SIAM Review, Volume 64, Issue 4, Page 1029-1029, November 2022. <br/> This issue of SIAM Review contains two papers in the Education section. The first, “A Generalized Dual Transform: Linear Algebra and Geometry of (Pseudo)Inverting a Matrix,” is presented by L. P. Withers, Jr. For a linear subspace $A$ of a vector space $V$, we may have a nonorthogonal basis of $A$. We could obtain an orthogonal basis (e.g., by the Gram--Schmidt orthogonalization procedure) and the orthogonality helps us to represent each element $bin A$ as a linear combination of the new basis in an easy way. How should we express $b$ as a linear combination of the original, unorthogonalized vectors? One suggestion is to construct a complementary list of vectors, called a dual list, such that each pair of vectors $a^i,a^j$ on that list are orthogonal and each vector has a length of one. The construction is called the dual transform. Involving the complementary subspace of $A$ in $V$ and orthogonal projections, we obtain again a simple formula for the representation of $b$. Next to generating dual vectors, the Gram--Schmidt orthogonalization procedure exhibits other interesting properties, which lead to a parallel so-called butterfly process for computing the dual transform. The article proceeds to explain how the dual transform is generalized via axioms and how the respective procedures are performed in the general setting. Several examples supplement the discussion. The paper is accessible to advanced undergraduate students with basic knowledge in linear algebra and complex analysis. The second article, “When Randomness Helps in Undersampling,” was written by Roel Snieder and Michael B. Wakin. In our digital era, we use many recordings: music, sounds of nature, and others. Many other signals such as telecommunication, temperature, and air pressure, are recorded for practical and scientific purposes. When signals are stored in computers, they are digitized by collecting and storing values of some functions at discrete times. A straightforward thought on how to accomplish that is to collect values uniformly in time and in all frequency components. However, the measurements might be feasible only for some times or frequencies; otherwise data acquisition or data transmission of a full collection might be too burdensome. When some times or frequencies are left out, it is said that the signal is undersampled. In that case, it is best to choose the times or frequencies to be left out randomly instead of uniformly. The authors focus on the problem of reconstructing a signal in the time domain using undersampled frequency components. The benefits of random undersampling are illustrated with an example of the air pressure recorded at a volcano in Costa Rica, but the authors cite other sources on seismic surveys as well as magnetic resonance imaging where benefits from undersampling are evidenced. The key to understand the phenomena is to analyze the effect of the sampling on the discret
SIAM评论，第64卷，第4期，1029-1029页，2022年11月。本期《SIAM评论》的教育部分有两篇论文。第一个，“广义对偶变换:(伪)逆矩阵的线性代数和几何”，由L. P. Withers, Jr.提出。对于向量空间V的线性子空间A，我们可以有A的非正交基。我们可以得到一个正交基(例如，通过Gram- Schmidt正交过程)，并且正交性帮助我们以一种简单的方式将A$中的每个元素$b表示为新基的线性组合。如何将b表示为原始的，未正交向量的线性组合?一个建议是构造一个向量的互补列表，称为对偶列表，使得列表上的每一对向量$a^i,a^j$是正交的，并且每个向量的长度为1。这种构造称为对偶变换。利用V中A的互补子空间和正交投影，我们得到了b的一个简单表示公式。除了生成对偶向量之外，Gram- Schmidt正交化过程还展示了其他有趣的特性，这导致了计算对偶变换的并行所谓的蝴蝶过程。文章接着解释了如何通过公理推广对偶变换，以及如何在一般情况下执行相应的过程。几个例子补充了讨论。本文适用于具有线性代数和复分析基础知识的高级本科生。第二篇文章是《当随机性有助于采样不足》，作者是Roel snyder和Michael B. Wakin。在我们的数字时代，我们使用许多录音:音乐，自然的声音，和其他。许多其他的信号，如电信、温度和气压，都是为了实用和科学的目的而记录下来的。当信号存储在计算机中时，通过收集和存储离散时间的一些函数值，将信号数字化。关于如何实现这一目标的一个简单的想法是在时间和所有频率分量中均匀地收集值。然而，测量可能只适用于某些时间或频率;否则，完整集合的数据采集或数据传输可能过于繁重。当某些时间或频率被忽略时，我们说信号是欠采样的。在这种情况下，最好随机选择要忽略的时间或频率，而不是均匀地选择。研究了利用欠采样频率分量在时域重构信号的问题。随机欠采样的好处以哥斯达黎加一座火山的气压记录为例进行了说明，但作者引用了地震调查和磁共振成像的其他来源，证明了欠采样的好处。理解这种现象的关键是分析采样对离散傅里叶变换的影响，该变换允许长度为$N$的离散时间信号表示为$N$复指数项的和。作者已经在https://mines.edu/~mwakin/software上提供了用于文章的信号处理代码和数据。这篇论文针对的是有数学背景和兴趣的工科本科生。
{"title":"Education","authors":"Darinka Dentcheva","doi":"10.1137/22n975585","DOIUrl":"https://doi.org/10.1137/22n975585","url":null,"abstract":"SIAM Review, Volume 64, Issue 4, Page 1029-1029, November 2022. <br/> This issue of SIAM Review contains two papers in the Education section. The first, “A Generalized Dual Transform: Linear Algebra and Geometry of (Pseudo)Inverting a Matrix,” is presented by L. P. Withers, Jr. For a linear subspace $A$ of a vector space $V$, we may have a nonorthogonal basis of $A$. We could obtain an orthogonal basis (e.g., by the Gram--Schmidt orthogonalization procedure) and the orthogonality helps us to represent each element $bin A$ as a linear combination of the new basis in an easy way. How should we express $b$ as a linear combination of the original, unorthogonalized vectors? One suggestion is to construct a complementary list of vectors, called a dual list, such that each pair of vectors $a^i,a^j$ on that list are orthogonal and each vector has a length of one. The construction is called the dual transform. Involving the complementary subspace of $A$ in $V$ and orthogonal projections, we obtain again a simple formula for the representation of $b$. Next to generating dual vectors, the Gram--Schmidt orthogonalization procedure exhibits other interesting properties, which lead to a parallel so-called butterfly process for computing the dual transform. The article proceeds to explain how the dual transform is generalized via axioms and how the respective procedures are performed in the general setting. Several examples supplement the discussion. The paper is accessible to advanced undergraduate students with basic knowledge in linear algebra and complex analysis. The second article, “When Randomness Helps in Undersampling,” was written by Roel Snieder and Michael B. Wakin. In our digital era, we use many recordings: music, sounds of nature, and others. Many other signals such as telecommunication, temperature, and air pressure, are recorded for practical and scientific purposes. When signals are stored in computers, they are digitized by collecting and storing values of some functions at discrete times. A straightforward thought on how to accomplish that is to collect values uniformly in time and in all frequency components. However, the measurements might be feasible only for some times or frequencies; otherwise data acquisition or data transmission of a full collection might be too burdensome. When some times or frequencies are left out, it is said that the signal is undersampled. In that case, it is best to choose the times or frequencies to be left out randomly instead of uniformly. The authors focus on the problem of reconstructing a signal in the time domain using undersampled frequency components. The benefits of random undersampling are illustrated with an example of the air pressure recorded at a volcano in Costa Rica, but the authors cite other sources on seismic surveys as well as magnetic resonance imaging where benefits from undersampling are evidenced. The key to understand the phenomena is to analyze the effect of the sampling on the discret","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"19 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Review, Volume 64, Issue 4, Page 919-919, November 2022. <br/> The first Research Spotlights article in this issue is concerned with filtering, a task of paramount importance in a great many applications such as numerical weather prediction and geophysical data assimilation. Authors Alessio Spantini, Ricardo Baptista, and Youssef M. Marzouk, in their article “Coupling Techniques for Nonlinear Ensemble Filtering,” describe discrete-time filtering as the act of characterizing the sequence of conditional distributions of the latent field at observation times, given all currently available measurements. Despite the existing literature on filtering, issues such as high-dimensional state spaces and sparse (in both space and time) observations still prove formidable in practice. The traditional approach of ensemble-based data assimilation is the ensemble Kalman filter (EnKF), involving a prediction (forecasting) step followed by an analysis step. However, the authors note an intrinsic bias of EnKF due to the linearity of the transformation, estimated under Gaussian assumptions, that is used in the analysis step, which limits its accuracy. To overcome this, they propose two non-Gaussian generalizations of the EnKF---the so-called stochastic and deterministic map filters---using nonlinear transformations derived from couplings between the forecast distribution and the filtering distribution. What is crucial is that the transformations “can be estimated efficiently...perhaps using only convex optimization,” that they “are easy to `localize' in high dimensions,” and that their computation “should not become increasingly challenging as the variance of the observation noise decreases.” Following a comprehensive description of their new approaches, the authors demonstrate numerically the superiority of their stochastic map filter approach over traditional EnKF. The subsequent discussion offers the reader several jumping off points for future research. Recovery of a sparse solution to a large-scale optimization problem is another ubiquitous problem arising in many applications such as image reconstruction, signal processing, and machine learning. The cost functional typically includes a regularization term in the form of an $ell_1$ norm term on the solution and/or regularized solution to enforce sparsity. Designing suitable algorithms for such recovery problems is the subject of our second Research Spotlights article. In “Sparse Approximations with Interior Point Methods,” authors Valentina De Simone, Daniela di Serafino, Jacek Gondzio, Spyridon Pougkakiotis, and Marco Viola set out to correct the misconception that first-order methods are to be preferred over second-order methods out of hand. Through case studies, they offer evidence that interior point methods (IPMs) which are constructed to “exploit special features of the problems in the linear algebra of IPMs” and which are designed “to take advantage of the expected sparsity of the optimal solution”
SIAM评论，第64卷，第4期，第919-919页，2022年11月。这一期的第一篇研究聚焦文章是关于滤波的，滤波是一项在许多应用中至关重要的任务，如数值天气预报和地球物理数据同化。作者Alessio Spantini, Ricardo Baptista和Youssef M. Marzouk在他们的文章“非线性集合滤波的耦合技术”中，将离散时间滤波描述为在给定所有当前可用的测量值的情况下，在观测时间表征潜在场的条件分布序列的行为。尽管已有关于滤波的文献，但诸如高维状态空间和稀疏(在空间和时间上)观测等问题在实践中仍然证明是艰巨的。基于集成的数据同化的传统方法是集成卡尔曼滤波(EnKF)，它包括一个预测(预报)步骤，然后是一个分析步骤。然而，作者指出，由于在高斯假设下估计的变换的线性，在分析步骤中使用了EnKF的固有偏差，这限制了其准确性。为了克服这个问题，他们提出了EnKF的两种非高斯推广——所谓的随机和确定性映射滤波器——使用从预测分布和滤波分布之间的耦合中导出的非线性变换。至关重要的是，转换“可以有效地估计……也许只使用凸优化”，它们“很容易在高维中‘定位’”，并且它们的计算“不会随着观测噪声的方差减少而变得越来越具有挑战性”。在全面描述了他们的新方法之后，作者在数值上证明了他们的随机映射滤波方法比传统的EnKF方法优越。随后的讨论为读者提供了未来研究的几个出发点。大规模优化问题的稀疏解的恢复是在图像重建、信号处理和机器学习等许多应用中出现的另一个普遍问题。代价函数通常包括一个正则化项，其形式为解决方案和/或正则化解决方案上的$ell_1$范数项，以加强稀疏性。为这样的恢复问题设计合适的算法是我们第二篇研究重点文章的主题。在《用内点法进行稀疏逼近》一书中，作者Valentina De Simone、Daniela di Serafino、Jacek Gondzio、Spyridon Pougkakiotis和Marco Viola着手纠正一阶方法优于二阶方法的错误观念。通过案例研究，他们提供了证据，证明内点法(ipm)是为了“利用ipm线性代数中问题的特殊特征”而构建的，并且是为了“利用最优解的期望稀疏性”而设计的，实际上可以成为解决这类优化问题的首选方法。他们的方法的关键是将原来的稀疏近似问题重新表述为一个看起来更大但具有可以资本化计算增益的特性的问题。对于四个代表性应用程序中的每一个，作者都展示了如何利用每次迭代中涉及的底层线性系统的特定问题结构的计算优势。这些努力是通过利用预期的稀疏性来补充的:使用启发式方法来减少接近零的变量，从而用条件较好的、较小的系统取代非常大的、条件不良的中间系统。他们的结论是，将时间投入到调整求解器以适应重新表述的变体所允许的结构上，并利用预期的稀疏性，可能是值得的，因为他们的演示表明，对于稀疏近似问题，ipm可能比最先进的一阶方法具有“明显的优势”。
{"title":"Research Spotlights","authors":"Misha E. Kilmer","doi":"10.1137/22n975561","DOIUrl":"https://doi.org/10.1137/22n975561","url":null,"abstract":"SIAM Review, Volume 64, Issue 4, Page 919-919, November 2022. <br/> The first Research Spotlights article in this issue is concerned with filtering, a task of paramount importance in a great many applications such as numerical weather prediction and geophysical data assimilation. Authors Alessio Spantini, Ricardo Baptista, and Youssef M. Marzouk, in their article “Coupling Techniques for Nonlinear Ensemble Filtering,” describe discrete-time filtering as the act of characterizing the sequence of conditional distributions of the latent field at observation times, given all currently available measurements. Despite the existing literature on filtering, issues such as high-dimensional state spaces and sparse (in both space and time) observations still prove formidable in practice. The traditional approach of ensemble-based data assimilation is the ensemble Kalman filter (EnKF), involving a prediction (forecasting) step followed by an analysis step. However, the authors note an intrinsic bias of EnKF due to the linearity of the transformation, estimated under Gaussian assumptions, that is used in the analysis step, which limits its accuracy. To overcome this, they propose two non-Gaussian generalizations of the EnKF---the so-called stochastic and deterministic map filters---using nonlinear transformations derived from couplings between the forecast distribution and the filtering distribution. What is crucial is that the transformations “can be estimated efficiently...perhaps using only convex optimization,” that they “are easy to `localize' in high dimensions,” and that their computation “should not become increasingly challenging as the variance of the observation noise decreases.” Following a comprehensive description of their new approaches, the authors demonstrate numerically the superiority of their stochastic map filter approach over traditional EnKF. The subsequent discussion offers the reader several jumping off points for future research. Recovery of a sparse solution to a large-scale optimization problem is another ubiquitous problem arising in many applications such as image reconstruction, signal processing, and machine learning. The cost functional typically includes a regularization term in the form of an $ell_1$ norm term on the solution and/or regularized solution to enforce sparsity. Designing suitable algorithms for such recovery problems is the subject of our second Research Spotlights article. In “Sparse Approximations with Interior Point Methods,” authors Valentina De Simone, Daniela di Serafino, Jacek Gondzio, Spyridon Pougkakiotis, and Marco Viola set out to correct the misconception that first-order methods are to be preferred over second-order methods out of hand. Through case studies, they offer evidence that interior point methods (IPMs) which are constructed to “exploit special features of the problems in the linear algebra of IPMs” and which are designed “to take advantage of the expected sparsity of the optimal solution” ","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"11 1","pages":""},"PeriodicalIF":10.2,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau","authors":"Alain Durmus, É. Moulines, M. Pereyra","doi":"10.1137/22m1522917","DOIUrl":"https://doi.org/10.1137/22m1522917","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"21 1","pages":"991-1028"},"PeriodicalIF":10.2,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81740714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Generalized Dual Transform: Linear Algebra and Geometry of (Pseudo)Inverting a Matrix","authors":"L. Withers","doi":"10.1137/19m1270410","DOIUrl":"https://doi.org/10.1137/19m1270410","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"115 1","pages":"1031-1061"},"PeriodicalIF":10.2,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85810021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some Comments on Preconditioning for Normal Equations and Least Squares","authors":"A. Wathen","doi":"10.1137/20m1387948","DOIUrl":"https://doi.org/10.1137/20m1387948","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"21 1","pages":"640-649"},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77377388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Delayed Gradient Methods for Symmetric and Positive Definite Linear Systems","authors":"Qinmeng Zou, F. Magoulès","doi":"10.1137/20m1321140","DOIUrl":"https://doi.org/10.1137/20m1321140","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"78 1","pages":"517-553"},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80829303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Deforming $||.||_{1}$ into $||.||_{infty}$ via Polyhedral Norms: A Pedestrian Approach","authors":"M. Gaudioso, J. Hiriart-Urruty","doi":"10.1137/21m1391481","DOIUrl":"https://doi.org/10.1137/21m1391481","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"4 1","pages":"713-727"},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73701075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Analyzing Pattern Formation in the Gray-Scott Model: An XPPAUT Tutorial","authors":"Demi L. Gandy, M. R. Nelson","doi":"10.1137/21m1402868","DOIUrl":"https://doi.org/10.1137/21m1402868","url":null,"abstract":"","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"90 1","pages":"728-747"},"PeriodicalIF":10.2,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85204860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Federal student loans are fixed-rate debt contracts with three main special features: (i) borrowers can use income-driven schemes to make payments proportional to their income above subsistence, (ii) after several years of good standing, the remaining balance is forgiven but taxed as ordinary income, and (iii) accrued interest is simple, i.e., not capitalized. For a very small loan, the cost-minimizing repayment strategy dictates maximum payments until full repayment, forgoing both income-driven schemes and forgiveness. For a very large loan, the minimal payments allowed by income-driven schemes are optimal. For intermediate balances, the optimal repayment strategy may entail an initial period of minimum payments to exploit the non-capitalization of accrued interest, but when the principal is being reimbursed maximal payments always precede minimum payments. Income-driven schemes and simple accrued interest mostly benefit borrowers with very large balances. MSC (2010): 91G20, 91G80.
{"title":"Minimizing the Repayment Cost of Federal Student Loans","authors":"P. Guasoni, Yu‐Jui Huang","doi":"10.1137/22M1505840","DOIUrl":"https://doi.org/10.1137/22M1505840","url":null,"abstract":". Federal student loans are fixed-rate debt contracts with three main special features: (i) borrowers can use income-driven schemes to make payments proportional to their income above subsistence, (ii) after several years of good standing, the remaining balance is forgiven but taxed as ordinary income, and (iii) accrued interest is simple, i.e., not capitalized. For a very small loan, the cost-minimizing repayment strategy dictates maximum payments until full repayment, forgoing both income-driven schemes and forgiveness. For a very large loan, the minimal payments allowed by income-driven schemes are optimal. For intermediate balances, the optimal repayment strategy may entail an initial period of minimum payments to exploit the non-capitalization of accrued interest, but when the principal is being reimbursed maximal payments always precede minimum payments. Income-driven schemes and simple accrued interest mostly benefit borrowers with very large balances. MSC (2010): 91G20, 91G80.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"105 1","pages":"689-709"},"PeriodicalIF":10.2,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79540102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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