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。}
{"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}
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 SS-integral points on PZ1∖{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 SS increases. In this paper, we implement a refinement of Kim’s method to explicitly compute various examples where SS has size 22 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。
{"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}
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.
{"title":"Minimal residual methods in negative or fractional Sobolev norms","authors":"Harald Monsuur, Rob Stevenson, Johannes Storn","doi":"10.1090/mcom/3904","DOIUrl":"https://doi.org/10.1090/mcom/3904","url":null,"abstract":"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.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135923277","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 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}
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 ℓ1ell ^1-stable but ℓqell ^q-unstable for any q>1q>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 11 embedded into the essential spectrum.
{"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<!-- 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<!-- 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}
{"title":"Minimization of differential equations and algebraic values of 𝐸-functions","authors":"Alin Bostan, Tanguy Rivoal, Bruno Salvy","doi":"10.1090/mcom/3912","DOIUrl":"https://doi.org/10.1090/mcom/3912","url":null,"abstract":"","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135428054","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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