小残差法

2区数学

Mathematics of Computation Pub Date : 2023-10-12 DOI: 10.1090/mcom/3904

Harald Monsuur, Rob Stevenson, Johannes Storn

引用次数: 0

A classification of genus 0 modular curves with rational points 有有理点的0个模曲线的分类

2区数学

Mathematics of Computation Pub Date : 2023-10-10 DOI: 10.1090/mcom/3907

None Rakvi

引用次数: 0

Uniqueness and stability for the solution of a nonlinear least squares problem 一类非线性最小二乘问题解的唯一性和稳定性

2区数学

Mathematics of Computation Pub Date : 2023-10-06 DOI: 10.1090/mcom/3918

Meng Huang, Zhiqiang Xu

{"title":"Uniqueness and stability for the solution of a nonlinear least squares problem","authors":"Meng Huang, Zhiqiang Xu","doi":"10.1090/mcom/3918","DOIUrl":"https://doi.org/10.1090/mcom/3918","url":null,"abstract":"In this paper, we focus on the nonlinear least squares: $mbox{min}_{mathbf{x} in mathbb{H}^d}| |Amathbf{x}|-mathbf{b}|$ where $Ain mathbb{H}^{mtimes d}$, $mathbf{b} in mathbb{R}^m$ with $mathbb{H} in {mathbb{R},mathbb{C} }$ and consider the uniqueness and stability of solutions. Such problem arises, for instance, in phase retrieval and absolute value rectification neural networks. For the case where $mathbf{b}=|Amathbf{x}_0|$ for some $mathbf{x}_0in mathbb{H}^d$, many results have been developed to characterize the uniqueness and stability of solutions. However, for the case where $mathbf{b} neq |Amathbf{x}_0| $ for any $mathbf{x}_0in mathbb{H}^d$, there is no existing result for it to the best of our knowledge. In this paper, we first focus on the uniqueness of solutions and show for any matrix $Ain mathbb{H}^{m times d}$ there always exists a vector $mathbf{b} in mathbb{R}^m$ such that the solution is not unique. But, in real case, such ``bad'' vectors $mathbf{b}$ are negligible, namely, if $mathbf{b} in mathbb{R}_{+}^m$ does not lie in some measure zero set, then the solution is unique. We also present some conditions under which the solution is unique. For the stability of solutions, we prove that the solution is never uniformly stable. But if we restrict the vectors $mathbf{b}$ to any convex set then it is stable.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302160","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}

引用次数: 0

Tamed stability of finite difference schemes for the transport equation on the half-line 半线上输运方程有限差分格式的驯服稳定性

2区数学

Mathematics of Computation Pub Date : 2023-10-04 DOI: 10.1090/mcom/3901

Lucas Coeuret

{"title":"Tamed stability of finite difference schemes for the transport equation on the half-line","authors":"Lucas Coeuret","doi":"10.1090/mcom/3901","DOIUrl":"https://doi.org/10.1090/mcom/3901","url":null,"abstract":"In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l Superscript 1\"> <mml:semantics> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">ell ^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-stable but <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l Superscript q\"> <mml:semantics> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mi>q</mml:mi> </mml:msup> <mml:annotation encoding=\"application/x-tex\">ell ^q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-unstable for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">q>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"1\"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding=\"application/x-tex\">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> embedded into the essential spectrum.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"2012 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547407","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}

引用次数: 0

Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity 半光滑非线性非线性Schrödinger方程时分裂方法的误差估计

2区数学

Mathematics of Computation Pub Date : 2023-10-04 DOI: 10.1090/mcom/3900

Weizhu Bao, Chushan Wang

{"title":"Error estimates of the time-splitting methods for the nonlinear Schrödinger equation with semi-smooth nonlinearity","authors":"Weizhu Bao, Chushan Wang","doi":"10.1090/mcom/3900","DOIUrl":"https://doi.org/10.1090/mcom/3900","url":null,"abstract":"We establish error bounds of the Lie-Trotter time-splitting sine pseudospectral method for the nonlinear Schrödinger equation (NLSE) with semi-smooth nonlinearity <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f left-parenthesis rho right-parenthesis equals rho Superscript sigma\"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mi>σ</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">f(rho ) = rho ^sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"rho equals StartAbsoluteValue psi EndAbsoluteValue squared\"> <mml:semantics> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>ψ</mml:mi> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">rho =|psi |^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the density with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"psi\"> <mml:semantics> <mml:mi>ψ</mml:mi> <mml:annotation encoding=\"application/x-tex\">psi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the wave function and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sigma greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">sigma >0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the exponent of the semi-smooth nonlinearity. Under the assumption of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H squared\"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">H^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-solution of the NLSE, we prove error bounds at <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis tau Superscript one half plus sigma Baseline plus h Superscript 1 plus 2 sigma Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>τ</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> </mml:mrow> <","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547406","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}

引用次数: 0

Computing quadratic points on modular curves 𝑋₀(𝑁) 计算模曲线上的二次点𝑋0(二进制)

2区数学

Mathematics of Computation Pub Date : 2023-10-03 DOI: 10.1090/mcom/3902

Nikola Adžaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, Borna Vukorepa

{"title":"Computing quadratic points on modular curves 𝑋₀(𝑁)","authors":"Nikola Adžaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, Borna Vukorepa","doi":"10.1090/mcom/3902","DOIUrl":"https://doi.org/10.1090/mcom/3902","url":null,"abstract":"In this paper we improve on existing methods to compute quadratic points on modular curves and apply them to successfully find all the quadratic points on all modular curves <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X 0 left-parenthesis upper N right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">X_0(N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of genus up to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"8\"> <mml:semantics> <mml:mn>8</mml:mn> <mml:annotation encoding=\"application/x-tex\">8</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and genus up to <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"10\"> <mml:semantics> <mml:mn>10</mml:mn> <mml:annotation encoding=\"application/x-tex\">10</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=\"upper N\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\"application/x-tex\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> prime, for which they were previously unknown. The values of <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> we consider are contained in the set <disp-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper L equals StartSet 58 comma 68 comma 74 comma 76 comma 80 comma 85 comma 97 comma 98 comma 100 comma 103 comma 107 comma 109 comma 113 comma 121 comma 127 EndSet period\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">L</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mn>58</mml:mn> <mml:mo>,</mml:mo> <mml:mn>68</mml:mn> <mml:mo>,</mml:mo> <mml:mn>74</mml:mn> <mml:mo>,</mml:mo> <mml:mn>76</mml:mn> <mml:mo>,</mml:mo> <mml:mn>80</mml:mn> <mml:mo>,</mml:mo> <mml:mn>85</mml:mn> <mml:mo>,</mml:mo> <mml:mn>97</mml:mn> <mml:mo>,</mml:mo> <mml:mn>98</mml:mn> <mml:mo>,</mml:mo> <mml:mn>100</mml:mn> <mml:mo>,</mml:mo> <mml:mn>103</mml:mn> <mml:mo>,</mml:mo> <mml:mn>107</mml:mn> <mml:mo>,</mml:mo> <mml:mn>109</mml:mn> <mml:mo>,</mml:mo> <mml:mn>113</mml:mn> <mml:mo>,</mml:mo> <mml:mn>121</mml:mn> <mml:mo>,</mml:mo> <mml:mn>127</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:ann","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135689172","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}

引用次数: 5

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