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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 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-25 DOI: 10.1090/mcom/3895
László Mérai, Igor Shparlinski
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
给定矩阵a∈gl d(Z) a inmathrm GL_d{(}mathbb Z{)。我们研究向量u n }mathbf u_n{的伪随机性,由形式为u n + 1≡a u n (mod p t), n = 0,1,…,}begin{equation*} mathbf {u}_{n+1} equiv A mathbf {u}_n pmod {p^t}, qquad n = 0, 1, ldots , end{equation*}模p t p^t与固定素数p p和足够大的整数t大于或等于1 t geqslant 1的线性递归关系生成。我们研究这样的序列在非常短的片段长度,这是无法通过以前使用的方法访问。我们的技术是基于N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654-670, 672]估计双Weyl和的方法和K. Ford的Vinogradov中值定理的完全显式形式[Proc. London mathematics]。Soc。(3) 85 (2002), pp. 565-633。这与I. E. Shparlinski [Proc. Voronezh State Pedagogical institute ., 197 (1978), 74-85 (in Russian)]的一些想法相结合,它允许我们构建u n mathbf u_n{坐标的多项式表示,并在多项式表示中控制其系数的p p进阶。}
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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 Q1 MATHEMATICS, APPLIED 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
Kim gave a new proof of Siegel’s Theorem that there are only finitely many S S -integral points on P Z 1 { 0 , 1 , } mathbb {P}^1_mathbb {Z}setminus {0,1,infty } . 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 S S increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where S S has size 2 2 which has been introduced by Betts and Dogra. In so doing, we exhibit new examples of a natural generalization of a conjecture of Kim.
Kim给出了西格尔定理的一个新的证明,证明在P Z 1∈{0,1,∞}mathbb P{^1_ }mathbb Z{}setminus {0,1, infty}上只有有限多个S -积分点。Kim的方法的一个优点是,它原则上允许人们实际找到这些点,但随着S的大小增加,计算变得非常复杂。在本文中,我们实现了Kim的方法的改进,以显式地计算由Betts和Dogra引入的S的大小为22的各种示例。在这样做的过程中,我们展示了Kim猜想的自然推广的新例子。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-12 DOI: 10.1090/mcom/3904
Harald Monsuur, Rob Stevenson, Johannes Storn
For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and that the norm of the residual provides an a posteriori error estimator. Furthermore, it allows for the treatment of general inhomogeneous boundary conditions. In many minimal residual formulations, however, one or more terms of the residual are measured in negative or fractional Sobolev norms. In this work, we provide a general approach to replace those norms by efficiently evaluable expressions without sacrificing quasi-optimality of the resulting numerical solution. We exemplify our approach by verifying the necessary inf-sup conditions for four formulations of a model second order elliptic equation with inhomogeneous Dirichlet and/or Neumann boundary conditions. We report on numerical experiments for the Poisson problem with mixed inhomogeneous Dirichlet and Neumann boundary conditions in an ultra-weak first order system formulation.
对于数值逼近,将偏微分方程重新表述为残差最小化问题的优点是所得到的线性系统是对称正定的,并且残差的范数提供了一个后验误差估计量。此外,它允许处理一般的非齐次边界条件。然而,在许多最小残差公式中,残差的一个或多个项以负或分数索博列夫范数测量。在这项工作中,我们提供了一种通用的方法,用有效的可求值表达式代替这些规范,而不牺牲所得到的数值解的拟最优性。我们通过验证具有非齐次Dirichlet和/或Neumann边界条件的模型二阶椭圆方程的四种表述的必要条件来举例说明我们的方法。本文报道了在超弱一阶系统公式中具有混合非齐次Dirichlet和Neumann边界条件的泊松问题的数值实验。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-10 DOI: 10.1090/mcom/3907
None Rakvi
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a non-CM elliptic curve defined over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Fix an algebraic closure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Q overbar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">{overline {mathbb Q}}</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="double-struck upper Q"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We get a Galois representation <disp-formula content-type="math/mathml"> [ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="rho Subscript upper E Baseline colon upper G a l left-parenthesis double-struck upper Q overbar slash double-struck upper Q right-parenthesis right-arrow upper G upper L 2 left-parenthesis ModifyingAbove double-struck upper Z With caret right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mi>E</mml:mi> </mml:msub> <mml:mo>:<!-- : --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>G</mml:mi> <mml:mi>a</mml:mi> <mml:mi>l</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo accent="false">¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">→<!-- → --></mml:mo> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo>^<!-- ^ --></mml
设E是在Q上定义的非cm椭圆曲线 mathbb {q} . 修正一个代数闭包Q¯ {overline {mathbb q}} Q的 mathbb {q} . 我们得到伽罗瓦表示法 [ ρ E : G a l ( Q ¯ / Q ) → G L 2 ( Z ^ ) rho _E colon {Gal}({overline {mathbb Q}}/mathbb {Q})to GL_2({widehat {mathbb {Z}}}) ] 通过为E (Q¯)的N N -扭转子群选择一个相容基体系来关联E (Q¯)。e ({overline {mathbb q}})。与gl2 (Z ^) GL_2({widehat {mathbb {z}}})满足−I∈G -I in G和det (G) = Z ^ x det (g)={widehat {mathbb {z}}}^{times } ,我们有模曲线(xg, π G) (xg,pi _G) / Q mathbb {q} 它松散地将椭圆曲线参数化使得ρ E的像 rho _E共轭于gt的一个子群。G^t。在这篇文章中,我们给出了所有这类有一个有理点的0 0个模曲线的完全分类。这种分类在有限的许多科中都有。此外,对于每一个模曲线态射π G: X G→X G L 2 (Z ^) pi _g colon x_g to x_{gl2 ({widehat {mathbb {z}}})} 可以显式计算。
{"title":"A classification of genus 0 modular curves with rational points","authors":"None Rakvi","doi":"10.1090/mcom/3907","DOIUrl":"https://doi.org/10.1090/mcom/3907","url":null,"abstract":"Let &lt;inline-formula content-type=\"math/mathml\"&gt; &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper E\"&gt; &lt;mml:semantics&gt; &lt;mml:mi&gt;E&lt;/mml:mi&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;E&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; be a non-CM elliptic curve 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 class=\"MJX-TeXAtom-ORD\"&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;. Fix an algebraic closure &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 overbar\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mover&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;Q&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo accent=\"false\"&gt;¯&lt;!-- ¯ --&gt;&lt;/mml:mo&gt; &lt;/mml:mover&gt; &lt;/mml:mrow&gt; &lt;mml:annotation encoding=\"application/x-tex\"&gt;{overline {mathbb Q}}&lt;/mml:annotation&gt; &lt;/mml:semantics&gt; &lt;/mml:math&gt; &lt;/inline-formula&gt; of &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 class=\"MJX-TeXAtom-ORD\"&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;. We get a Galois representation &lt;disp-formula content-type=\"math/mathml\"&gt; [ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"rho Subscript upper E Baseline colon upper G a l left-parenthesis double-struck upper Q overbar slash double-struck upper Q right-parenthesis right-arrow upper G upper L 2 left-parenthesis ModifyingAbove double-struck upper Z With caret right-parenthesis\"&gt; &lt;mml:semantics&gt; &lt;mml:mrow&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;ρ&lt;!-- ρ --&gt;&lt;/mml:mi&gt; &lt;mml:mi&gt;E&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mo&gt;:&lt;!-- : --&gt;&lt;/mml:mo&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mi&gt;G&lt;/mml:mi&gt; &lt;mml:mi&gt;a&lt;/mml:mi&gt; &lt;mml:mi&gt;l&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mover&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;Q&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo accent=\"false\"&gt;¯&lt;!-- ¯ --&gt;&lt;/mml:mo&gt; &lt;/mml:mover&gt; &lt;/mml:mrow&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mo&gt;/&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;Q&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo stretchy=\"false\"&gt;)&lt;/mml:mo&gt; &lt;mml:mo stretchy=\"false\"&gt;→&lt;!-- → --&gt;&lt;/mml:mo&gt; &lt;mml:mi&gt;G&lt;/mml:mi&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mn&gt;2&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;mml:mo stretchy=\"false\"&gt;(&lt;/mml:mo&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mover&gt; &lt;mml:mrow class=\"MJX-TeXAtom-ORD\"&gt; &lt;mml:mi mathvariant=\"double-struck\"&gt;Z&lt;/mml:mi&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;^&lt;!-- ^ --&gt;&lt;/mml","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254746","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
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 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-06 DOI: 10.1090/mcom/3918
Meng Huang, Zhiqiang Xu
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.
本文主要研究非线性最小二乘法:$mbox{min}_{mathbf{x} In mathbb{H}^d}| Amathbf{x}|-mathbf{b}|$其中$A mathbb{H}^{m乘以d}$, $mathbf{b} In mathbb{R}^m$与$mathbb{H} In mathbb{R},mathbb{C} }$,并考虑解的唯一性和稳定性。例如,在相位检索和绝对值校正神经网络中就会出现这样的问题。对于$mathbf{b}=|Amathbf{x}_0|$对于mathbb{H}^d$中的$mathbf{x}_0的情况,已经开发了许多结果来表征解的唯一性和稳定性。然而,对于$mathbf{b} neq |Amathbf{x}_0| $对于mathbb{H}^d$中的任何$mathbf{x}_0 $的情况,据我们所知,它没有现有的结果。在本文中,我们首先关注解的唯一性,并证明对于任意矩阵$A mathbb{H}^{m 乘以d}$,总存在一个向量$mathbf{b} 在mathbb{R}^m$中使得解不唯一。但是,在实际情况中,这样的“坏”向量$mathbf{b}$是可以忽略不计的,也就是说,如果$mathbf{b} 在mathbb{R}_{+}^m$中不存在于某个度量零集中,那么解是唯一的。我们还给出了解唯一的一些条件。对于解的稳定性,我们证明了解绝不是一致稳定的。但是如果我们限制向量$mathbf{b}$到任意凸集,那么它是稳定的。
{"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 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-04 DOI: 10.1090/mcom/3901
Lucas Coeuret
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 1 ell ^1 -stable but q ell ^q -unstable for any q > 1 q>1 . 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 1 1 embedded into the essential spectrum.
本文证明了在精确的谱假设下,具有数值边界条件的正半线上的标量左向输运方程的有限差分近似对任意q >都是稳定的,但不稳定的;1 >该证明依赖于对一类特定的Toeplitz算子的有限秩扰动的格林函数的精确描述,这些算子的本质谱属于封闭的单位圆盘,并且在本质谱中嵌入了一个模数为11的简单特征值。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-04 DOI: 10.1090/mcom/3900
Weizhu Bao, Chushan Wang
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<!-- ρ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mi>σ<!-- 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<!-- ρ --></mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>ψ<!-- 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<!-- 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<!-- 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<!-- 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<!-- σ --></mml:mi> </mml:mrow> <
针对半光滑非线性方程f(ρ) = ρ σ f(rho ) = rho ^sigma ,其中ρ = | ψ | 2 rho =|psi ^2是带ψ的密度 psi 波函数和σ &gt;0 sigma &gt;0为半光滑非线性的指数。在NLSE的H^2解的假设下,我们证明了在O(τ 1 2 + σ + H 1 + 2 σ) O(tau ^{frac {1}{2}+sigma } + h^{1+2sigma })和O(τ + h2) O(tau + h^{2}L^2 - 0 &gt的范数;σ≤1 20 &gt;sigma leq frac {1}{2} σ≥1 2 sigma geq frac {1}{2} ,以及O(τ 12 + h) O(tau ^frac {1}{2} + h) h ^1 - σ≥1的范数 sigma geq frac {1}{2} ,其中h h和τ tau 分别为网格尺寸和时间步长。另外,当1 2 &gt;σ &gt;1 frac {1}{2}&gt;sigma &gt;1和在NLSE的h3h ^3解的假设下,我们给出了在O(τ σ + H 2 σ) O(tau ^{sigma } + h^{2sigma }) H^1 -范数。在我们的证明中采用了两个关键因素:一是为了避免对0 &gt情况下的数值解进行先验估计,采用了数值流的无条件l2l ^2稳定性;σ≤1 20 &gt; sigma leq frac {1}{2} 建立一个l∞l^infty -条件H^1 H^1 -稳定性得到l∞l^infty 在σ≥12的情况下,用数学归纳法得到了数值解的-界和误差估计 sigma ge frac {1}{2} ;二是引入正则化技术,避免了半光滑非线性的奇异性,从而得到改进的局部截断误差。最后,用数值结果证明了我们的误差范围。
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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 Q1 MATHEMATICS, APPLIED Pub Date : 2023-10-03 DOI: 10.1090/mcom/3902
Nikola Adžaga, Timo Keller, Philippe Michaud-Jacobs, Filip Najman, Ekin Ozman, Borna Vukorepa
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
本文改进了现有的求模曲线上二次点的方法,并应用这些方法成功地求出了在所有模曲线x0 (N) X_0(N)上的格数不超过8 8和格数不超过10 10的所有N N素数的二次点,这些二次点以前是未知的。我们所考虑的N N的值包含在集合L = {58、68、74、76、80、85、97、98、100、103、107、109、113、121、127}中。begin{equation*} mathcal {L}={58, 68, 74, 76, 80, 85, 97, 98, 100, 103, 107, 109, 113, 121, 127 }. end{equation*}我们得到了除X 0(103) X_0(103)上定义在Q(2885) mathbb Q(sqrt 2885)上的一对伽罗瓦共轭点外,对于N∈{L} N, X 0(N) X_0(N)上的所有非尖次二次点in{}mathcal L都是复乘法(CM)点。我们还计算了由这些点参数化的椭圆曲线的j j不变量,并确定了CM点的几何自同态环。{}
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引用次数: 5
Randomizing the trapezoidal rule gives the optimal RMSE rate in Gaussian Sobolev spaces 随机化梯形规则给出了高斯Sobolev空间中最优的RMSE率
2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-09-29 DOI: 10.1090/mcom/3910
Takashi Goda, Yoshihito Kazashi, Yuya Suzuki
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引用次数: 0
Minimization of differential equations and algebraic values of 𝐸-functions 最小化的微分方程和代数值𝐸-functions
2区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2023-09-28 DOI: 10.1090/mcom/3912
Alin Bostan, Tanguy Rivoal, Bruno Salvy
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Mathematics of Computation
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