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熟知Grothendieck都知道，他有代数几何三部曲：EGA、SGA、FGA。其中知名度最高的无疑是EGA和SGA，他们可以说是代数几何的圣经，是无数重要且知名概念和理论的源头。而关于SGA，别说中文翻译了（目前我只有SGA2的中译版），就算是法语版全系列，也不好集齐，更别说还是可复制、用Latex重写过的美化版本。部分书似乎已经在网上绝迹，我也没细查。我当年也耗费了不少时间才集齐了SGA 1-7，现在分享出来给有需要或者想要收藏的人。PS：因为文件比较大，也比较多，因此分成了四个压缩包分卷上传。更新：作者不再提供文件下载。

分析学大师Elias M. Stein（曾是陶哲轩的老师），写了四本分析学系列教材，统称为普林斯顿分析学讲座（Princeton Lectures in Analysis）。他们分别是：I Fourier Analysis：An Introduction II Complex Analysis III Real Analysis: Measure Theory, Integration, and Hilbert Spaces IV Functional Analysis: Introduction to Further Topics in Analysis当时集齐这四本书花了我不少时间，似乎这四本书知名度不一，我下的第一本是复分析教材Complex Analysis。现在我将这些好东西拿出来分享给有需要的人。PS：如果需要中译版的，目前只能找到《实分析》和《复分析》两本，链接：伊莱亚斯 M. 斯坦恩（Elias M. Stein）《复分析》与《实分析》教材更新：作者不再提供附件下载。

Zariski的名字估计学代数几何的人都耳熟能详，先是入门时期的交换代数教材，然后就是深入研究时期随处可见的Zariski拓扑。本帖我们分享的便是著名的Zariski交换代数教材。Oscar Zariski & Pierre Samuel写的交换代数经典教材Commutative Algebra，该教材也是学习代数几何的经典入门前置教材之一，用于补充交换代数相关的前置知识。毕竟众所周知，代数几何的基础是抽象代数，尤其是交换代数，因此想要学习代数几何，就必须要有交换代数方面的扎实基础。交换代数方面的经典教材不少，包括Atiyah的那本Introduction To Commutative Algebra，那本书篇幅较小，更为简略感觉更加适合新人小白。而Zariski的Commutative Algebra则内容更加完备、更为系统性，该教材分为两本，基本上把代数几何相关的交换代数内容全都梳理了一遍。因此，Zariski的这本教材不仅可以作为初学者的交换代数入门教材，还能作为交换代数的词典用于查阅交换代数相关的知识。Zariski的这本教材，我记得当年网络上能找到的只是Commutative Algebra I的影印版，非但不是pdf版而且还不能复制。接着找Commutative Algebra II也是花了很多心思才弄到。而这个Commutative Algebra I和Commutative Algebra II的pdf可复制版能搞到可谓是相当不容易，我记得当年最后是我自己整OCR扫描弄出来的。在此之前，我先是淘宝付费买过pdf可复制版，再是自己买了本纸质书（也不好找），总之折腾了很久。更新：关于Atiyah的教材请看Atiyah交换代数经典入门教材：Introduction to Commutative Algebra，而关于Matsumura的教材则请移步Matsumura交换代数入门教材：Commutative Ring Theory。最新更新：作者不再提供附件下载。

Loring W Tu的微分几何入门教材An Introduction to Manifolds，中译名为《流形导论》。这本教材十分适合对微分几何感兴趣的萌新小白作为入门教材，想当年高二的时候，我就是因为看Jürgen Jost的Riemannian Geometry and Analysis看不懂，转而看Loring W Tu的An Introduction to Manifolds补充基础。Loring W Tu的书可以说非常对我胃口，这本书首先内容完备，把微分几何所有重要的基础概念给你讲一遍，而且语言简洁明了、思路清晰、通俗易懂。初三到高中时期，我看过不少微分几何的教材，包括陈省身的《微分几何讲义》，最后还是Loring W Tu的An Introduction to Manifolds让我真正学懂了微分几何😄。本教材从最基础的欧几里得空间光滑函数开始讲起，并不需要太多的前置知识即可开啃😁，只需要有大学本科数分高代的一些基础即可。而且其中的数学英文也并不需要太高的水平，因此也适合初步开始读英文文献的小白用于锻炼自己的英文数学阅读能力。Loring W Tu除了这本流形导论，还有一个微分几何的教材叫做Differential Geometry Connections, Curvature, and Characteristic Classes，我当年为了看懂几何分析相关的Jürgen Jost的的Riemannian Geometry and Analysis，也看了一下这本教材，可以说受益匪浅。关于Loring W Tu的另一本教材，见Loring W Tu微分几何教材：Differential Geometry Connections, Curvature, and Characteristic Classes。而关于Jürgen Jost的教材则见Jürgen Jost黎曼几何与几何分析教材：Riemannian Geometry and Geometric Analysis。PS：作者不再提供附件下载。

在上贴中分析学大师Elias M. Stein的分析系列教材，我分享了Elias M. Stein的分析全系列英文版，然而有人说想看中文版。经过我的查找，发现网络上流出的Stein中译书很少，最后只找到了比较知名的《复分析》和《实分析》。PS：由于文件较大，两本书分成了3个压缩包分卷上传。作者不再提供附件下载。

在上帖中，我分享了Zariski的交换代数教材：Zariski交换代数经典教材Commutative Algebra系列（pdf可复制版）。其实交换代数方面，除了Zariski的教材，还有Atiyah的Introduction to Commutative Algebra，以及Matsumura的Commutative Ring Theory可以作为交换代数的入门教材。Atiyah的教材是这三本教材中最简单的，Zariski的教材虽然很完备，但是篇幅过长，而且内容太过经典了，没有Atiyah的教材那样更加贴近新时代。而Matsumura的教材篇幅要比Atiyah的长一些，而且似乎感觉Atiyah的表达更加通俗易懂一些，毕竟Atiyah是众所周知的大师级人物。下面我们来回忆一下Atiyah的一些人物轶事。Atiyah作为与Serre齐名的伟大数学家，他最著名的工作即是与辛格一起证明了指标定理（Atiyah-Singer Index Theorem）。而Atiyah也与Grothendieck关系匪浅，见下图😁而Atiyah对物理也同样非常感兴趣，他与很多物理学家合作研究过，包括知名的唯一获得过菲尔兹奖的理论物理学家——威腾。其中我印象最深刻的就是他与威腾的关系Q：和他（威滕）一起工作是什么样的？阿蒂亚：2001年，他在加州理工学院做访问教授，并邀请我去那里。在那里我感觉自己又回到了研究生时代。每天早上，我会去学院找威滕，大概讨论一个小时左右。他会给我布置家庭作业。然后我用接下来的23个小时努力完成作业。与此同时，他也会去做很多其他的事情。我们之间的合作十分紧密。对于我来说，这是一段非常不可思议的经历，就像是和一位非常优秀的导师一起工作一样。我的意思是，他总是能比我先知道问题的答案。如果我们观点不一致，那么结果总是他是对的而我是错的。这让人非常尴尬！Atiyah虽然晚年因为各种事情而名誉受损，比如说最出圈的——声称证明了黎曼猜想，但这不影响他是一个伟大杰出的数学家。关于Matsumura的教材则请移步Matsumura交换代数入门教材：Commutative Ring Theory。PS：作者不再提供附件下载。

熟知Grothendieck都知道，他有代数几何三部曲：EGA、SGA、FGA。其中知名度最高的无疑是EGA和SGA，他们可以说是代数几何的圣经，是无数重要且知名概念和理论的源头。相较于SGA，EGA受众可能更大些，看的人也更多些。毕竟SGA只是讨论班，而EGA则相当于代数几何的百科词典。在上帖中，我已经分享了SGA法语原版全系列（链接：代数几何教皇Grothendieck经典著作：代数几何讨论班法语原版全系列），EGA法语全系列相较于SGA在当年要好收集一些，但也不容易。在当时已经有中文翻译版了，还有英文版翻译，我都看过，最后觉得还是法语版最好，英文版次之。因为有些术语翻译成中文，真的不太好理解，见英语不好，读不懂英文数学教材怎么办？不过之后我还是会把中文翻译版和英文翻译版都发出来。接着我还发一发Grothendieck的其他著作，包括收获与播种、伽罗华长征、一个纲领的提纲（Esquisse d'un Programme）等。EGA有四系列，为EGA 1-4，但总共分为8册书，EGA 3有两本，EGA 4有四本。由于文件较大，我分成两贴将这些东西发完。本贴先发EGA 1-3，需要下载三个压缩包分卷。下一贴链接：代数几何教皇Grothendieck经典著作：代数几何原理法语原版全系列（2）更新：需要EGA英译版版全系列的请看下帖：代数几何教皇Grothendieck经典著作：代数几何原理EGA英译版全系列。需要EGA 1（1971第二版）的，请看Grothendieck经典著作：代数几何原理EGA 1（1971第二版）法语+英译。更新：作者不再提供文件下载。

法国数学家让-皮埃尔·塞尔（Jean Pierre Serre）是迄今为止最年轻的菲尔兹奖得主，他获奖时年仅27岁，被国际数学领域誉为“在世最伟大”的数学家之一，他在代数拓扑学、多复变函数论、代数几何与数论方面取得了开创性的、历史性的巨大贡献。 Serre与被誉为代数几何的上帝的Grothendick是亲密无间的学术伙伴，他们两个的数学风格可以说是截然相反的，相较于Grothendick更喜欢构造宏大的理论，Serre更喜欢解决具体的问题。而两者的合作碰撞出了无数的火花，诞生了无数经典的理论。具体可见 明星崛起 - 宛如来自空无的召唤。而代数凝聚层（Faisceaux algébriques cohérents）这本书是Serre的经典著作之一，讲述的是层论方法在代数几何中的应用。本书原版是法语写的，后来被翻译成中英文版本。现在我将自己收藏已久的中英法三个版本，都分享出来给有需要的人，欢迎感兴趣的收藏收藏！PS：第一个附件为法语版，第二个为英语版，第三个为中文版。 作者不再提供附件下载。

这是本图论的入门教材，Graph Theory Fifth Edition，隶属于著名的GTM系列，作者是Reinhard Diestel。这是本对新人友好的教材，之前本科上离散数学的课时，因为涉及到图论，而学校的课堂又太水让我心生不满，于是便找了本图论的教材来看，就是这本GTM173。这本教材详略得当，并且图文并茂，十分符合图论的风格。并且开始就从图论开始讲，没有过多的废话直接切入主题。整本教材内容完备，基本上把图论相关的基础知识都覆盖了，因此对图论感兴趣的初学者可以尝试读读这本书。我毕竟不是做这个方向的，因此并不能发表太多的评价。之所以分享这本书也是因为，刚好网站里有人想要图论相关的入门教材但没有，因此分享一下。更新：作者不再提供文件下载。

这是Grothendick著名的关于同调代数的文章Tôhoku paper的英文翻译版，原文是法语版，标题为Sur quelques points d'algèbre homologique。英文翻译为：Some aspects of homological algebra。该文章概述了很多同调代数的重要概念，其中基本都跟代数几何有联系，并且里面不少概念其实是Grothendick本人提出来的，如abelian categories。可以说这篇文章是同调代数的经典文章，在数学圈内也时常有人推荐看这篇文章，毕竟这可是祖师爷亲自从同调代数的基础概念一步步讲起，这对学同调代数或者代数几何的人都有很大裨益。我收藏这篇文章的时候都2021年了，现在拿出来推荐给大家！之后我还会把法语原版也发出来。更新：作者不再提供文件下载。

经典泛函分析教材，作者是Mr. Andrew Pinchuck。这是本非常适合小白入门的泛函分析教材，里面的内容讲述通俗易懂、清晰明了。并且从最基础的线性空间讲起，并不需要太多的前置知识即可开始学习。这本书也是我人生中看的第一本英文书，同时也是我第一本看完的英文数学书。这算是我的数学启蒙教材之一，得益于这本书对萌新的友好，当时才初三、高一时期的我对这本书可谓是喜欢至极。现在我拿出来给大家推荐，希望能帮助到更多有需要的人！更新：作者不再提供文件下载。

在前面两帖Zariski交换代数经典教材Commutative Algebra系列（pdf可复制版）和 Atiyah交换代数经典入门教材：Introduction to Commutative Algebra 中，我分享了Zariski和Atiyah的交换代数教材。在本帖中，我把Matsumura的教材也分享出来。在这里我重新回顾一下这三本教材的区别。首先，Zariski的教材很完备，但是篇幅过长，而且内容太过经典了，没有另外两本那么与时俱进。因此Zariski的教材更加适合作为交换代数的词典用于查阅。当然如果你不需要按部就班从头到尾的看完一本书，Zariski的教材选择性的跳着看，完全可以作为入门教材。我高中的时候就是看Zariski的教材的。Atiyah的教材是这三本教材中最简单的，也是篇幅最短的。而Matsumura的教材篇幅要比Atiyah的长一些，并且Matsumura的教材有一些Atiyah中没有的概念，因此也值得一读，不过Atiyah教材的表达要更加通俗易懂一些。因此，我的建议是三本教材都读一读，但没必要全部看完，把需要掌握的基础概念都掌握了就行。读文献时有些术语找不到，还有Stack Project可以查（话说我把Stack Project的一些内容翻译成中文放在百科里了，就几个定义🙃），当然如果是一些比较经典的概念倒是可能在Zariski的教材能查到。PS：作者不再提供附件下载。

IntroductionThe rapid evolution of big data technologies and artificial intelligence has radically transformed many aspects of society, businesses, people and the environment, enabling individuals to manage, analyze and gain insights from large volumes of data (Dwivedi et al., 2023). The AI Spark Big Model is one effective technology that has played a critical role in addressing significant data challenges and sophisticated ML operations. For example, the adoption of Apache Spark in various industries has resulted in the growth of a number of unique and diverse Spark applications such as machine learning, processing streaming data and fog computing (Ksolves Team, 2022). As Pointer (2024) stated, in addition to SQL, streaming data, machine learning, and graph processing, Spark has native API support for Java, Scala, Python, and R. These evolutions made the model fast, flexible, and friendly to developers and programmers. Still, the AI Spark Big Model has some challenges: the interpretability of the model, the scalability of the model, the ethical implications, and integration problems. This paper addresses the negative issues linked to the implementation of these models and further explores the  potential future developments that Spark is expected to undergo.Challenges in the AI Spark Big ModelOne critical problem affecting the implementation of the Apache Spark model involves problems with serialization, precisely, the cost of serialization often associated with Apache Spark (Simplilearn, 2024). Serialization and deserialization are necessary in Spark as they help transfer data over the network to the various executors for processing. However, these processes can be expensive, especially when using languages such as Python, which do not serialize data as effectively as Java or Scala. This inefficiency can have a significant effect on the performance of Spark applications. In Spark architecture, applications are partitioned into several segments sent to the executors (Nelamali, 2024). To achieve this, objects need to be serialized for network transfer. If Spark encounters difficulties in serializing objects, it results in the error: org. Apache. Spark. SparkException: Task not serializable. This error can occur in many situations, for example, when some objects used in a Spark task are not serializable or when closures use non-serializable variables (Nelamali, 2024). Solving serialization problems is essential for improving the efficiency and stability of Spark applications and their ability to work with data and execute tasks in distributed systems.Figure 1: Figure showing the purpose of Serialization and deserialization The second challenge affecting the implementation of Spark involves the management of memory. According to Simplilearn, 2024, the in-memory capabilities of Spark offer significant performance advantages because data processing is done in memory, but at the same time, they have drawbacks that can negatively affect application performance. Spark applications usually demand a large amount of memory, and poor memory management results in frequent garbage collection pauses or out-of-memory exceptions. Optimizing memory management for big data processing in Spark is not trivial and requires a good understanding of how Spark uses memory and the available configuration parameters (Nelamali, 2024). Among the most frequent and annoying problems is the OutOfMemoryError, which can affect the Spark applications in the cluster environment. This error can happen in any part of Spark execution but is more common in the driver and executor nodes. The driver, which is in charge of coordinating the execution of tasks, and the executors, which are in charge of the data processing, both require a proper distribution of memory to avoid failures (Simplilearn, 2024). Memory management is a critical aspect of the Spark application since it affects the stability and performance of the application and, therefore, requires a proper strategy for allocating and managing resources within the cluster.The use of Apache Spark is also greatly affected by the challenges of managing large clusters. When data volumes and cluster sizes increase, the problem of cluster management and maintenance becomes critical. Identifying and isolating job failures or performance issues in large distributed systems can be challenging (Nelamali, 2024). One of the problems that can be encountered is when working with large data sets; actions sometimes produce errors if the total size of the results exceeds the value of Spark Driver Max Result Size set by Spark. Driver. maxResultSize. When this threshold is surpassed, it triggers the error: org. Apache. Spark. SparkException: Job aborted due to stage failure: The total size of serialized results of z tasks (x MB) is more significant than Spark Driver maxResultSize (y MB) (Nelamali, 2024). These errors highlight the challenges of managing big data processing in Spark, where complex solutions for cluster management, resource allocation, and error control are needed to support large-scale computations.Figure 2: The Apache Spark ArchitectureAnother critical issue that has an impact on the Apache Spark deployment is the Small Files Problem. Spark could be more efficient when dealing with many small files because each task is considered separate, and the overhead can consume most of the job's time. This inefficiency makes Spark less preferable for use cases that involve many small log files or similar data sets. Moreover, Spark also depends on the Hadoop ecosystem for file handling (HDFS) and resource allocation (YARN), which adds more complexity and overhead. Nelamali, 2024 argues that although Spark can operate in standalone mode, integrating Hadoop components usually improves Spark's performance. The implementation of Apache Spark is also affected by iterative algorithms as there is a problem of support for complex analysis. However, due to the system's architecture being based on in-memory processing, in theory, Spark should be well-suited for iterative algorithms. However, it can be noticed that it can be inefficient sometimes (Sewal & Singh, 2021). This inefficiency is because Spark uses resilient distributed datasets (RDDs) and requires users to cache intermediate data in case it is used for subsequent computation. After each iteration, there is data writing and reading, which performs operations in memory, thus noting higher times of execution and resources requested and consumed, which affects the expected boost in performance. Like Spark, which has MLlib for extensive data machine learning, some libraries may not be as extensive or deep as those in the dedicated machine learning platforms (Nguyen et al., 2019). Some users may be dissatisfied with Spark’s provision since MLlib may present basic algorithms, hyper-parameter optimization, and compatibility with other extensive ML frameworks. This restriction tends to make Spark less suitable for more elaborate analytical work, and a person may have to resort to the use of other tools as well as systems to obtain a certain result.The Future of Sparka. Enhanced Machine Learning (ML)Since ML assumes greater importance in analyzing BD, Spark’s MLlib is updated frequently to manage the increasing complexity of ML procedures (Elshawi et al., 2018). This evolution is based on enhancing the number of the offered algorithms and tools that would refine performance, functionality, and flexibility. Future enhancements is more likely to introduce deeper learning interfaces that can be directly integrated into the Spark platform while implementing more neural structures in the network. Integration of TensorFlow and PyTorch, along with the optimized library for GPU, will be helpful in terms of time and computational complexity required for training and inference associated with high dimensional data and large-scale machine learning problems. Also, the focus will be on simplifying the user interface through better APIs, AutoML capabilities, and more user-friendly interfaces for model optimization and testing (Simplilearn, 2024). These advancements will benefit data scientists and engineers who deal with big data and help democratize ML by providing easy ways to deploy and manage ML pipelines in distributed systems. Better support for real-time analysis and online education will also help organizations gain real-time insights, thus improving decision-making.b. Improved Performance and Efficiency Apache Spark's core engine is continuously improving to make it faster and more efficient as it continues to be one of the most popular technologies in the ample data space. Some of the areas of interest are memory management and other higher levels of optimization, which minimize the overhead of computation and utilization of resources (Simplilearn, 2024). Memory management optimization will reduce the time taken for garbage collection and enhance the management of in-memory data processing, which is vital for high throughput and low latency in big data processing. Also, improvements in the Catalyst query optimizer and Tungsten execution engine will allow for better execution of complicated queries and data transformations. These enhancements will be beneficial in cases where large amounts of data are shuffled and aggregated, often leading to performance issues. Future attempts to enhance support for contemporary hardware, like faster storage devices such as NVMe and improvements in CPU and GPU, will only increase Spark's capacity to process even more data faster (Armbrust et al., 2015). Moreover, future work on AQE will enable Spark to adapt the execution plans at runtime by using statistics, which will enhance data processing performance. Altogether, these improvements will guarantee that Spark remains a high-performance and scalable tool that will help organizations analyze large datasets.c. Integration with the Emerging Data Sources With the growth of the number of data sources and their types, Apache Spark will transform to process many new data types. This evolution will enhance the support for the streaming data originating from IoT devices that give real-time data that requires real-time analyses. Improved connectors and APIs shall improve data ingestion and processing in real-time, hence improving how quickly Spark pulls off high-velocity data (Dwivedi et al., 2023). In addition, the exact integration with the cloud will also be improved in Spark, where Cloud platforms will take charge of ample data storage and processing. This involves more robust integration with cloud-native storage, data warehousing, and analytics services from AWS, Azure, and Google Cloud. Also, Spark will leverage other types of databases, such as NoSQL, graph, and blockchain databases, to enable the user to conduct analytics on different types and structures of data. Thus, Spark will allow organizations to offer the maximum value from the information they deal with, regardless of its source and form, providing more comprehensive and timely information.d. Cloud-Native Features Since cloud computing is becoming famous, Apache Spark is also building inherent compatibility for cloud-based environments that makes its use in cloud environments easier. The updates focusing on the cloud surroundings are the Auto-Scaling Services for the provisioning and configuring tools that simplify the deployment of Spark Clusters on cloud solutions (Simplilearn, 2024). These tools will allow integration with cloud-native storage and compute resources and allow users to grow their workloads on the cloud. New possibilities in resource management will enable the user to control and allocate cloud resources more effectively according to their load, releasing resources in case of low utilization and adapting costs and performance characteristics in this way. Spark will also continue to provide more backing to serverless computing frameworks, enabling users to execute Spark applications without handling the underlying infrastructure. This serverless approach will allow for automatic scaling, high availability, and cost optimization since users only pay for the time the computing resources are used. Improved support for Kubernetes, one of the most popular container orchestration systems, will strengthen Spark's cloud-native features and improve container management, orchestration, and integration with other cloud-native services (Dwivedi et al., 2023). These enhancements will help to make Spark more usable and cost-effective for organizations that are using cloud infrastructure to support big data analytics while at the same time reducing the amount of overhead required to do so.e. Broader Language Support Apache Spark is expected to become even more flexible as the support for other programming languages is expected to be added to the current list of Scala, Java, Python, and R languages used in Spark development. Thus, by including languages like Julia, which is famous for its numerical and scientific computing performance, Spark can draw developers working in specific niches that demand high data processing (Simplilearn, 2024). Also, supporting languages like JavaScript could bring Spark to the large community of web developers, allowing them to perform big data analytics within a familiar environment. The new language persists in compatibility to integrate Spark's various software environments and processes that the developers deem essential. Besides, this inclusiveness increases the span of control, thereby making extensive data analysis more achievable, while the increased number of people involved in the Spark platform ideas fosters creativity as more people get a chance to participate as well as earn from the platform (Dwivedi et al., 2023). Thus, by making Spark more available and setting up the possibility to support more programming languages, it would be even more embedded into the vast data platform, and more people would come forward to develop the technology.f. Cross-Platform and Multi-Cluster Operations In the future, Apache Spark will experience significant developments aimed at enhancing the long-awaited cross-system interoperability and organizing several clusters or the cluster of one hybrid or multiple clouds in the future (Dwivedi et al., 2023). Such improvements will help organizations avoid having Spark workloads run on one platform or cloud vendor alone, making executing more complex and decentralized data processing tasks possible. The level of interoperability will be enhanced in a way that there will be data integration and data sharing between the on-premise solutions, private clouds and public clouds to enhance data consonance (Simplilearn, 2024). These developments will offer a real-time view of the cluster and resource consumption, which will help to mitigate the operational overhead of managing distributed systems. Also, strong security measures and compliance tools will guarantee data management and security in different regions and environments (Dwivedi et al., 2023). With cross-platform and multi-cluster capabilities, Spark will help organizations fully leverage their data architecture, allowing for more flexible, scalable, and fault-tolerant big data solutions that meet the organization's requirements and deployment topology.g. More robust Growth of community and Ecosystem Apache Spark's future is, therefore, closely linked with the health of the open-source ecosystem, which is central to the development of Apache Spark through contributions and innovations. In the future, as more developers, researchers, and organizations use Spark, we can expect to see the development of new libraries and tools that expand its application in different fields (Simplilearn, 2024). Community-driven projects may promote the creation of specific libraries for data analysis, machine learning, and other superior functions, making Spark even more versatile and efficient. These should provide new features and better performance, encourage best practice and comprehensive documentation and make the project approachable for new members if and when they are needed. The cooperation will also be healthy in developing new features for real-time processing and utilising other resources and compatibility with other technologies, as noted by Armbrust et al., 2015. The further development of the Ecosystem will entail more active and creative users who can test and improve the solutions quickly. This culture of continual improvement and expansion of new services will ensure that Spark continues to evolve; it will remain relevant today and in the future for big data analytics and will remain desirable for the market despite the dynamics of the technological landscape.ConclusionDespite significant progress, Apache Spark has numerous difficulties associated with big data and machine learning problems when using flexible and fault-tolerant structures: serialization, memory, and giant clusters. Nonetheless, there are a couple of factors that have currently impacted Spark. Nevertheless, the future of Spark is quite bright, with expectations of having better features in machine learning, better performance, integration with other data sources, and the development of new features in cloud computing. More comprehensive language support, single/multiple clusters, more cluster operations, and growth of the Spark community and Ecosystem will further enhance its importance in big data and AI platforms. Thus, overcoming these challenges and using future progress, Spark will go on to improve and offer improved and more efficient solutions in different activities related to data processing and analysis.References Armbrust, M., Xin, R. S., Lian, C., Huai, Y., Liu, D., Bradley, J. K., ... & Zaharia, M. (2015, May). Spark SQL: Relational data processing in Spark. In Proceedings of the 2015 ACM SIGMOD international conference on management of data (pp. 1383-1394). Dwivedi, Y. K., Sharma, A., Rana, N. P., Giannakis, M., Goel, P., & Dutot, V. (2023). Evolution of artificial intelligence research in Technological Forecasting and Social Change: Research topics, trends, and future directions. Technological Forecasting and Social Change, p. 192, 122579. Elshawi, R., Sakr, S., Talia, D., & Trunfio, P. (2018). Extensive data systems meet machine learning challenges: towards big data science as a service. Big data research, 14, 1-11. Ksolves Team (2022). Apache Spark Benefits: Why Enterprises are Moving To this Data Engineering Tool. Available at: https://www.ksolves.com/blog/big-data/spark/apache-spark-benefits-reasons-why-enterprises-are-moving-to-this-data-engineering-tool#:~:text=Apache%20Spark%20is%20rapidly%20adopted,machine%20learning%2C%20and%20fog%20computing. Nelamali, M. (2024). Different types of issues while running in the cluster. https://sparkbyexamples.com/spark/different-types-of-issues-while-running-spark-projects/ Nguyen, G., Dlugolinsky, S., Bobák, M., Tran, V., López García, Á., Heredia, I., ... & Hluchý, L. (2019). Machine learning and deep learning frameworks and libraries for large-scale data mining: a survey. Artificial Intelligence Review, 52, 77-124. Pointer. K. (2024). What is Apache Spark? The big data platform that crushed Hadoop. Available at: https://www.infoworld.com/article/2259224/what-is-apache-spark-the-big-data-platform-that-crushed-hadoop.html#:~:text=Berkeley%20in%202009%2C%20Apache%20Spark,machine%20learning%2C%20and%20graph%20processing. Sewall, P., & Singh, H. (2021, October). A critical analysis of Apache Hadoop and Spark for big data processing. In 2021 6th International Conference on Signal Processing, Computing and Control (ISPCC) (pp. 308–313). IEEE. Simplilearn (2024). The Evolutionary Path of Spark Technology: Lets Look Ahead! Available at: https://www.simplilearn.com/future-of-spark-article#:~:text=Here%20are%20some%20of%20the,out%2Dof%2Dmemory%20errors. Tang, S., He, B., Yu, C., Li, Y., & Li, K. (2020). A survey on spark ecosystem: Big data processing infrastructure, machine learning, and applications. IEEE Transactions on Knowledge and Data Engineering, 34(1), 71-91.

问题描述：正常使用pip install xxx安装会弹出错误，导致下载失败。必须增加trust host字段，才能下载成功：pip --trusted-host pypi.python.org install在cmd运行python -c "import ssl; print(ssl.get_default_verify_paths())"在默认路径里没有找到ca证书，而在Lib\site-packages\certifi文件夹中，却发现了cacert.pem文件。故而认为原因是ca证书丢失或者寻找ca证书路径出错，因此尝试修改pip的默认ca证书路径。pip.ini文件中有大量的pip配置信息，因此需要先找到该文件。在cmd通过pip -v config list发现，在多个路径中，都没有找到pip.ini文件。且了解到，pip会有一个默认的pip.conf文件（其实就是pip.ini），因此断定默认pip.ini配置文件丢失。解决办法：在python根目录中，新建pip.ini，在里面写上[global] index-url = https://mirrors.aliyun.com/pypi/simple/ [install] trusted-host=mirrors.aliyun.com 之后cmd输入pip -v config list，能够看到证明修改成功。现在在cmd直接输入pip install xxx已经不再弹出错误。

There are several ways to write and render beautiful math on web. However, some methods can't be directly applied to Vue.js/Nuxt.js. In this article, we will explain how to use katex and mathjax to render math in Vue.js/Nuxt.js.KatexTo automatically render all math in all the pages, you need to use CDN to load katex :<!-- index.html --> <!DOCTYPE html> <html lang="en"> <head> <meta charset="UTF-8"> <link rel="icon" href="/poem-studio-favicon-black.svg"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous"> <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script> <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous" onload="renderMathInElement(document.body);"></script> <title>Manitori</title> </head> <body> <div id="app"></div> <script type="module" src="/src/main.js"></script> </body> </html>If you are using Nuxt.js, then you need to change your nuxt.config.ts ://nuxt.config.ts export default defineNuxtConfig({ app: { head: { link: [ {rel:'stylesheet', href:"https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css", integrity:"sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww", crossorigin:"anonymous"} ], script: [ { defer:true, src:"https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js", integrity:"sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd", crossorigin:"anonymous" }, { defer:true, src:"https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js", integrity:"sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk", crossorigin:"anonymous", onload:"renderMathInElement(document.body);" }, ] } } })If you want to specify the options of the renderMathInElement function, you could call renderMathInElement in another <script> :<!DOCTYPE html> <html lang="en"> <head> <meta charset="UTF-8"> <link rel="icon" href="/poem-studio-favicon-black.svg"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.css" integrity="sha384-wcIxkf4k558AjM3Yz3BBFQUbk/zgIYC2R0QpeeYb+TwlBVMrlgLqwRjRtGZiK7ww" crossorigin="anonymous"> <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/katex.min.js" integrity="sha384-hIoBPJpTUs74ddyc4bFZSM1TVlQDA60VBbJS0oA934VSz82sBx1X7kSx2ATBDIyd" crossorigin="anonymous"></script> <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.10/dist/contrib/auto-render.min.js" integrity="sha384-43gviWU0YVjaDtb/GhzOouOXtZMP/7XUzwPTstBeZFe/+rCMvRwr4yROQP43s0Xk" crossorigin="anonymous"></script> <script> document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { // customised options // • auto-render specific keys, e.g.: delimiters: [ {left: '$$', right: '$$', display: true}, {left: '$', right: '$', display: false}, {left: '\\(', right: '\\)', display: false}, {left: '\\[', right: '\\]', display: true} ], // • rendering keys, e.g.: throwOnError : false }); }); </script> <title>Manitori</title> </head> <body> <div id="app"></div> <script type="module" src="/src/main.js"></script> </body> </html>Note that you'd better change document.body to a specific area document.getElementById(Id), otherwise, it may cause some fatal error, see Vue - TypeError: Cannot read properties of null (reading 'insertBefore'). To render math in a specific area, you need to call renderMathInElement separately in each page. For example:<script lang="ts" setup> onMounted(()=>{ nextTick(()=>{ var node = document.getElementById(Id) document.addEventListener("DOMContentLoaded", function() { renderMathInElement(node, { // customised options // • auto-render specific keys, e.g.: delimiters: [ {left: '$$', right: '$$', display: true}, {left: '$', right: '$', display: false}, {left: '\\(', right: '\\)', display: false}, {left: '\\[', right: '\\]', display: true} ], // • rendering keys, e.g.: throwOnError : false }); }); }) }) </script>In Vue.js, you may need to asynchronously render math, so you can follow this example:<script lang="ts" setup> var node = document.getElementById(Id) Promise.resolve() .then(()=>{ nextTick(()=>{ document.addEventListener("DOMContentLoaded", function() { renderMathInElement(node, { // customised options // • auto-render specific keys, e.g.: delimiters: [ {left: '$$', right: '$$', display: true}, {left: '$', right: '$', display: false}, {left: '\\(', right: '\\)', display: false}, {left: '\\[', right: '\\]', display: true} ], // • rendering keys, e.g.: throwOnError : false }); }); }) }) </script>MathjaxIt is easy to automatically render all math using mathjax. Like katex, you'd better use CDN to load mathjax :<!-- index.html --> <!DOCTYPE html> <html lang="en"> <head> <meta charset="UTF-8"> <link rel="icon" href="/poem-studio-favicon-black.svg"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <script type="text/javascript" id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@4.0.0-beta.6/tex-chtml.js"> </script> <script> MathJax = { tex: { inlineMath: [['$', '$'], ['\\(', '\\)']] } }; </script> <title>Manitori</title> </head> <body> <div id="app"></div> <script type="module" src="/src/main.js"></script> </body> </html>If you are using Nuxt.js, then change your nuxt.config.ts like this://nuxt.config.ts export default defineNuxtConfig({ app: { head: { script: [ { type: "text/javascript", id: "MathJax-script", async: true, src: "https://cdn.jsdelivr.net/npm/mathjax@4.0.0-beta.6/tex-chtml.js", }, { innerHTML: "MathJax = {tex: {inlineMath: [['$', '$'],['$$', '$$']]}};", }, ] })However, in Vue.js you also need to asynchronously render math using mathjax, otherwise, The rendered math formula will revert back to original text. You can call MathJax.typesetPromise() to achieve this. For example:<script lang="ts" setup> Promise.resolve() .then(()=>{ nextTick(() => { MathJax.typesetPromise(); }); }) </script>Or you could use setTimeout instead of nextTick:setTimeout(() => { MathJax.typesetPromise(); }, 3000);Following our methods, you can easily integrate Vue.js/Nuxt.js with katex and mathjax😄!

现代诗也叫"白话诗"，最早可追源到清末，是诗歌的一种，与古典诗歌相比而言，虽都为感于物而作，但一般不拘格式和韵律。现代诗形式自由，意涵丰富，意象经营重于修辞运用，完全突破了古诗"温柔敦厚，哀而不怨"的特点，更加强调自由开放和直率陈述与进行"可感与不可感之间"的沟通。中国代表人物：徐志摩、北岛、顾城、海子等徐志摩（1897—1931年）徐志摩，现代诗人、散文家。新月派代表诗人，新月诗社成员。在剑桥两年深受西方教育的熏陶及欧美浪漫主义和唯美派诗人的影响，奠定其浪漫主义诗风。代表作品有《再别康桥》《翡冷翠的一夜》。1. 再别康桥轻轻的我走了，正如我轻轻的来；我轻轻的招手，作别西天的云彩。那河畔的金柳，是夕阳中的新娘；波光里的艳影，在我的心头荡漾。软泥上的青荇，油油的在水底招摇；在康桥的柔波里，我甘心做一条水草！那榆荫下的一潭，不是清泉，是天上虹 揉碎在浮藻间，沉淀着彩虹似的梦。寻梦？撑一支长蒿，向青草更青处漫溯，满载一船星辉，在星辉斑斓里放歌。但我不能放歌，悄悄是别离的笙箫；夏虫也为我沉默，沉默是今晚的康桥！悄悄的我走了，正如我悄悄的来；我挥一挥衣袖，不带走一片云彩。2. 我不知道风是在哪一个方向吹我不知道风是在哪一个方向吹——我是在梦中，在梦的轻波里依洄。我不知道风是在哪一个方向吹——我是在梦中，她的温存，我的迷醉。我不知道风是在哪一个方向吹——我是在梦中，甜美是梦里的光辉。我不知道风是在哪一个方向吹——我是在梦中，她的负心，我的伤悲。我不知道风是在哪一个方向吹——我是在梦中，在梦的悲哀里心碎！我不知道风是在哪一个方向吹——我是在梦中，黯淡是梦里的光辉。3. 偶然我是天空里的一片云，偶尔投影在你的波心──你不必讶异，更无须欢喜──在转瞬间消灭了踪影。你我相逢在黑夜的海上，你有你的，我有我的，方向；你记得也好，最好你忘掉在这交会时互放的光亮！4. 沙扬娜拉--赠日本女郎最是那一低头的温柔，像一朵水莲花不胜凉风的娇羞，道一声珍重，道一声珍重，那一声珍重里有蜜甜的忧愁--沙扬娜拉!顾城（1956—1993年）顾城，中国朦胧诗派的重要代表诗人，被称为当代的"唯灵浪漫主义"诗人。顾城在新诗、旧体诗和寓言故事诗上都有很高的造诣，其《一代人》中的一句"黑夜给了我黑色的眼睛/我却用它寻找光明"成为中国新诗的经典名句。5. 一代人黑夜给了我黑色的眼睛我却用它寻找光明6. 远和近你一会看我一会看云我觉得你看我时很远你看云时很近7. 小巷小巷又弯又长没有门没有窗我拿把旧钥匙敲着厚厚的墙8. 门前我多么希望，有一个门口早晨，阳光照在草上我们站着扶着自己的门扇门很低，但太阳是明亮的草在结它的种子风在摇它的叶子我们站着，不说话就十分美好有门，不用开开是我们的，就十分美好早晨，黑夜还要流浪我们把六弦琴交给他我们不走了我们需要土地需要永不毁灭的土地我们要乘着它度过一生土地是粗糙的，有时狭隘然而，它有历史有一份天空，一份月亮一份露水和早晨我们爱土地我们站着用木鞋挖着泥土门也晒热了我们轻轻靠着，十分美好墙后的草不会再长大了，它只用指尖，触了触阳光戴望舒（1905—1950年）戴望舒，中国现代著名的诗人，为中国现代象征派诗歌的代表。因《雨巷》成为传诵一时的名作，他被称为“雨巷诗人”。9. 雨巷撑着油纸伞，独自彷徨在悠长，悠长又寂寥的雨巷，我希望逢着一个丁香一样的结着愁怨的姑娘。她是有丁香一样的颜色，丁香一样的芬芳，丁香一样的忧愁，在雨中哀怨，哀怨又彷徨。她彷徨在寂寥的雨巷，撑着油纸伞像我一样，像我一样地，默默彳亍着，冷漠，凄清，又惆怅。她静默地走近走近，又投出太息一般的眼光，她飘过像梦一般的像梦一般的凄婉迷茫。像梦中飘过一支丁香地，我身旁飘过这女郎；她静静地远了，远了，到了颓圮的篱墙，走尽这雨巷。在雨的哀曲里，消了她的颜色，散了她的芬芳，消散了，甚至她的太息般的眼光，丁香般的惆怅。撑着油纸伞，独自彷徨在悠长，悠长又寂寥的雨巷，我希望飘过一个丁香一样的结着愁怨的姑娘。10. 烦忧说是寂寞的秋的清愁，说是辽远的海的相思。假如有人问我的烦忧，我不敢说出你的名字。我不敢说出你的名字，假如有人问我的烦忧：说是辽远的海的相思，说是寂寞的秋的清愁。林徽因（1904—1955年）林徽因， 中国著名建筑师、诗人、作家。文学上，著有散文、诗歌、小说、剧本、译文和书信等，代表作《你是人间四月天》，《莲灯》，《九十九度中》等。其中，《你是人间四月天》最为大众熟知，广为传诵。11. 你是人间的四月天我说你是人间的四月天；笑响点亮了四面风；轻灵在春的光艳中交舞着变。你是四月早天里的云烟，黄昏吹着风的软，星子在无意中闪，细雨点洒在花前。那轻，那娉婷你是，鲜妍百花的冠冕你戴着，你是天真，庄严，你是夜夜的月圆。雪化后那篇鹅黄，你象；新鲜初放芽的绿，你是；柔嫩喜悦水光浮动着你梦期待中白莲。你是一树一树的花开，是燕在梁间呢喃，——你是爱，是暖，是希望，你是人间的四月天！12. 雨后天我爱这雨后天，这平原的青草一片！我的心没底止的跟着风吹，风吹：吹远了香草，落叶，吹远了一缕云，象烟——象烟。13. 情愿我情愿化成一片落叶，让风吹雨打到处飘零；或流云一朵，在澄蓝天，和大地再没有些牵连。但抱紧那伤心的标志，去触遇没着落的怅惘；在黄昏，夜班，蹑着脚走，全是空虚，再莫有温柔；忘掉曾有这世界；有你；哀悼谁又曾有过爱恋；落花似的落尽，忘了去这些个泪点里的情绪。到那天一切都不存留，比一闪光，一息风更少痕迹，你也要忘掉了我曾经在这世界里活过。卞之琳(1910—2000年)卞之琳，现当代诗人("汉园三诗人"之一)、文学评论家、翻译家。为中国的文化教育事业做了很大贡献。诗《断章》是他不朽的代表作。对莎士比亚很有研究，西语教授，并且在现代诗坛上做出了重要贡献。被公认为新文化运动中重要的诗歌流派新月派和现代派的代表诗人。14. 断章你站在桥上看风景，看风景人在楼上看你。明月装饰了你的窗子，你装饰了别人的梦。胡适(1891—1962年)胡适，著名思想家、文学家、哲学家。 以倡导"白话文、领导新文化运动闻名于世。" 胡适一生的学术活动主要在文学、哲学、史学、考据学、教育学、红学几个方面。他在学术上影响最大的是提倡"大胆的假设、小心的求证"的治学方法。15. 梦与诗都是平常经验，都是平常影象，偶然涌到梦中来，变幻出多少新奇花样！都是平常情感，都是平常言语，偶然碰着个诗人，变幻出多少新奇诗句！醉过才知酒浓，爱过才知情重；——你不能做我的诗，正如我不能做你的梦。16. 也是微云也是微云，也是微云过后月光明。只不见去年得游伴，也没有当日的心情。不愿勾起相思，不敢出门看月。偏偏月进窗来，害我相思一夜。17. 秘魔崖月夜依旧是月圆时，依旧是空山，静夜；我独自月下归来，──这凄凉如何能解！翠微山上的一阵松涛惊破了空山的寂静。山风吹乱的窗纸上的松痕，吹不散我心头的人影。艾青（1910—1996年）艾青，中国现代诗人。被认为是中国现代诗的代表诗人之一。诗作努力反映民族和人民的苦难与命运，反映现实的生活和斗争，突出表现对光明的热烈向往和讴歌，风格朴素雄浑。长篇小说《绿洲笔记》等艾青被称为“一生追求光明的作家”18. 我爱这土地假如我是一只鸟，我也应该用嘶哑的喉咙歌唱：这被暴风雨所打击的土地，这永远汹涌着我们的悲愤的河流，这无止息地吹刮着的激怒的风，和那来自林间的无比温柔的黎明……——然后我死了，连羽毛也腐烂在土地里面。为什么我的眼里常含泪水？因为我对这土地爱得深沉……19. 树一棵树，一棵树彼此孤离地兀立着风与空气告诉着它们的距离但是在泥土的覆盖下它们的根生长着在看不见的深处它们把根须纠缠在一起20. 失去的岁月不像丢失的包袱可以到失物招领处找得回来，失去的岁月甚至不知丢失在什么地方——有的是零零星星地消失的，有的丢失了十年二十年，有的丢失在喧闹的城市，有的丢失在遥远的荒原，有的是人潮汹涌的车站，有的是冷冷清清的小油灯下面；丢失了的不像是纸片，可以拣起来倒更像一碗水投到地面被晒干了，看不到一点影子；时间是流动的液体——用筛子、用网，都打捞不起；时间不可能变成固体，要成了化石就好了，即使几万年也能在岩层里找见i时间也像是气体，像急驰的列车头上冒出的烟！失去了的岁月好像一个朋友，断掉了联系，经受了一些苦难，忽然得到了消息；说他早已离开了人间余光中（1928—2017）余光中，一生从事诗歌、散文、评论、翻译，自称为自己写作的"四度空间"。至今驰骋文坛已逾半个世纪，涉猎广泛，被誉为"艺术上的多妻主义者"。其文学生涯悠远、辽阔、深沉，为当代诗坛健将、散文重镇、著名批评家、优秀翻译家。21. 乡 愁小时候乡愁是一枚小小的邮票我在这头母亲在那头长大後乡愁是一张窄窄的船票我在这头新娘在那头後来啊乡愁是一方矮矮的坟墓我在外头母亲在里头而现在乡愁是一湾浅浅的海峡我在这头大陆在那头22.等你，在雨中等你 在雨中 在造虹的雨中蝉声沉落 蛙声升起一池的红莲如红焰 在雨中你来不来都一样 竟感觉每朵莲都像你尤其隔著黄昏 隔著这样的细雨永恒 刹那 刹那 永恒等你 在时间之外在时间之内 等你 在刹那 在永恒如果你的手在我的手里此刻如果你的清芬在我的鼻孔 我会说 小情人诺 这只手应该采莲 在吴宫这只手应该摇一柄桂浆 在木兰舟中一颗星悬在科学馆的飞檐耳坠子一般的悬著瑞士表说都七点了　忽然你走来步雨後的红莲 翩翩 你走来像一首小令从一则爱情的典故里你走来从姜白石的词中 有韵地 你走来23. 今生今世今生今世我最忘情的哭声有两次一次，在我生命的开始一次，在你生命的告终第一次，我不会记得，是听你说的第二次，你不会晓得，我说也没用但两次哭声的中间啊有无穷无尽的笑声一遍一遍又一遍回荡了整整三十年你都晓得，我都记得有无穷无尽的笑声一遍一遍又一遍回荡了整整30年你都晓得，我都记得郑愁予（1933—）郑愁予，当代诗人。他的《错误》、《水手刀》、《残堡》、《小小的岛》、《如雾起时》等诗，不仅令人着迷，而且使人陶醉。被称为"浪子诗人"，"中国的中国诗人"。24. 错误我打江南走过那等在季节里的容颜如莲花的开落东风不来，三月的柳絮不飞你底心如小小寂寞的城恰若青石的街道向晚音不响，三月的春帷不揭你底心是小小的窗扉紧掩我达达的马蹄是美丽的错误我不是归人，是个过客……25. 小小的岛你住的小小的岛我正思念那儿属於热带，属於青青的国度浅沙上，老是栖息著五色的鱼群小鸟跳响在枝上，如琴键的起落那儿的山崖都爱凝望，披垂著长藤如发那儿的草地都善等待，铺缀著野花如过果盘那儿浴你的阳光是蓝的，海风是绿的则你的健康是郁郁的，爱情是徐徐的云的幽默与隐隐的雷笑林丛的舞乐与冷冷的流歌你住的那小小的岛我难描绘难绘那儿的午寐有轻轻的地震如果，我去了，将带著我的笛杖那时我是牧童而你是小羊要不，我去了，我便化做萤火虫以我的一生为你点盏灯北岛北岛，中国当代诗人，为朦胧诗代表人物之一，是民间诗歌刊物《今天》的创办者，曾多次获诺贝尔文学奖提名。北岛是当代影响最大的中国诗人之一。代表诗作有《回答》《一切》。26. 回答卑鄙是卑鄙者的通行证，高尚是高尚者的墓志铭，看吧，在那镀金的天空中，飘满了死者弯曲的倒影。冰川纪过去了，为什么到处都是冰凌？好望角发现了，为什么死海里千帆相竞？我来到这个世界上，只带着纸、绳索和身影，为了在审判之前，宣读那些被判决的声音。告诉你吧，世界我--不--相--信！纵使你脚下有一千名挑战者，那就把我算作第一千零一名。我不相信天是蓝的，我不相信雷的回声，我不相信梦是假的，我不相信死无报应。如果海洋注定要决堤，就让所有的苦水都注入我心中，如果陆地注定要上升，就让人类重新选择生存的峰顶。新的转机和闪闪星斗，正在缀满没有遮拦的天空。那是五千年的象形文字，那是未来人们凝视的眼睛。27. 一切一切都是命运一切都是烟云一切都是没有结局的开始一切都是稍纵即逝的追寻一切欢乐都没有微笑一切苦难都没有泪痕一切语言都是重复一切交往都是初逢一切爱情都在心里一切往事都在梦中一切希望都带着注释一切信仰都带着呻吟一切爆发都有片刻的宁静一切死亡都有冗长的回声28. 一束在我和世界之间你是海湾，是帆是缆绳忠实的两端你是喷泉，是风是童年清脆的呼喊在我和世界之间你是画框，是窗口是开满野花的田园你是呼吸，是床头是陪伴星星的夜晚在我和世界之间你是日历，是罗盘是暗中滑行的光线你是履历，是书签是写在最后的序言在我和世界之间你是纱幕，是雾是映入梦中的灯盏你是口笛，是无言之歌是石雕低垂的眼帘在我和世界之间你是鸿沟，是池沼是正在下陷的深渊你是栅栏，是墙垣是盾牌上永久的图案刘半农（1891-1934）刘半农，是“五四”新文化运动的积极倡导者，尝试派代表诗人之一。中国新文化运动先驱，文学家、语言学家和教育家。主要作品有诗集《扬鞭集》、《瓦釜集》和《半农杂文》。29. 教我如何不想她天上飘着些微云，地上吹着些微风。啊！微风吹动了我头发，教我如何不想她？月光恋爱着海洋，海洋恋爱着月光。啊！这般蜜也似的银夜，教我如何不想她？水面落花慢慢流，水底鱼儿慢慢游。啊！燕子你说些什么话？教我如何不想她？枯树在冷风里摇。野火在暮色中烧。啊！西天还有些儿残霞，教我如何不想她？30. 落叶秋风把树叶吹落在地上，它只能悉悉索索，发几阵悲凉的声响。它不久就要化作泥；但它留得一刻，还要发一刻的声响，虽然这已是无可奈何的声响了，虽然这已是它最后的声响了。刘大白( 1880~1932年 )刘大白，中国诗人，与鲁迅先生是同乡好友，现代著名诗人，文学史家。1919年他应经亨颐之聘在浙一师与陈望道、夏丏尊、李次九一起改革国语教育，被称为"四大金刚"。1925年为复旦大学校歌作词。复旦校歌歌词介于文言与白话之间，由复旦师生传唱至今。31. 是谁把是谁把心里相思，种成红豆?待我来碾豆成尘，看还有相思没有?是谁把空中明月，捻得如钩?待我来抟钩作镜，看永久团圆能否?32. 邮吻我不是不能用指头儿撕，我不是不能用剪刀儿剖，祇是缓缓地轻轻地很仔细地挑开了紫色的信唇；我知道这信唇里面，藏着她秘密的一吻。从她底很郑重的折叠里，我把那粉红色的信笺，很郑重地展开了。我把她很郑重地写的一字字一行行，一行行一字字地很郑重地读了。我不是爱那一角模糊的邮印，我不是爱那幅精致的花纹，祇是缓缓地轻轻地很仔细地揭起那绿色的邮花；我知道这邮花背后，藏着她秘密的一吻。闻一多（1899—1946年）闻一多，中国现代伟大的爱国主义者，坚定的民主战士，中国民主同盟早期领导人，中国共产党的挚友，新月派代表诗人和学者。1925年3月在美国留学期间创作《七子之歌》。33. 七子之歌澳门你可知“妈港”不是我的真名姓？我离开你的襁褓太久了，母亲！但是他们掳去的是我的肉体，你依然保管着我内心的灵魂。三百年来梦寐不忘的生母啊！请叫儿的乳名，叫我一声“澳门”！母亲！我要回来，母亲！香港我好比凤阙阶前守夜的黄豹，母亲呀，我身分虽微，地位险要。如今狞恶的海狮扑在我身上，啖着我的骨肉，咽着我的脂膏；母亲呀，我哭泣号啕，呼你不应。母亲呀，快让我躲入你的怀抱！母亲！我要回来，母亲！台湾我们是东海捧出的珍珠一串，琉球是我的群弟我就是台湾。我胸中还氲氤着郑氏的英魂，精忠的赤血点染了我的家传。母亲，酷炎的夏日要晒死我了；赐我个号令，我还能背城一战。母亲！我要回来，母亲！威海卫再让我看守着中华最古的海，这边岸上原有圣人的丘陵在。母亲，莫忘了我是防海的健将，我有一座刘公岛作我的盾牌。快救我回来呀，时期已经到了。我背后葬的尽是圣人的遗骸！母亲！我要回来，母亲！广州湾东海和广州是我的一双管钥，我是神州后门上的一把铁锁。你为什么把我借给一个盗贼？母亲呀，你千万不该抛弃了我！母亲，让我快回到你的膝前来，我要紧紧地拥抱着你的脚踝。母亲！我要回来，母亲！九龙我的胞兄香港在诉他的苦痛，母亲呀，可记得你的幼女九龙？自从我下嫁给那镇海的魔王，我何曾有一天不在泪涛汹涌！母亲，我天天数着归宁的吉日，我只怕希望要变作一场空梦。母亲！我要回来，母亲！旅顺，大连我们是旅顺，大连，孪生的兄弟。我们的命运应该如何的比拟？两个强邻将我来回的蹴蹋，我们是暴徒脚下的两团烂泥。母亲，归期到了，快领我们回来。你不知道儿们如何的想念你！母亲！我们要回来，母亲！冰心（1900—1999年）冰心，中国诗人，现代作家，翻译家，儿童文学作家，社会活动家，散文家。笔名冰心取自“一片冰心在玉壶”。发表的《寄小读者》散文，成为中国儿童文学的奠基之作。34. 母亲母亲呵！天上的风雨来了，鸟儿躲到它的巢里；心中的风雨来了，我只躲到你的怀里。35. 纸船——寄母亲我从不肯妄弃了一张纸，总是留着——留着，叠成一只一只很小的船儿，从舟上抛下在海里。有的被天风吹卷到舟中的窗里，有的被海浪打湿，沾在船头上。我仍是不灰心的每天的叠着，总希望有一只能流到我要它到的地方去。母亲，倘若你梦中看见一只很小的白船儿，不要惊讶它无端入梦。这是你至爱的女儿含着泪叠的，万水千山求它载着她的爱和悲哀归去。八，二十七，一九二三太平洋舟中。邵洵美 ( 1906—1968年 )邵洵美，新月派诗人、散文家、出版家、翻译家。晚年从事外国文学翻译工作，译有马克·吐温、雪莱、泰戈尔等人的作品。其诗集有《天堂与五月》、《花一般的罪恶》。36. 季候初见你时你给我你的心，里面是一个春天的早晨。再见你时你给我你的话，说不出的是炽烈的火夏。三次见你你给我你的手，里面藏着个叶落的深秋。最后见你是我做的短梦，梦里有你还有一群冬风。朱湘（1904-1933）朱湘是二十年代清华园的四个学生诗人之一，与饶孟侃(字子理)、孙大雨(字子潜)和杨世恩(字子惠)并称为"清华四子"，后来与其他三子成为了中国现代诗坛上的重要诗人。在校期间，他的艺术天分已经崭露出来，当时就是清华校园的文学名人。37. 采莲曲小船呀轻飘，杨柳呀风里颠摇；荷叶呀翠盖，荷花呀人样娇娆。日落，微波，金丝闪动过小河。左行右撑，莲舟上扬起歌声。菡萏呀半开，蜂蝶呀不许轻来，绿水呀相伴，清净呀不染尘埃，溪间，采莲，水珠滑走过荷钱。拍紧拍轻，桨声应答着歌声。藕心呀丝长，羞涩呀水底深藏；不见呀蚕茧，丝多呀蛹裹在中央？溪头采藕，女郎要采又夷犹。波沉，波升，波上抑扬着歌声。莲蓬呀子多：两岸呀榴树婆娑，喜鹊呀喧噪，榴花呀落上新罗。溪中采蓬，耳鬓边晕着微红。风定风生，风飔荡漾着歌声。升了呀月钩，明了呀织女牵牛；薄雾呀拂水，凉风呀飘去莲舟。花芳衣香消溶入一片苍茫；时静，时闻，虚空里袅着歌音。38. 摇篮歌春天的花香真正醉人，一阵阵温风拂上人身，你瞧日光它移的多慢，你听蜜蜂在窗子外哼：睡呀，宝宝，蜜蜂飞的真轻。天上瞧不见一颗星星，地上瞧不见一盏红灯；什么声音也都听不到，只有蚯蚓在天井里吟：睡呀，宝宝，蚯蚓都停了声。一片片白云天空上行，像是些小船飘过湖心，一刻儿起，一刻儿又沉，摇着船舱里安卧的人：睡呀，宝宝，你去跟那些云。不怕它北风树枝上鸣，放下窗子来关起房门；不怕它结冰十分寒冷，炭火生在那白铜的盆：睡呀，宝宝，挨着炭火的温。陈梦家(1911~1966)陈梦家，中国现代著名古文字学家、考古学家、诗人。在三十年代的诗名很大，曾与闻一多、徐志摩、朱湘一起被目为"新月诗派的四大诗人"。著有诗集《梦家诗集》等，是后期新月派享有盛名的代表诗人和重要成员。39. 那一晚那一晚天上有云彩没有星，你搀了我的手牵动我的心。天晓得我不敢说我爱你，为了我是那样年青。那一晚你同我在黑巷里走，肩靠肩，你的手牵住我的手。天晓得我不敢说我爱你，把这句话压在心头。那一晚天那样暗人那样静，只有我和你身偎身那样近。天晓得我不敢说我爱你，平不了这乱跳的心。那一晚是一生难忘的错恨，上帝偷取了年青人的灵魂。如今我一万声说我爱你，却难再挨近你的身。40. 一朵野花一朵野花在荒原里开了又落了，不想这小生命，向着太阳发笑，上帝给他的聪明他自己知道，他的欢喜，他的诗，在风前轻摇。一朵野花在荒原里开了又落了，他看见青天，看不见自己的渺小，听惯风的温柔，听惯风的怒号，就连他自己的梦也容易忘掉。舒婷舒婷，中国当代女诗人，朦胧诗派的代表人物。她的诗歌充盈着浪漫主义和理想的色彩，对祖国、对人生、对爱情、对土地的爱，既温馨平和又潜动着激情。41. 向北方一朵初夏的蔷薇划过波浪的琴弦向不可及的水平远航乌云像癣一样布满天空的颜面鸥群却为她铺开洁白的翅膀去吧我愿望的小太阳如果你沉没了就睡在大海的胸膛在水母银色的帐顶永远有绿色的波涛喧响让我也漂去吧让阳光熨贴的风把我轻轻吹送顺着温暖的海流漂向北方42. 致橡树我如果爱你——绝不像攀援的凌霄花借你的高枝炫耀自己；我如果爱你——绝不学痴情的鸟儿为绿荫重复单调的歌曲；也不止像泉源常年送来清凉的慰藉；也不止像险峰增加你的高度，衬托你的威仪。甚至日光。甚至春雨。不，这些都还不够！我必须是你近旁的一株木棉，作为树的形象和你站在一起。根，紧握在地下；叶，相触在云里。每一阵风吹过，我们都互相致意，但没有人听懂我们的言语。你有你的铜枝铁干像刀、像剑，也像戟；我有我红硕的花朵像沉重的叹息，又像英勇的火炬。我们分担寒潮、风雷、霹雳；我们共享雾霭、流岚、虹霓。仿佛永远分离，却又终身相依。这才是伟大的爱情，坚贞就在这里：爱——不仅爱你伟岸的身躯，也爱你坚持的位置，足下的土地。43. 四月的黄昏四月的黄昏里流曳着一组组绿色的旋律在峡谷低回在天空游移要是灵魂里溢满了回响又何必苦苦寻觅要歌唱你就歌唱吧 但请轻轻 轻轻 温柔地四月的黄昏仿佛一段失而复得的记忆也许有一个约会至今尚未如期也许有一次热恋而不能相许要哭泣你就哭泣吧 让泪水流啊 流啊 默默地海子（1964-1989）海子，当代青年诗人。 海子带着对诗歌精神的信念走入诗歌，走入永恒。他直接成为这种精神的象征。海子的诗歌精神即浪漫精神。它要求通过一次性行动突出原始生命的内核和本质。海子的诗歌就是这种行动，它给我们展现了一个宠廓的前景，我们开始从当下的现实抬起头来，眺望远方。天空和大海的巨大背景逐渐在我们身后展开。44. 答 复麦地别人看见你觉得你温暖 美丽我则站在你痛苦质问的中心被你灼伤我站在太阳 痛苦的芒上麦地神秘的质问者啊当我痛苦地站在你的面前你不能说我一无所有你不能说我两手空空45. 风很美风很美小小的风很美自然界的乳房很美水很美水啊无人和你说话的时刻很美46. 面朝大海, 春暖花开从明天起, 做一个幸福的人喂马, 劈柴, 周游世界从明天起, 关心粮食和蔬菜我有一所房子, 面朝大海, 春暖花开从明天起, 和每一个亲人通信告诉他们我的幸福那幸福的闪电告诉我的我将告诉每一个人给每一条河每一座山取一个温暖的名字陌生人, 我也为你祝福愿你有一个灿烂的前程愿你有情人终成眷属愿你在尘世获的幸福我也愿面朝大海, 春暖花开47. 春天, 十个海子春天, 十个海子全都复活在光明的景色中嘲笑这一野蛮而悲伤的海子你这么长久地沉睡到底是为了什么?春天, 十个海子低低地怒吼围着你和我跳舞、唱歌扯乱你的黑头发, 骑上你飞奔而去, 尘土飞扬你被劈开的疼痛在大地弥漫在春天, 野蛮而复仇的海子就剩这一个, 最后一个这是黑夜的儿子, 沉浸于冬天, 倾心死亡不能自拔, 热爱着空虚而寒冷的乡村那里的谷物高高堆起, 遮住了窗子它们一半而于一家六口人的嘴, 吃和胃一半用于农业, 他们自己繁殖大风从东吹到西, 从北刮到南, 无视黑夜和黎明你所说的曙光究竟是什么意思48. 日记姐姐，今夜我在德令哈，夜色笼罩姐姐，我今夜只有戈壁草原尽头我两手空空悲痛时握不住一颗泪滴姐姐，今夜我在德令哈这是雨水中一座荒凉的城除了那些路过的和居住的德令哈┄┄今夜这是唯一的，最后的，抒情。这是唯一的，最后的，草原。我把石头还给石头让胜利的胜利今夜青稞只属于她自己一切都在生长今夜我只有美丽的戈壁 空空姐姐，今夜我不关心人类，我只想你洛夫(1928.5.11~ )洛夫，国际著名诗人、世界华语诗坛泰斗、诺贝尔文学奖提名者、中国最著名的现代诗人，被诗歌界誉为"诗魔"。49. 烟之外在涛声中唤你的名字而你的名字已在千帆之外潮来潮去左边的鞋印才下午右边的鞋印已黄昏了六月原是一本很感伤的书结局如此之凄美——落日西沉你依然凝视那人眼中展示的一片纯白他跪向你向昨日那朵美了整个下午的云海哟，为何在众灯之中独点亮那一盏茫然还能抓住什么呢？你那曾被称为云的眸子现有人叫作烟50. 子夜读信子夜的灯是一条未穿衣棠的小河你的信像一尾鱼游来读水的温暖读你额上动人的鳞片读江河如读一面镜读镜中你的笑如读泡沫51. 众荷喧哗众荷喧哗而你是挨我最近最静，最最温婉的一朵要看，就看荷去吧我就喜欢看你撑着一把碧油伞从水中升起我向池心轻轻扔过去一拉石子你的脸便哗然红了起来惊起的一只水鸟如火焰般掠过对岸的柳枝再靠近一些只要再靠我近一点便可听到水珠在你掌心滴溜溜地转你是喧哗的荷池中一朵最最安静的夕阳蝉鸣依旧依旧如你独立众荷中时的寂寂我走了，走了一半又停住等你等你轻声唤我52. 母亲母亲卑微如青苔庄严如晨曦柔如江南的水声坚如千年的寒玉举目时她是皓皓明月垂首时她是莽莽大地您的伟大凝结了我的血肉您的伟大塑造了我的灵魂您的一生是一次爱的航行您用优美的年轮编成一册散发油墨清香的日历年年我都在您的深情里度过在您的肩膀和膝头嬉戏您是一棵大树春天倚着您幻想夏天倚着您繁荣秋天倚着您成熟冬天倚着您沉思您那高大宽广的树冠使四野永不荒野母亲您给了我生命您是抚育我成长的土地在悲伤时您是慰藉在沮丧时您是希望在软弱时您是力量在您小小海湾中躲避风雨您为我开阔了视野您是我永远的挚友生命的动力您怀着爱怜谨慎地俯身守护您尽情袒露明亮的胸襟您旺盛的精力笑容坚强您沸腾的血液奔流不息让我沉浸在您的欢乐中让我享受在您的温馨中让我陶醉在您的双臂间让我偎依在您的怀抱里悠悠的云里有淡淡的诗淡淡的诗里有绵绵的爱绵绵的爱里有深深的情深深的情里有浓浓的意如果母亲是雨那我就是雨后的虹如果母亲是月那我就是捧月的星母亲是我生长的根我是母亲理想的果我长大了母亲的黑发却似枫叶上的寒霜星星点点闪着银光我深深地吻着那些岁月的痕迹捧掬我一颗心献给您愿芳香醇厚的甜蜜萦绕您的生活愿我银铃般的笑声盈满您的眉间愿我全部的祝福揉进您的心田一片绿叶饱含它对根的情谊一首颂歌浓缩我对您的敬爱让您心中的花朵盛开如云让芬芳伴您走过悠悠岁月纪弦(1913~2013)纪弦，是台湾诗坛的三位元老之一(另两位为覃子豪与钟鼎文)，在台湾诗坛享有极高的声誉。纪弦不仅创作极丰，而且在理论上亦极有建树。他是现代派诗歌的倡导者，诗风明快，善嘲讽，乐戏谑。他的诗极有韵味，且注重创新，令后学者竞相仿效，成为台湾诗坛的一面旗帜。53. 不再唱的歌当我的与众不同成为一种时髦，而众人都和我差不多了，我便不再唱这支歌了。别问我为什么，亲爱的。我的路是千山万水。我的花是万紫千红。54. 你的名字用了世界上最轻最轻的声音，轻轻地唤你的名字每夜每夜。写你的名字，画你的名字，而梦见的是你的发光的名字：如日，如星，你的名字。如灯，如钻石，你的名字。如缤纷的火花，如闪电，你的名字。如原始森林的燃烧，你的名字。刻你的名字！刻你的名字在树上。刻你的名字在不凋的生命树上。当这植物长成了参天的古木时，啊啊，多好，多好，你的名字也大起来。大起来了，你的名字。亮起来了，你的名字。于是，轻轻轻轻轻轻轻地呼唤你的名字。席慕蓉（1943——）席慕蓉，当代画家、诗人、散文家。著有诗集、散文集、画册及选本等五十余种，《七里香》、《无怨的青春》、《一棵开花的树》等诗篇脍炙人口，成为经典。席慕容的作品多写爱情、人生、乡愁，写得极美，淡雅剔透，抒情灵动，饱含着对生命的挚爱真情，影响了整整一代人的成长历程。55.一棵开花的树如何让你遇见我在我最美丽的时刻为这我已在佛前求了五百年求佛让我们结一段尘缘佛於是把我化做一棵树长在你必经的路旁阳光下慎重地开满了花朵朵都是我前世的盼望当你走近请你细听那颤抖的叶是我等待的热情而当你终於无视地走过在你身後落了一地的朋友啊那不是花瓣那是我凋零的心56. 抉 择假如我来世上一遭只为与你相聚一次只为了亿万光年里的那一刹那一刹那里所有的甜蜜与悲凄那麽 就让一切该发生的都在瞬间出现吧我俯首感谢所有星球的相助让我与你相遇与你别离完成了上帝所作的一首诗然後 再缓缓地老去57. 初相遇美丽的梦和美丽的诗一样都是可遇而不可求的常常在最没能料到的时刻里出现我喜欢那样的梦在梦里 一切都可以重新开始一切都可以慢慢解释心里甚至还能感觉到所有被浪费的时光竟然都能重回时的狂喜和感激胸怀中满溢著幸福只因为你就在我眼前对我微笑 一如当年我真喜欢那样的梦明明知道你已为我跋涉千里却又觉得芳草鲜美 落英缤纷好像你我才初初相遇58. 青 春所有的结局都已写好所有的泪水也都已启程却忽然忘了是怎麽样的一个开始在那个古老的不再回来的夏日无论我如何地去追索年轻的你只如云影掠过而你微笑的面容极浅极淡逐渐隐没在日落後的群岚遂翻开那发黄的扉页命运将它装订得极为拙劣含著泪 我一读再读却不得不承认青春是一本太仓促的书59. 前 缘人若真能转世 世间若真有轮回那麽 我的爱 我们前世曾经是什麽你 若曾是江南采莲的女子我 必是你皓腕下错过的那朵你 若曾是逃学的顽童我 必是从你袋中掉下的那颗崭新的弹珠在路旁的草丛中目送你毫不知情地远去你若曾是面壁的高僧我必是殿前的那一柱香焚烧著 陪伴过你一段静默的时光因此 今生相逢 总觉得有些前缘未尽却又很恍忽 无法仔细地去分辨无法一一地向你说出60. 莲的心事我 是一朵盛开的夏荷多希望你能看见现在的我风霜还不曾来侵蚀秋雨也未滴落青涩的季节又已离我远去我已亭亭 不忧 也不惧现在 正是我最美丽的时刻重门却已深锁在芬芳的笑靥之後谁人知我莲的心事无缘的你啊不是来得太早 就是太迟

In this section, we focus on Section 2 in [Sch], following [Hu], [Hu1], and [Hu2]. Moreover, we need to compare Huber's adic spaces with Berkovich's analytic spaces and Tate's rigid analytic spaces. Hence, we will briefly introduce the notion of Berkovich's analytic spaces in §1.3 and the notion of rigid analytic varieties in §1.4.§1. Adic SpacesDefinition 1.1.  A morphism $f:X\rightarrow Y$ of adic spaces is adic if, for every $x\in X$, there exist open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that the ring homomorphism $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ of $f$-adic rings is adic.§1.1. Morphisms of finite type. The material can be seen in [SP] and [Hu1].First, we review the definition of morphisms of schemes of finite type/presentation (see [SP], Definition 29.15.1, Lemma 29.15.2, and Definition 29.21.1, and Lemma 29.21.2).Definition 1.2. Let $f:X\rightarrow Y$ be a morphism of schemes.We say that $f$ is locally of finite type if, for all affine opens $U,V$ of $X,Y$ with $f(U)\subset V$, the ring map $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ is of finite type.We say that $f$ is of finite type if it is quasi-compact and locally of finite type.We say that $f$ is locally of finite presentation if, for all affine opens $U,V$ of $X,Y$ with $f(U)\subset V$, the ring map $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ is of finite presentation.We say that $f$ is of finite presentation if it is quasi-compact, quasi-separated, and locally of finite presentation. Compared with the above definition, we reach to the case of adic spaces.Definition 1.3 ([Hu1, Definition 1.2.1]). Let $f:X\rightarrow Y$ be a morphism of adic spaces.We say that $f$ is locally of finite type if, for every $x\in X$, there exists open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that the ring homomorphism $(\mathscr{O}_{Y}(V),\mathscr{O}^{+}_{Y}(V))\rightarrow(\mathscr{O}_{X}(U),\mathscr{O}^{+}_{X}(U))$ of affinoid rings is topologically of finite type.We say that $f$ is of finite type if it is quasi-compact and locally of finite type.We say that $f$ is locally of finite presentation if, for every $x\in X$, there exists open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that the ring homomorphism $(\mathscr{O}_{Y}(V),\mathscr{O}^{+}_{Y}(V))\rightarrow(\mathscr{O}_{X}(U),\mathscr{O}^{+}_{X}(U))$ of affinoid rings is topologically of finite type and, if the topology of $\mathscr{O}_{Y}(V)$ is discrete, the ring map $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(U)$ is of finite presentation.Then $\{\textrm{morphisms locally of finite presentation}\}\subset\{\textrm{morphisms locally of finite type}\}\subset\{\textrm{adic}\newline\textrm{morphisms}\}$.§1.2. Unramified, smooth, and étale morphisms.For definitions of morphisms of finite type and finite presentation, see §1.1.First, we review the notions of unramified, smooth, and étale ring maps (see [SP], 10.138, 10.148, and 10.150, and 10.151).Definition 1.4. Let $R\rightarrow S$ be a ring map. We say $R\rightarrow S$ is formally smooth/formally unramified/formally étale or $S$ is formally smooth/formally unramified/formally étale over $R$ if for every solid commutative diagramwhere $I\subset A$ is a square zero ideal, there exists at least one/at most one/a unique dotted map $S\rightarrow A$ making the diagram commute.The definitions of smooth and étale ring maps make use of the naive cotangent complex, but we will simplify this.Definition 1.5. Let $R\rightarrow S$ be a ring map.We say $R\rightarrow S$ is smooth/étale or $S$ is smooth/étale over $R$ if $R\rightarrow S$ is of finite presentation and formally smooth/formally étale.We say $R\rightarrow S$ is unramified or $S$ is unramified over $R$ if $R\rightarrow S$ is of finite type and formally unramified.Compared with the definitions above, we reach to the case of adic spaces via changing some arrows.Definition 1.6 ([Hu1, Definition 1.6.5]). A morphism $f:X\rightarrow Y$ of adic spaces is unramified/smooth/étale if $f$ is locally of finite type/locally of finite presentation/locally of finite presentation and if, for any affinoid ring $A$, any ideal $I\subset A^{\vartriangleright}$ with $I^{2}=0$, and any morphism ${\rm{Spa}}(A)\rightarrow Y$, the map ${\rm{Hom}}_{Y}({\rm{Spa}}(A),X)\rightarrow{\rm{Hom}}_{Y}({\rm{Spa}}(A/I),X)$ is injective/surjective/bijective.A morphism $f:X\rightarrow Y$ of adic spaces is unramified/smooth/étale at a point $x\in X$ if there exist open affinoid subspaces $U,V$ of $X,Y$ with $x\in U$ and $f(U)\subset V$ such that $f|_{U}:U\rightarrow V$ is unramified/smooth/étale.Note that the second statement of (i) above can be described as follows. For every solid commutative diagram in the following, there exist at most one/at least one/a unique one dotted map making the diagram commute.§1.3. Berkovich’s analytic spaces.We will introduce the notion of Berkovich's analytic spaces following [Ber] and [Ber1]. Berkovich's analytic spaces is one of the non-archimedean analogues of complex analytic spaces. The definition of analytic spaces in [Ber1] is more general than the definition in [Ber] (the analytic spaces in [Ber] corresponds to the good analytic spaces in [Ber1]). So we will make use of the definition in [Ber1].§1.3.1 Underlying topological spaces.First, we introduce some structures on topological spaces for further use (see [Ber1, §1, 1.1]). All compact, locally compact, and paracompact spaces are assumed to be Hausdorff.Definition 1.7.A topological space is paracompact if it is Hausdorff and every open cover of it admits a locally finite refinement.A topological space $X$ is locally Hausdorff if every point $x\in X$ admits an open Hausdorff neighborhood.Remark 1.8. Note that in [Tam], a paracompact space also requires that the locally finite refinement in (i) above is an open cover.Let $X$ be a topological space and let $\tau$ be a collection of subsets of $X$ provided with the induced topology. We put $\tau|_{Y}:=\{V\in\tau;V\subset Y\}$ for any subset $Y\subset X$.Definition 1.9. We say that the collection $\tau$ above is a quasi-net on $X$ if, for every point $x\in X$, there exist $V_{1},...,V_{n}\in\tau$ such that $x\in V_{1}\cap\cdot\cdot\cdot\cap V_{n}$ and the set $V_{1}\cup\cdot\cdot\cdot\cup V_{n}$ is a neighborhood of $x$, i.e. $V_{1}\cup\cdot\cdot\cdot\cup V_{n}$ contains an open set $U\subset X$ with $x\in U$. Furthermore, $\tau$ is said to be a {\rm{net on $X$}} if it is a quasi-net and, for any $U,V\in\tau$, $\tau|_{U\cap V}$ is a quasi-net on $U\cap V$.Definition 1.10 ([Dug, p255]). Let $X$ be a topological space and $S\subset X$ be a subset. $S$ is said to be locally closed if every point $s\in S$ has a neighborhood $U$ such that $S\cap U$ is closed in $U$.§1.3.2 The category of analytic spaces.Throughout, we fix a nonarchimedean field $k$ whose valuation can be trivial. The category of $k$-affinoid spaces is dual to the category of $k$-affinoid algebras (see [Ber, §2.1]). The $k$-affinoid spaces associated with a $k$-affinoid algebra $\mathscr{A}$ is denoted by $X:=\mathscr{M}(\mathscr{A})$.If for each nonarchimedean field $K$ over $k$, we are given a class $\Phi_{K}$ of $K$-affinoid spaces, the system $\Phi=\{\Phi_{K}\}$ is assumed to satisfy the following conditions:(i) $\mathscr{M}(K)\in\Phi_{K}$.(ii) $\Phi_{K}$ is stable under isomorphisms and direct products. In other words, for $X\in\Phi_{K}$, if $X'$ is a $K$-affinoid space with $X\cong X'$, then we have $X'\in\Phi_{K}$, and for $X,Y\in\Phi_{K}$, we have $X\times Y\in\Phi_{K}$.(iii) If $\varphi:Y\rightarrow X$ is a finite morphism of $K$-affinoid spaces with $X\in\Phi_{K}$, then $Y\in\Phi_{K}$.(iv) If $(V_{i})_{i\in I}$ is a finite affinoid covering of a $K$-affinoid space $X$ with $V_{i}\in\Phi_{K}$, then $X\in\Phi_{K}$.(v) If $K\hookrightarrow L$ is an isometric embedding of nonarchimedean fields over $k$, then for any $X\in\Phi_{K}$, one has $X{\widehat{\otimes}_{K}L}\in\Phi_{L}$.Definition 1.11. The class $\Phi_{K}$ is said to be dense if each point of each $X\in\Phi_{K}$ admits a fundamental system of affinoid neighborhoods $V\in\Phi_{K}$. The system $\Phi$ is said to be dense if all $\Phi_{K}$ are dense.The affinoid spaces from $\Phi_{K}$ (resp. $\Phi$) and their affinoid algebras will be called $\Phi_{K}$-affinoid (resp. $\Phi$-affinoid).From (ii) and (iii) above, we deduce that $\Phi_{K}$ is stable under fiber products. In other words, for $X,Y,Z\in\Phi_{K}$ with morphisms $X\rightarrow Z$ and $Y\rightarrow Z$, we have $X\times_{Z}Y\in\Phi_{K}$.Let $X$ be a locally Hausdorff space and let $\tau$ be a net of compact subsets on $X$.Definition 1.12. A $\Phi_{K}$-atlas $\mathscr{A}$ on $X$ with the net $\tau$ is a map that assigns, to each $U\in\tau$, a $\Phi_{K}$-affinoid algebra $\mathscr{A}_{U}$ together with a homeomorphism $U\xrightarrow{\sim}\mathscr{M}(\mathscr{A}_{U})$ and, to each pair $U,V\in\tau$ with $U\subset V$, a bounded homomorphism $\mathscr{A}_{V}\rightarrow\mathscr{A}_{U}$ of $\Phi_{K}$-affinoid algebras that identifies $(U,\mathscr{A}_{U})$ with an affinoid domain in $(V,\mathscr{A}_{V})$.Definition 1.13. A triple $(X,\mathscr{A},\tau)$ of the above form is said to be a $\Phi_{K}$-analytic space.§1.4. Rigid analytic varieties.The notion of rigid analytic variety is also one of the nonarchimedean analogues of complex analytic space. It originated in John Tate's thesis, [Tat]. In this subsection, we briefly introduce it following [BGR] and [BS].§1.4.1 $G$-topological spaces. As a technical trick, we generalize the usual topology to the so-called Grothendieck topology, [SGA4]. Roughly speaking, a $G$-topological space is a set that admits a Grothendieck topology. We will first introduce Grothendieck topology following the definition in [BS], where the "Grothendieck topology" means the "Grothendieck pretopology" in [SGA4].Definition 1.14. Let $\mathscr{C}$ be a (small) category. A Grothendieck topology $T$ consists of the category ${\rm{Cat}}(T)=\mathscr{C}$ and a set ${\rm{Cov}}(T)$ of families $(U_{i}\rightarrow U)_{i\in I}$ of morphisms in $\mathscr{C}$, called open coverings, such that the following axioms are satisfied:If $U'\rightarrow U$ is an isomorphism in $\mathscr{C}$, then the one-element family $(U'\rightarrow U)\in{\rm{Cov}}(T)$.If $(U_{i}\rightarrow U)_{i\in I}$ and $(V_{ij}\rightarrow U_{i})_{j\in I}$ are open coverings, then $(V_{ij}\rightarrow U)_{i,j\in I}\in{\rm{Cov}}(T)$.If $(U_{i}\rightarrow U)_{i\in I}$ is an open covering and $V\rightarrow U$ is a morphism in $\mathscr{C}$, then the fiber products $V\times_{U}U_{i}$ exist in $\mathscr{C}$ and $(V\times_{U}U_{i}\rightarrow V)_{i\in I}\in{\rm{Cov}}(T)$.Remark 1.15. Note that this is slightly different to the definition in [Poon], which requires that a Grothendieck topology consists of the set ${\rm{Cov}}(T)$ only. Moreover, the pair $(\mathscr{C},T)$ is usually called a site. However, to suite our needs in rigid geometry, we stick with the terminology in [BS].We specialize the definition above to the case that is more suited to our needs. And from now on, we will exclusively consider the Grothendieck topology of such a special type, unless explicitly stated otherwise.Definition 1.16. Let $X$ be a set. A Grothendieck topology (also called $G$-topology) $\mathfrak{T}$ on $X$ consists ofa category of subsets of $X$, called admissible open subsets or $\mathfrak{T}$-open subsets of $X$, with inclusions as morphisms, anda set ${\rm{Cov}}(\mathfrak{T})$ of families $(U_{i}\rightarrow U)_{i\in I}$ of inclusions with $\bigcup_{i\in I}U_{i}=U$, called admissible coverings or $\mathfrak{T}$-coverings.Remark 1.17. Note that in this case, the fiber products will come as intersections of sets.We call $X$ a $G$-topological space and write more explicitly as $X_{\mathfrak{T}}$ when $\mathfrak{T}$ is needed to be specified.§1.4.2 Presheaves and sheaves on $G$-topological spaces. The notion of Grothendieck topology defined in § 1.4.1 enables us to adapt presheaf or sheaf to such a general situation.Definition 1.18 ([BS, 5.1, Definition 2]). Let $\mathfrak{C}$ be a category and let $\mathfrak{T}$ be a Grothendieck topology in the sense of Definition 1.14. A presheaf $\mathscr{F}$ on $\mathfrak{T}$ with values in $\mathscr{C}$ is a functor $$\mathscr{F}:{\rm{Cat}}(\mathfrak{T})^{opp}\longrightarrow\mathfrak{C}.$$If $\mathfrak{C}$ is a category admitting products, then the presheaf $\mathscr{F}$ is said to be a sheaf if the sequence $$\mathscr{F}(U)\rightarrow\prod_{i\in I}\mathscr{F}(U_{i})\mathrel{\mathop{\rightrightarrows}} \prod_{i,j\in I}\mathscr{F}(U_{i}\times_{U}U_{j})$$ is exact for any open covering $(U_{i}\rightarrow U)_{i\in I}$ in ${\rm{Cov}}(\mathfrak{T})$.Remark 1.19. Note that the definition of Grothendieck topology assures the existence of the fiber products $U_{i}\times_{U}U_{j}$ in $\textrm{Cat}(\mathfrak{T})$.Morphisms of presheaves or sheaves are just natural transformations of functors.Definition 1.20. A morphism of presheaves $f:\mathscr{F}\rightarrow\mathscr{G}$ is a morphism of functors from $\mathscr{F}$ to $\mathscr{G}$. A morphism of sheaves $f:\mathscr{F}\rightarrow\mathscr{G}$ is a morphism of presheaves $f:\mathscr{F}\rightarrow\mathscr{G}$.Hence, we can define presheaves and sheaves on a $G$-topological space.Definition 1.21 ([BGR, 9.2.1, Definition 1]). A presheaf $\mathscr{F}$ with values in a category $\mathscr{C}$ on a $G$-topological space $X$ is a contravariant functor $$\mathscr{F}:{\rm{Cat}}(\mathfrak{T})\longrightarrow\mathscr{C},$$ where $\mathfrak{T}$ is a Grothendieck topology on $X$. If $\mathscr{C}$ is a category admitting products, then $\mathscr{F}$ is a sheaf on the $G$-topological space $X$ if it is a sheaf in the sense of Definition 1.18.The following kind of Grothendieck topology is of special interest to us.Definition/Proposition 1.22 ([BGR, §5.1, Proposition 5]). Let $K$ be a field and let $X$ be an affinoid $K$-space. Then the strong Grothendieck topology on $X$ is a Grothendieck topology on $X$ that satisfies the following conditions:$(G_{0})$ $\varnothing$ and $X$ are admissible open subsets of $X$.$(G_{1})$ Let $U\subset X$ be an admissible open subset with an admissible covering $(U_{i})_{i\in I}$ and let $V\subset U$ a subset. If $U_{i}\cap V$ is admissible open in $X$ for each $i\in I$, then $V$ is admissible open in $X$.$(G_{2})$ If $\mathfrak{U}=(U_{i})_{i\in I}$ is a covering of an admissible open $U\subset X$ with an admissible refinement such that each $U_{i}$ is admissible open in $X$, then $\mathfrak{U}$ is an admissible covering of $U$.§1.4.3 Locally $G$-ringed spaces and analytic varieties.The definition of rigid analytic varieties makes use of the notion of locally $G$-ringed spaces. The so-called $G$-ringed spaces are analogous to our familiar ringed spaces.Definition 1.23 ([BGR, §9.1.1]). A $G$-ringed space is a pair $(X,\mathscr{O}_{X})$ consisting of a $G$-topological space $X$ and a sheaf $\mathscr{O}_{X}$ of rings on $X$, called the structure sheaf of $X$. A locally $G$-ringed space is a $G$-ringed space $(X,\mathscr{O}_{X})$ such that all stalks $\mathscr{O}_{X,x},x\in X$, are local rings. If the structure sheaf $\mathscr{O}_{X}$ is a sheaf of algebras over a fixed ring $R$, then such a $G$-ringed space $(X,\mathscr{O}_{X})$ is said to be over $R$.Definition 1.24 ([BGR, §9.1.1]). A map $f:X\rightarrow Y$ between $G$-topological spaces is said to be continuous if the following conditions are satisfied:(i) If $V\subset Y$ is an admissible subsets, then $f^{-1}(V)$ is an admissible subsets of $X$.(ii) If $(V_{i})_{i\in I}$ is an admissible covering of an admissible subset $V\subset Y$, then $(f^{-1}(V_{i}))_{i\in I}$ is an admissible covering of the admissible subset $f^{-1}(V)$.We need appropriate morphisms for $G$-ringed spaces. In fact, we have the following definitions analogous to that of morphisms of ringed spaces and locally ringed spaces.Definition 1.25 ([BGR, 9.3.1]). A morphism of $G$-ringed spaces $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ is a pair $(f,f^{*})$ where $f:X\rightarrow Y$ is a continuous map of $G$-topological spaces and $f^{*}$ is a collection $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(f^{-1}(V))$ of ring maps for any admissible open subset $V\subset Y$ that are compatible with restriction maps.A morphism of locally $G$-ringed spaces $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ is a morphism of $G$-ringed space $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ such that all induced ring maps $f^{*}_{x}:\mathscr{O}_{Y,f(x)}\rightarrow\mathscr{O}_{X,x}$ for $x\in X$ are local.Let $R$ be a fixed ring. An $R$-morphism $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ of $G$-ringed spaces over $R$ is a morphism of $G$-ringed spaces $f:(X,\mathscr{O}_{X})\rightarrow(Y,\mathscr{O}_{Y})$ such that, in addition, $f^{*}$ is a collection $\mathscr{O}_{Y}(V)\rightarrow\mathscr{O}_{X}(f^{-1}(V))$ of $R$-algebra homomorphisms for all admissible open subsets $V\subset Y$.Remark 1.26. We follow the convention of ringed spaces that we denote a $G$-ringed space $(X,\mathscr{O}_{X})$ simply by $X$ and we denote a morphism of $G$-ringed spaces by suppressing the morphism of structure sheaves.In the following, let $k$ be a fixed complete nonarchimedean field. Next, we are in a position to introduce global analytic varieties.Definition 1.27 ([BGR, 9.3.1, Definition 4]). A rigid analytic variety over $k$ (also called a $k$-analytic variety) is a locally $G$-ringed space $(X,\mathscr{O}_{X})$ over $k$ such that the following axioms are verified:(i) The Grothendieck topology of $X$ satisfies properties $G_{0}$, $G_{1}$, and $G_{2}$ described in Proposition 1.22.(ii) There exists an admissible covering $(X_{i})_{i\in I}$ of $X$ with $(X_{i},\mathscr{O}_{X}|_{X_{i}})$ being a $k$-affinoid variety for each $i\in I$.§2. Almost mathematicsIn this section, we focus on Faltings' almost mathematics which first arose in his paper [Hodg], which is the first of a series works on the subject of $p$-adic Hodge theory, ending with [Falt]. The motivating point of $p$-adic Hodge theory can be traced back to Tate's classical paper [Tat1]. We will use Gabber's book [Gab] as a basic reference. The content will be useful in understanding Section 4 in Scholze's paper [Sch].References [BGR] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis. A systematic approach to rigid analyticgeometry, Grundlehren der Mathematischen Wissenschaften, Bd. 261, Springer, Berlin-Heidelberg-New York, 1984. [BS] Siegfried Bosch, Lectures on Formal and Rigid Geometry, Lect.Notes Mathematics vol. 2105, Springer, Cham, 2014. [Poon] Bjorn Poonen, Rational Points on Varieties, Graduate Studies in Mathematics Volume: 186, American Mathematical Society, 2017. [SGA4] M. Artin, A. Grothendieck, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Lecture Notes in Math. 269, 270, 305, Berlin-Heidelberg-New York, Springer. 1972-1973. [Gab] O. Gabber and L. Ramero, Almost ring theory, volume 1800 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003. [Hodg] G.Faltings, p-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), 255-299. [Falt] G.Faltings, Almost étale extensions, Astérisque 279 (2002), 185-270. [Tat] J. Tate, Rigid analytic spaces, Invent. Math. 12 (1971), 257-289. [Tat1] J. Tate, p-divisible groups, Proc. conf. local fields (1967), 158-183. [Dug] James Dugundji, Topology, Allyn and Bacon, Inc., 470 Atlantic Avenue, Boston, 1966. [Tam] Tammo Tom Dieck, Algebraic Topology, European Mathematical Society, 2008. [Ber] V.G. Berkovich, Spectral Theory and analytic Geometry over NonArchimedean fields, Math. Surv. Monogr. vol. 33, Am. Math. Soc., Providence, RI, 1990. [Ber1] V.G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Etud. Sci. 78 (1993). [SP] The Stacks Project Authors, Stacks Project. Available at http://math.columbia.edu/algebraic_geometry/stacks-git/. [Sch] Peter Scholze, Perfectoid Spaces, IHES Publ. math. 116 (2012), 245-313. [Hu] R. Huber, Continuous valuations, Math. Z. 212 (1993), 455-477. [Hu1] R. Huber, Étale cohomology of rigid analytic varieties and adic spaces, Aspects of Mathematics, E30., Friedr. Vieweg & Sohn, Braunschweig, Springer Fachmedien Wiesbaden, 1996. [Hu2] R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), 513-551.

Do you also want to highlight the code block in your website like other platforms? There are two Javascript libraries that help you acheive syntax highlighting. They are Prism.js and highlight.js. In this tutorial, we will show how to use Prism.js to do syntax highlighting with Nuxt.js.First, you can use npm to install PrismJS:npm install prismjsNext, go to composables directory and create a new file prism.js which looks like:import Prism from 'prismjs' import 'prismjs/themes/prism-tomorrow.css' // You can choose other themes export default PrismNow, prismJS is successfully configured and you just need to call the function highlightAll() every time you need. For example:<script lang="ts" setup> onMounted(()=>{ Prism.highlightAll() }) $fetch('https://www.example.com') .then((res)=>{ nextTick(()=>{ Prism.highlightAll() }) }) </script>Moreover, note that with the above configuration, prismJS can only highlight Javascript and a few other languages. Therefore, if you want to highlight more languages, you need to import their components into prism.js one by one. For example, if you want to highlight Typescript:import Prism from 'prismjs' import 'prismjs/themes/prism-tomorrow.css' // You can choose other themes import "prismjs/components/prism-typescript" export default PrismIf you have a lot of languages that are needed to highlight, then it would be annoying to import all their components to prism.js. To avoid this problem, you can use the Autoloader plugin. However, it is difficult to use prismJS plugins with Nuxt.js. The easiest way to load plugins is to use CDN, which will be explained in our next article.

In How to render math in Vue or Nuxt?, we explain the way to use Mathjax to render beautiful math formula in Nuxt.js. However, using CDN to load Mathjax is easy, but it degrades the performance of our pages. In other words, Mathjax may load slow if it is loaded by CDN.One of the solutions is to load Mathjax locally. You can use npm to install Mathjax to your project. npm install mathjax@4.0.0-beta.6However, it is difficult to integrate with Mathjax in Nuxt. Not only because there is little information about this, but following the Mathjax documentation may not work at all for Nuxt.js. Therefore, there is only one way that is easy enough: load Mathjax's JS file locally. In other words, you continue to use CDN to load Mathjax, but load local Mathjax's JS file. Just follow the steps:Download Mathjax's source files from MathJax v4.0.0-beta.6.Put all the Mathjax files in the directory public/mathjax. Note that if you are using VSCode to write your Nuxt project, don't forget to see VSCode error: To enable project-wide JavaScript/TypeScript language features, exclude large folders... to avoid an important error.In your nuxt.config.ts head: { link: [{ rel: 'icon', type: 'image/x-icon', href: '/favicon.png' }], script: [ { type: "text/javascript", id: "MathJax-script", async: true, src: "/mathjax/tex-chtml.js", tagPosition: "bodyClose", }, { innerHTML: "MathJax = {tex: {inlineMath: [['$', '$'],['$$', '$$']]}};", }, ] }

1. Introduction代数几何是数学的核心领域，也是如今国际数学界的主流。代数几何与许多数学分支都存在广泛的联系，比如数论、微分几何、代数拓扑、复几何、表示论、同调代数、交换代数、偏微分方程等等，这些分支的发展同时也对代数几何起到促进作用。数学史上的许多重大的事件，比如，费马大定理、莫德尔猜想、韦伊猜想的证明都跟代数几何有关。同时，代数几何存在广泛的应用，比如密码学、弦理论、大数据、统计学习理论等等。代数几何之下有众多分支，比如复代数几何，热带几何，算术几何，远阿贝尔几何，$p$进霍奇理论（complex algebraic geometry, tropical geometry, arithmetic geometry, anabelian geometry, p-adic hodge theory），每个分支代表代数几何研究的一个大方向，而在每个大方向下，又有各种以不同的问题为导向的子方向。在这篇文章中，我们将会对代数几何，包括它的分支算术代数几何，做一个简短的介绍。2. An Introduction to Arithmetic Geometry算术几何是算术代数几何的简称，它是代数几何的一个分支，主要研究与数论有关的问题，比如丢番图方程。著名的费马大定理其实就是丢番图方程的一种。Definition 2.1. Diophantine equations are equations whose solutions are required to be integers.Example 2.2. The equations in Fermat's Last Theorem : $x^{n} + y^{n} = z^{n}$ for all integers $n\geq 2$ are Diophantine equations.Example 2.3. The equations $ax + by = c$ are called linear Diophantine equations.Example 2.4. The equations $x^{2} + y^{2} = z^{2}$ are called Pythagorean equations.从上可以看出椭圆曲线与丢番图方程之间存在某种联系，因此数论上的问题就可以转移到几何上的椭圆曲线进行研究。接下来，我们将给出椭圆曲线的定义，但是在此之前我们先做一些约定。我们记$K$为一个任意的域，$f(x)\in K[x]$ 为$K$上的一个三次多项式，假设这个多项式有不同的根，由于这个域并不一定是代数闭域，因此有些不同的根存在于这个域的代数闭包 $\overline{K}$上。同时，我们假设域$K$不是特征2的。Definition 2.5. The solutions to the equation $y^{2} = f(x)$ , where $x$ and $y$ are in some extension $K'$ of $K$, are called the $K'$-points of the elliptic curve defined by the equation.Example 2.6. The locus of the equations $y^{2} = x^{3} - n^{2}x$ is a special case of elliptic curve.Figure 1. Elliptic curves从上面的定义和这个例子，我们可以看出椭圆曲线的方程形式上像一个丢番图方程。事实上，当我们限定椭圆曲线方程的解为整数解时，方程就成为了丢番图方程。既然说到了椭圆曲线，我们不得不提及一下跟椭圆曲线有关联的椭圆函数。椭圆函数是19 世纪数学最光辉的成就之一，它当初是由求椭圆弧长诱导出来的，与椭圆积分也有很密切的联系，毕竟椭圆积分就是用来求椭圆弧长的。顺带一提，椭圆周长目前没有办法求精确值，其周长表达式没法表达成初等函数的形式，它只有椭圆积分表达式以及级数展开式。在定义椭圆函数之前，我们需要先定义复数域$\mathbb{C}$上的lattice。Definition 2.7. A lattice $L$ in the complex plane is the set of all integral linear combinations of two given complex numbers $\omega_{1}$ and $\omega_{2}$, where $\omega_{1}$ and $\omega_{2}$ are linear independent.Example 2.8. If we take $\omega_{1}$ = 1 and $\omega_{2}$ = $i$, we will get a lattice of Gaussian integers $\{mi+n| m , n\in \mathbb{Z}\}$.Definition 2.9. A meromorphic function on $\mathbb{C}$ is said to be an elliptic function relative to a given lattice $L$, if $f(z+l)=f(z)$ for all $l\in L$.从定义可以看出，椭圆函数是一个双周期的函数。这使人联想到实数情况的单周期函数。一个$\mathbb{R}$上的周期函数，可以看成一个圆上的函数，而一个$L$的椭圆函数则可以看成一个圆环上的函数。我们可以证得关于一个lattice 的所有椭圆函数的集合构成一个域$\mathcal{E}_{L}$，它是所有亚纯函数的域的子域，因为任意两个椭圆函数的和差积商都是椭圆函数。接下来，我们继续讨论椭圆曲线。椭圆曲线与模形式有紧密的关联，而它们之间的联系成为了证明费马大定理的关键。由于作者并不能完全看懂费马大定理的证明，因此这里不做过多阐述。我们知道当年最后完成费马大定理证明的数学家是Wiles，而Wiles在他的paper 中证明了所有有理数集上的半稳定的椭圆曲线都是modular的，从而使费马大定理成为一个推论被证明。值得一提的是，Wiles在十岁的时候在一本叫做《最后定理》的书中了解到了费马大定理，他很受震撼并打算成为第一个解决费马大定理的人，最后正如他自己所说，很多数学家用自己的一生尝试解决费马大定理都没有成功，最后只有他成功了。关于椭圆曲线、椭圆函数、模形式、费马大定理的证明，想了解更多的读者可以参考[1], [11]。讲完费马大定理，接下来我们来讲讲费马大定理背后的故事，即费马大定理之所以最后能够被Wiles证明，主要是归功于某些数学家的关键性工作。其中两位即是日本数学家Shimura 和Taniyama，他们提出的谷山—志村猜想成为了证明费马大定理的关键一步。还有一位数学大师，在讲他之前我们需要先做一些铺垫。上个世纪，算术几何中不仅仅只有费马大定理，还有韦伊猜想（有限域上的黎曼猜想）、莫德尔猜想。韦伊猜想被Deligne所证明，而莫德尔猜想被Faltings所证明。Deligne和Faltings都是如今数学界的泰斗级人物，不论是Wiles、Deligne还是Faltings ，他们的证明都离不开一个人的奠基性工作，他就是被很多人认为是20世纪最伟大的数学家Grothendieck。Grothendieck被称作代数几何的教皇，有一句很经典的描述他的话就是：“20世纪代数几何涌现了很多天才和菲尔兹奖，但是上帝只有Grothendieck一个。”Grothendieck的工作使代数几何这门古老的学科重新焕发出新的生命力，这也使代数几何进入如今的黄金时期。Grothendieck的哲学直接被数学所吸收，以至于现在数学的新人根本无法想象Grothendieck时代前这个领域的模样。从二十世纪中叶开始，整个代数几何领域越来越抽象和普遍的研究倾向，大部分都得归功于Grothendieck的影响。Grothendieck 的影响之大，几乎所有数学分支都能感受到。如今的代数几何已经是后Grothendieck时代了，代数几何涌现出了很多后起之秀，比如说日本数学家Shinichi Mochizuki、德国数学家Peter Scholze。接下来，我们继续介绍算术几何的有关内容。上文中我们提到了可以通过研究椭圆曲线和模形式，进而研究数论问题。而椭圆曲线其实只是代数曲线中的一种特殊情况，代数曲线是算术几何的一个重要研究课题。别看名字很高大上，它其实很常见，比如说在欧几里得平面上的代数曲线，就是我们用多项式方程$f(x,y) = 0$所定义的平面曲线。而想要定义一般的代数曲线就不那么简单了，这需要用到Grothendieck发展的概形的理论。在定义一般的曲线之前，我们需要不少的预备知识，因此在这里我们只做简单的描述，想要了解更多细节的读者可以参考[2]。首先，在定义概形之前，我们需要定义层的概念。我们有阿贝尔群层、环层、模层等等，取决于层所取的范畴。关于范畴论的概念不熟悉的读者可以参考[7]。Definition 2.10 ([2], [16]). Let $X$ be a topological space. A presheaf $\mathcal{F}$ of abelian group on $X$ is a contravariant functor $$ \mathcal{F}:\textbf{Top}^{\textrm{opp}}\rightarrow \textbf{Ab}$$ from the category of open sets of $X$ to the category of abelian groups.If $\mathcal{F}$ is a presheaf on $X$, the set $\mathcal{F}(U)$ consists of the sections of $\mathcal{F}$ over the open set $U$. If $s\in \mathcal{F}(U)$, we write $s|_{V}$ for an element of $\mathcal{F}(V)$ corresponding to $s$.Definition 2.11. A presheaf $\mathcal{F}$ on a topological space $X$ is a sheaf, if it satisfies the following conditions:(Uniqueness) if $U$ is an open set of $X$, and $\{V_{i}\}$ is an open covering of $U$, then for an element $s\in \mathcal{F}(U)$ such that $s|_{V_{i}}$ = 0 for all $i$, we have $s = 0$.if $U$ is an open set of $X$, and $\{V_{i}\}$ is an open covering of $U$. If we have elements $s_{i}\in \mathcal{F}(V_{i})$ for each $i$, such that for each $i, j$, $s_{i}|_{V_{i} \cap V_{j}} = s_{j}|_{V_{i}\cap V_{j}}$, then there is an element $s \in \mathcal{F}(U)$ such that $s|_{V_{i}} = s_{i}$ for each $i$.Definition 2.12. Let $\mathcal{F}$ be a presheaf on $X$, if $P$ is a point of $X$, we define the stalk $\mathcal{F}_{P}$ of $\mathcal{F}$ at $P$ to be direct limit of the groups $\mathcal{F}(U)$ $$\lim\limits_{\longrightarrow}\mathcal{F}(U)$$ for all open sets $U$ containing $P$.一个预层上某个点的茎$\mathcal{F}_{P}$，其实就是一个等价类的集合，我们可以记茎中任意一个元素为$\langle U,s\rangle$，并称它为$\mathcal{F}$截面的芽。其中$U$为$P$ 点的开邻域，$s\in\mathcal{F}(U)$。接下来，我们记$A$为一个环，$Spec(A)$为该环所有素理想的集合，称为谱。如果$\alpha$是环$A$的任意一个理想，我们记$V(\alpha)\subseteq Spec(A)$为所有包含理想$\alpha$ 的素理想的集合。我们令$V(\alpha)$为$Spec(A)$中的闭集，从而在$Spec(A)$上定义了一个Zariski拓扑。接着，我们再定义拓扑空间$Spec(A)$上的环层$\mathcal{O}$。 这样下来，$(Spec(A),\mathcal{O})$成为一个局部赋环空间。接下来我们给出赋环空间的定义。回顾一下，一个环$A$被称为局部环，如果它只有唯一一个极大理想$\mathfrak{m}_{A}$。Definition 2.13. A ringed space is a pair $(X,\mathcal{O}_{X})$, where $X$ is a topological space and $\mathcal{O}_{X}$ is a sheaf of rings on $X$ called the structure sheaf. A ringed space is a locally ringed space, if for each $P\in X$, the stalk $\mathcal{O}_{X,P}$ is a local ring.有了上面这些储备，我们终于可以定义概形。首先我们定义仿射概形，之后就是一般的概形。Definition 2.14. An affine scheme is a locally ringed space $(X,\mathcal{O}_{X})$, which is isomorphic to a spectrum $\textrm{Spec }A$ of some ring $A$. A scheme is a locally ringed space $(X,\mathcal{O}_{X})$ in which every point $p$ of $X$ has an open neighborhood $U$ such that $(U,\mathcal{O}_{X}|_{U})$ is an affine scheme.从以上的定义，我们可以看出概形跟流形有异曲同工之妙。对于一个流形来说，它局部上都是一个欧几里得空间。而对于一个概形来说，它局部上都是一个仿射概形，同时因为同构关系，概形局部上的仿射概形可以看成某个环的谱。这样下来，流形由一个个欧几里得空间拼起来，而概形由一个个环的谱拼起来。而事实上，微分几何里的流形是可以用局部赋环空间表示的（更多细节请参考[10], [15]）。现在我们有了概形，就可以定义一般意义上的代数曲线了。在此之前，我们先定义概形的一些基本性质。Definition 2.15. Let $X$ be a scheme. We say that $X$ is integral if for each open affine set $U\subset X$, $\mathcal{O}_{X}(U)$ is an integral domain.Definition 2.16. Let $f:X\rightarrow Y$ be a morphism of schemes. The diagonal morphism of $X$ is a morphism $\triangle:X\rightarrow X\times_{Y}X$ such that $\textrm{pr}_{1}\circ\triangle=\textrm{pr}_{2}\circ\triangle=\textrm{id}_{X}$. We say that $f$ is separated or that $X$ is separated over $Y$ if the diagonal morphism of $X$ is a closed immersion.Definition 2.17. Let $f:X\rightarrow Y$ be a morphism of schemes. We say that $f$ is proper or that $X$ is proper over $Y$ if $f$ is separated, of finite type, and universally closed.Definition 2.18. Let $X$ be a scheme. The dimension of $X$ is the dimension of its underlying topological space $\textrm{sp}(X)$, which we will denote by $\textrm{dim }X$.Definition 2.19. An algebraic curve is an integral scheme of dimension 1, proper over a field $K$, all of whose local rings are regular.因此，一个代数曲线其实就是一个一维的概形。流形也如此，一维的流形也叫做曲线。以上我们完成了对代数曲线的定义，通过代数曲线我们可以研究数论问题。但是，研究代数曲线是需要工具的。在这些工具中，就有algebraic stack和moduli theory。Algebraic stack是stack的特殊情况，stack是对概形的进一步推广。而stack可以看成某种群胚纤维化范畴(category fibred in groupoid)，可以运用Descent à la Grothendieck来定义。而moduli theory就是研究某一类数学对象的参数空间，比如曲线的模空间、椭圆曲线的模空间。由于目前这些理论不是作者的研究方向，作者不作过多阐述。2.1 The $p$-adic numbers field $\mathbb{Q}_{p}$ and the $p$-adic integers ring $\mathbb{Z}_{p}$接下来，我们来简单说明一下$p$进数域$\mathbb{Q}_{p}$是如何构造出来的。首先，我们以有理数域$\mathbb{Q}$为例，粗略解释一下完备化（completion）的过程：我们取有理数域所有柯西序列构成的集合，定义逐项加法和乘法后可以证明它构成一个交换环，接着模掉所有零序列构成的理想，我们就得到一个完备的域，它是有理数域的域扩张。一个域的完备化不是唯一的，对应不同定义于域上的绝对值，我们可以定义不同的柯西序列，进而构造出不同的完备化。在这里，我们给出任意域上的绝对值与完备域的定义。Definition 2.20. Let $K$ be a field. An absolute value on $K$ is a map $\left|\cdot\right|:K\rightarrow\mathbb{R}_{\geq0}$ such that $\left|x\right|=0\Leftrightarrow x=0$, $\left|xy\right|=\left|x\right|\left|y\right|$, and $\left|x+y\right|\leq\left|x\right|+\left|y\right|$. We say that $K$ is complete if it is complete with respect to the distance $d(x,y)=\left|x-y\right|$ induced by the absolute value $\left|\cdot\right|$ on it.接下来我们先定义有理数域上的$p$进序数。Definition 2.21. Let $p$ be any prime number. We define the $p$-adic ordinal ord$_{p}a$ of an non-zero integer $a$ to be the highest power of $p$ which divides $a$, i.e. the greatest $m$ such that $p^{m}|a$ or $a\equiv0(\textrm{mod }p^{m})$.我们约定当整数$a=0$时，ord$_{p}a=\infty$。接着对于任意$x=a/b\in\mathbb{Q}$，我们定义$\textrm{ord}_{p}x=\textrm{ord}_{p}a-\textrm{ord}_{p}b$。如果将ord看成一个函数，那么它是良定义的，因为如果将$x$写成$x=ac/bc$，我们有$\textrm{ord}_{p}x=\textrm{ord}_{p}ac-\textrm{ord}_{p}bc=\textrm{ord}_{p}a-\textrm{ord}_{p}b$。接着我们定义$p$进绝对值：$$\left| x \right|_{p} = \begin{cases} \frac{1}{p^{\textrm{ord}_{p}x}}, & \textrm{if} \ x\neq 0\\ 0,  & \textrm{if} \ x = 0. \end{cases}$$我们先阐述复数域$\mathbb{C}$的构造过程，首先我们作有理数域$\mathbb{Q}$的完备化（关于通常的绝对值$\left|\cdot\right|$）$\widehat{\mathbb{Q}}$得到实数域$\mathbb{R}$，然后取实数域的代数闭包$\overline{\mathbb{R}}$ 得到复数域。$p$进数域$\mathbb{Q}_{p}$其实就是有理数域$\mathbb{Q}$的$p$进完备化（关于$p$进绝对值 $\left|\cdot\right|_{p}$）$\widehat{\mathbb{Q}}$。然而当我们取$p$进数域的代数闭包$\overline{\mathbb{Q}}_{p}$时，发现它不是完备的，因此我们对其再作一次完备化，最后得到$\mathbb{C}_{p}$。它是最小的包含有理数域的既是代数闭的，又是完备的域。于是，我们有如下关系：$$\begin{cases} \mathbb{C}_{p}=\widehat{\overline{\mathbb{Q}}}_{p}=\widehat{\overline{\widehat{\mathbb{Q}}}}, \textrm{p-adic analog} \\ \mathbb{C}=\overline{\mathbb{R}}=\overline{\widehat{\mathbb{Q}}}, \textrm{usual case} \end{cases}$$接着$p$进整数环$\mathbb{Z}_{p}$即是$p$进数域$\mathbb{Q}_{p}$的离散赋值环：$$\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}\mid \left|x\right|_{p}\leq1\}.$$3. Grothendieck's Theory接下来，我们来回顾一下上世纪Grothendieck所做的工作。其实代数几何如今整体上能分成两个方向，一个是以Grothendieck发展的抽象理论为基础的方向，另一个是与微分几何结合主要研究复几何的方向（参考[14]）。Grothendieck所做的工作当然远远不止上文所说的概形，还有étale cohomology（平展上同调）, crystalline cohomology（晶体上同调）, $l$-adic cohomology（$l$进上同调）, topos（拓扑范）, motives, Grothendieck topology, Grothendieck universe等等。除此之外，Grothendieck 还有三本被誉为代数几何圣经的著作，分别是EGA(Éléments de géométrie algébrique),SGA(Séminaire de géométrie algébrique)和FGA(Fondements de la Géometrie Algébrique)，翻译成中文就是《代数几何原理》、《代数几何讨论班》和《代数几何基础》。首先我们来说说Grothendieck著名的motives理论，该理论的哲学即是将所有的性质相似的上同调，诸如奇异上同调、德拉姆上同调、平展上同调和晶体上同调，统一起来。下面我们给出上同调的定义，该定义涉及到阿贝尔范畴。所谓的阿贝尔范畴，它的原型是阿贝尔群范畴，上世纪Grothendieck将其重要的性质抽象出来，只剩下足够计算同调代数的东西。Definition 3.1. A cochain complex $\mathcal{C}= \{\mathcal{C}^{n},d^{n}\}$ in an abelian category $\mathfrak{U}$ is a collection of objects $C^{i},i\in \mathbb{Z}$ , and morphisms $d^{i} : C^{i} \rightarrow C^{i+1}$, such that $d^{i}\circ d^{i+1} = 0$. The morphisms $d=\{d^{i}\}$ are called the differential (or coboundary operator).The $i$th cohomology object of the complex $\mathcal{C}$ is defined to be $H^{i}(\mathcal{C}) = \textrm{Ker }d^{i}/\textrm{Im }d^{i-1}$.根据范畴的不同，我们可以定义上同调群、上同调模，接着就可以定义singular cohomology（奇异上同调）、de Rham cohomology（德拉姆上同调）、Galois cohomology（伽罗华上同调）、Čech cohomology （切赫上同调）等等。在集合论中，我们有类与集合的概念。所谓的类由所有享有共同性质的数学对象构成，但是它不一定是一个集合，如果它不是一个集合，我们称这个类是真类。接下来，我们给出Grothendieck universe 的定义，它是在上世纪由Grothendieck提出来的，用来避免不构成集合的真类。如果读者想要了解更多相关内容，可以参考[5], [6]。Definition 3.2. A Grothendieck universe is a non-empty set $\mathcal{U}$ that satisfied the following conditions:if $x\in \mathcal{U}$ and $y\in x$, then $y\in \mathcal{U}$.if $x,y\in \mathcal{U}$, then $\{ x,y\}\in \mathcal{U}$.if $x \in \mathcal{U}$, then $\mathcal{P}(x) \in \mathcal{U}$, where $\mathcal{P}(x)$ denotes the set of all subsets of $x$.if $(x_{i},i\in I)$ is a family of elements of $\mathcal{U}$ and $I \in \mathcal{U}$, then $\bigcup_{i\in I}x_{i} \in \mathcal{U}$.4. Modern Mathematics以上内容其实都已经是以前发展的理论了，基本上都是20世纪的内容，已经有点旧了。接下来，我们讲一下21世纪比较新的内容：Shinichi Mochizuki和Peter Scholze的工作。Shinichi Mochizuki(望月新一)就是那位声称证明了abc猜想的数学家，我们习惯叫他为望月大神。他刚开始主要是做hyperbolic curve相关的研究的，后来他开始通过运用自己以前的研究成果来研究远阿贝尔几何（anabelian geometry）。远阿贝尔几何最初是Grothendieck提出来的一个宏伟的理论，如今它被望月新一进一步发展，构建了一个名叫宇宙际理论（Inter-universal Teichmüller Theory）的东西，用于证明abc猜想，可惜世界上没有多少数学家能够看得懂他的证明，因此关于他的证明主流数学界仍不认可。不同的是，Peter Scholze的工作则更为主流数学界所接受，很多人都更愿意做Peter Scholze的方向。Peter Scholze就是那个国际奥林匹克数学竞赛拿金牌，高中开始学习研究生数学的数学家，很年轻。在他的博士论文中，他发展出了一个叫状似完备空间（perfectoid spaces）的新东西，成为了当代算术几何最具影响力的数学家之一。4.1. Rigid GeometryPeter Scholze 所做的perfectoid spaces与刚性几何（Rigid Geometry）有关，接下来我们将对刚性几何的部分内容做介绍。想要了解更多的读者请参考[3], [4]。首先我们需要研究非阿基米德的绝对值。对于与绝对值相关的valuation，在本文中我们将不予讨论。我们着重讨论非阿基米德的绝对值的特别之处。Definition 4.1. A (non-archimedean) absolute value $\upsilon$ on a field $K$ is a map $\left| \cdot \right|$ : K $\rightarrow$ $\mathbb{R}_{\geq0}$, such that for all $x,y\in K$ the following conditions verified:$\left| x \right|$ = 0 $\Leftrightarrow$ $x=0$.$\left| xy \right|$ = $\left| x \right|$$\left| y \right|$$\left| x+y \right| \leq \max\{\left| x \right|, \left| y \right|\}$Proposition 4.2. Let $x,y\in K$, we have $\left| x+y \right|$ = $\max\{\left| x \right|, \left| y \right|\}$, if $\left| x \right| \neq \left| y \right|$.Proof. Without loss of generality, we assume $\left| x \right| < \left| y \right|$. Then $\left| x+y \right|$ $<$ $\max\{\left| x \right|, \left| y \right|\}$ =$ \left| y \right|$ implies$$\ \left| y \right| = \left| (y+x)-x \right| \leq \max\{\left| x+y \right|, \left| x \right|\} < \left| y \right|$$which is contradictory. So we must have $\left| y \right| = \left| y+x \right| = \max\{\left| x \right|, \left| y \right|\}$ as claimed.通过绝对值，我们定义任意域$K$上的距离为$d(x,y) = \left| x-y \right|$，然后该距离诱导出$K$上的一个拓扑。有了$K$中任意两点的距离，根据非阿基米德的三角不等式，对于所有$x,y,z \in K$，我们可以得出：$$d(y,z) \leq \max\{d(x,y),d(x,z)\}$$根据命题4.2，该不等式两边相等，如果不等式右边的两个距离不相等。这意味着：在域$K$中的任意三角形，都是等腰三角形。更进一步，我们可以证出：域$K$中任意一个圆盘中的点都可以作为该圆盘的中心。因此，如果$K$中的两个圆盘有非空交集，那么它们就是共心的。下面我们给出证明。Definition 4.3. For a centre $a\in K$ and a radius $r\in \mathbb{R}_{> 0}$, we define the disk without boundary to be the set $$D^{-}(a,r) = \{ x \in K\mid d(x,a)<r \}$$ And we define the disk with boundary to be the set $$D^{+}(a,r) = \{ x \in K\mid d(x,a)\leq r\}$$Proposition 4.4. Each point of disk without boundary in K is the centre of the disk.Proof. Assume that $a$ is the centre of a disk, $b$ is a point different from $a$. For any $x\in D^{-}(a,r)$, we have $$ d(x,b) = \left| x-b \right| = \left| (x-a)+(a-b) \right| \leq \max\{\left| x-a \right|,\left| a-b \right|\} < r $$类似的，我们可以证明对于有边界的圆盘，其中的任意一点都可以是它的中心。4.2 Perfectoid Geometry接下来我们粗略地说一下，Perfectoid spaces, [4]，这篇文章里面的一些内容，鉴于作者水平有限，不能一一详述。首先，perfectoid是perfect+oid，意思就是more or less perfect，类完美。首先，我们回顾一下什么是完美域（perfect fields）。Definition 4.5. Let $K$ be a field. We say that $K$ is perfect if either $K$ has characteristic $0$, or if $K$ has characteristic $p>0$, the Frobenius $$ \Phi:K\rightarrow K, x\mapsto x^{p}$$ is an isomorphism.Perfectoid spaces这篇文章的动机源于以下Fontaine-Wintenberger的一个定理：Theorem 4.6. The absolute Galois groups of $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and $\mathbb{F}_{p}((t))$ are canonically isomorphic.Remark 4.7. $$\mathbb{Q}_{p}(p^{1/p^{\infty}})=\lim_{\substack{\longrightarrow \\ n>0}}\mathbb{Q}_{p}(p^{1/p^{n}})=\bigcup_{n>0}\mathbb{Q}_{p}(p^{1/p^{n}}).$$$\mathbb{Q}_{p}(p^{1/p^{\infty}})$是一个特征0的域，它的剩余类域$\mathbb{F}_{p}$是特征$p$，这种域被称为混合特征的（mixed characteristic）。而$\mathbb{F}_{p}((t))$ 是一个特征$p$的域。意思是如果将所有$X^{p^{n}}-p\in\mathbb{Q}_{p}[X]$的根加到$\mathbb{Q}_{p}$里面，它会看起来像一个特征$p$的域$\mathbb{F}_{p}((t))$。想要更好地理解$\mathbb{Q}_{p}(p^{1/p^{n}})$是什么意思，可以参考$\mathbb{C}\cong\mathbb{R}(i)\cong\mathbb{R}[X]/(X^{2}+1)$这个例子。同时，我们有这样一个tower:$$\mathbb{Q}_{p}\subseteq \mathbb{Q}_{p}(p^{1/p})\subseteq \mathbb{Q}_{p}(p^{1/p^{2}})\subseteq \cdot\cdot\cdot \subseteq \mathbb{Q}_{p}(p^{1/p^{n}})\subseteq \cdot\cdot\cdot \subseteq \mathbb{Q}_{p}(p^{1/p^{\infty}}).$$定理4.6可以在更加一般的框架下研究，这就引申出了perfectoid fields。 首先，我们给出非阿基米德域的定义，它其实就是一个拓扑由一个非阿基米德绝对值生成的拓扑域。Definition 4.8. A non-archimedean field is a topological field $K$ whose topology is induced by a non-trivial valuation of rank 1.Definition 4.9. A perfectoid field is a complete non-archimedean field $K$ with residue characteristic $p>0$ whose associated rank-1-valuation is non-discrete and the Frobenius $\Phi:K^{\circ}/p\rightarrow K^{\circ}/p,x\mapsto x^{p}$ is surjective.Example 4.10. The $p$-adic completion $\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}$ of $\mathbb{Q}_{p}(p^{1/p^{\infty}})$ and the $t$-adic completion $\widehat{\mathbb{F}_{p}((t))(t^{1/p^{\infty}})}:=\mathbb{F}_{p}((t))((t^{1/p^{\infty}}))$ of $\mathbb{F}_{p}((t))(t^{1/p^{\infty}})$ are perfectoid fields.$$\widehat{\mathbb{Q}_{p}(p^{1/p^{\infty}})}=\widehat{\mathbb{Z}_{p}[p^{1/p^{\infty}}]}[\frac{1}{p}]=(\lim_{\longleftarrow} \mathbb{Z}_{p}[p^{1/p^{\infty}}]/p^{n})[\frac{1}{p}],$$$$\widehat{\mathbb{F}_{p}((t))(t^{1/p^{\infty}})}=\widehat{\mathbb{F}_{p}[t^{1/p^{\infty}}]}[\frac{1}{t}]=(\lim_{\longleftarrow} \mathbb{F}_{p}[t^{1/p^{\infty}}]/t^{n})[\frac{1}{t}].$$Perfectoid field叫做类完美域，当它为特征$p$时，它是一个完美域。同时，这里有一个tilt的过程，它可以看成一个函子叫做tilt funtor:$$K\mapsto K^{\flat}$$将一个任意特征的perfectoid field打到一个特征$p$的perfectoid field。同时，我们有$$K^{\flat}=\lim_{\substack{\longleftarrow \\ x\mapsto x^{p}}}K.$$接着我们有了更加一般的定理，它推广了定理4.6。Theorem 4.11. The absolute Galois groups of $K$ and $K^{\flat}$ are canonically isomorphic.总之，这篇文章中，Peter Scholze提出一种框架，它能将任意特征的问题简化为特征$p$的问题，因为特征$p$往往更好研究，同时也有很多好的性质和结论。References Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd ed., Springer-Verlag New York, Inc., 1993. Robin Hartshorne, Algebraic Geometry, Springer, New York, NY, Springer Science+Business Media New York, 1977. Siegfried Bosch, Lectures on formal and rigid geometry, volume 2105 of Lecture Notes in Mathematics. Springer, Cham, 2014. Peter Scholze, Perfectoid Spaces, IHES Publ. math. 116 (2012), pp. 245–313. Grothendieck with Artin, M. and Verdier, J. L. Théorie des Topos et Cohomologie Étale des Schémas. Séminaire de Géométrie Algébrique du Bois-Marie 1963-1964 (SGA 4), Springer-Verlag Berlin Heidelberg, 1973. Pierre Deligne, Cohomologie Étale, Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 1/2, Springer-Verlag Berlin Heidelberg, 1977. Peter J. Hilton and Urs Stammbach, A Course in Homological Algebra, Springer-Verlag New York, 1997. Fredrik Meyer, Notes on algebraic stacks, https://blog.fredrikmeyer.net/uio-math, 2013. G. Everest and Thomas Ward, An Introduction to Number Theory, Springer-Verlag London, 2005. Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, New York, NY, 2011. Andrew John Wiles, Modular elliptic curves and Fermat's Last Theorem, Annals of Mathematics, 141 (1995), 443-552. Michael Artin, Allyn Jackson, David Mumford, and John Tate, Coordinating Editors, Alexandre Grothendieck, Notices of the AMS 51, 2016. Joe Harris and Ian Morrison, Moduli of Curves, Springer-Verlag New York, Springer Science+Business Media New York, 1998. Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, Wiley-Interscience; 1st edition (August 16, 1994), 1978. J.S. Milne, Algebraic Geometry (v6.02), www.jmilne.org/math/ , 2017. Glen E. Bredon, Sheaf Theory, Springer-Verlag New York, Springer Science+Business Media New York, 1997.

在京北市的八月，阳光透过树叶的缝隙，洒下斑驳的光影，为这座城市增添了几分宁静与祥和。然而，对于陈木而言，这个夏天注定不平凡。陈木，一个普通的图书管理员，每天的工作就是整理书架、帮助读者查找资料、登记借阅信息。他的生活简单而规律，除了工作，他几乎不与人交往，也没有什么特别的爱好。然而，在这个平凡的外表下，陈木却有着一个不为人知的秘密——他对数学有着近乎狂热的热爱。陈木的数学之路始于他的童年。他的父亲是一位数学老师，家中的书架上摆满了各种数学书籍。在父亲的熏陶下，陈木从小就对数字和公式产生了浓厚的兴趣。他喜欢沉浸在数学的世界里，那里充满了逻辑和秩序，每一个问题都有答案，每一个答案都有逻辑。随着年龄的增长，陈木对数学的热爱并没有减退，反而越来越强烈。然而，陈木并没有选择成为一名数学家。他的父亲在他高中毕业那年因病去世，留下了一大笔债务。为了维持生计，陈木放弃了上大学的机会，选择了一份稳定的工作。尽管如此，他并没有放弃对数学的追求。每天下班后，他都会在自己的小屋里，点上一盏台灯，沉浸在数学的世界中。陈木的小屋并不大，只有一间卧室和一个小厨房。卧室里除了一张床和一个衣柜，最显眼的就是那张堆满了数学书籍和草稿纸的书桌。墙上贴满了各种数学公式和图表，这些都是陈木在研究过程中的灵感和成果。他的数学研究并没有特定的方向，他涉猎广泛，从几何到代数，从数论到拓扑，他都有所涉猎。一、寄出希望8月22日，陈木坐在他那堆满书籍和草稿纸的小屋里，手中紧握着一封精心准备的信件。这封信，是他多年心血的结晶，里面详细介绍了他新发明的数学工具以及基于此工具创立的全新数学体系。更重要的是，信中隐晦地透露了他已经解决了数学界公认的难题——哥德巴赫猜想。这一成就，如果得到证实，将无疑让他在数学界声名大噪。陈木没有选择快捷的电子邮件或快递服务，而是特意选择了北京本市的平邮。在他看来，这种方式更加庄重，也更能体现他对数学的尊重与敬畏。他满怀期待地将信件投进了邮筒，心中默默祈祷着它能尽快到达菲尔兹奖得主、世界知名数学家邱教授的手中。时间一天天过去，陈木的心情也由最初的期待转为焦虑，再到后来的失落与不解。他无数次设想信件可能遭遇的各种情况：是邮局出了差错？还是邱教授太忙，暂时无暇顾及？甚至，他心中也闪过一丝疑虑：会不会是邱教授根本就看不懂我的理论？这种自我怀疑如同毒蛇般缠绕着他，让他夜不能寐，食不知味。实际上，那封信根本没能送到邱教授手上，邱教授日理万机，平日里不知道多少封信会寄到邱教授这里。邱教授的助手，打开信封扫了一眼标题，就直接把这封信丢进了垃圾桶里。二、网络风波面对长时间的沉默，陈木终于按捺不住内心的冲动。他决定不再被动等待，而是主动出击。他将自己的研究成果整理成文稿，并配以详细的图解和说明，然后将其发布到了某乎上。他希望通过这种方式，引起更多人的关注和讨论，甚至有可能直接吸引到邱教授的注意。起初，确实有一些对数学感兴趣的“名词党”网友被他的新颖观点所吸引，纷纷留言表示支持和赞赏。但很快，质疑的声音也开始涌现。有专业人士指出他的理论存在逻辑上的漏洞和矛盾；数学专业的学者们则直接质疑他的数学基础是否扎实；更有甚者，直接揭露了他的“民科”身份，认为他不过是一个对数学一知半解却妄图颠覆传统的门外汉。面对这些质疑和嘲讽，陈木的心被深深刺痛。他无法接受自己视若珍宝的研究成果被如此轻易地否定，更无法忍受自己被贴上“民科”的标签。他开始变得偏执和固执己见，认为那些质疑他的人都是无知的庸才，无法理解自己天才般的创见。于是，他直接大胆回应别人的质疑：我不是针对哪一个人，我是说在座的各位都是垃圾。这句话马上引起轩然大波，网络上一边倒的对陈木进行口诛笔伐。而暴风雨中心的陈木，他沉浸在自我构建的幻想世界中，坚信自己的数学水平已经登峰造极无人能及，更还幻想着网友们最终被啪啪打脸，面红耳赤又无可奈何的景象。三、现实的打击随着网络舆论的发酵和扩散，陈木的名字和他的“研究成果”彻底出圈了，开始在大众互联网上引起广泛的关注和讨论。然而，这些讨论大多数都是负面的和质疑的。一些激进的网友甚至开始追踪他的网络足迹，试图找到他的真实身份并对其进行线下攻击。终于有一天晚上陈木的家门被猛烈地敲响。他打开门一看只见一群愤怒的网友站在门外他们的眼神中充满了敌意和不满。他们指责陈木是骗子和疯子要求他立即承认自己的错误并道歉。陈木试图解释和反驳但他们的声音却像潮水般将他淹没。在这场激烈的冲突中，陈木被众人暴打了一顿，这场冲突对身体的伤害不大，更多的是对精神的侮辱。他的身体多处受伤，还不算太严重，而心灵却遭受了重创。他躺在冰冷的地面上望着天花板上的灯光感到前所未有的绝望和无助。他意识到自己的世界已经崩塌了自己曾经追求的一切都已经化为泡影。四、古书的启示经历了这场灾难性的打击后陈木变得沉默寡言、萎靡不振。他不再相信任何人也不再追求任何梦想。每天他都像一具行尸走肉般生活着浑浑噩噩、度日如年。直到有一天他在自家院子的角落里无意间挖到了一本古旧的书籍。这本书封面泛黄、字迹模糊但字里行间却透露出一种超脱世俗的智慧与力量。陈木好奇地翻开书页开始阅读起来。随着阅读的深入他仿佛被一股神秘的力量所吸引，让他的心灵得到了前所未有的震撼与洗涤。书中记载的并非世俗的数学或科学理论，而是关于修仙悟道、超脱生死的至高境界。陈木被书中的内容深深吸引，他开始思考人生的意义和价值以及自己在这个宇宙中的位置和作用。他意识到自己所追求的不过是镜花水月般的虚幻之物，而真正的智慧与力量其实就蕴藏在宇宙万物之中等待着有心人去发现与领悟。五、修仙之路受到古书的启示和启发，陈木决定放弃世俗的一切牵绊与欲望全身心地投入到修仙之路中。他开始学习书中的修炼方法和技巧日复一日地打坐冥想、炼丹炼器、参悟天道……修行无岁月，时间一天天流逝，转眼间百年过去了，陈木突破到了元婴境，成为了传说中的元婴大能。从家里出去，陈木惊讶的发现自己家的时间流速跟外界不一样。百年过去，外面才紧紧过去一天！这堪称强大的时间法则莫非是来自这本古书？陈木站在半空，直接展开神识，他惊奇的发现自己的神识居然可以无限延伸，覆盖到到整个世界。他能洞察到世界上的每一个角落，每个细微之处发生的点点滴滴，都逃不过他的感知。他看到了远在华青大学的邱教授，年迈之年仍在努力专研最新的学问，同时也看到了无数默默无闻的数学学者，为了自己所热爱的数学，甘愿放弃世俗的欲望，只为能找到那个心目中的“她”。陈木感慨，过去的自己是多么的愚蠢和无知，以前的他根本不是做数学的。而如今，身为元婴境修士的他，也已知晓哥德巴赫猜想如何证明，但正所谓天机不可泄露，他不能将这些东西直接告诉人们，不然强行拔高人类的科学水平，会导致无法想象的因果效应。而如今这个小小的地球也已经容不下他，再过不了多久，地球上强力的法则之力就会将他强行排斥出地球。随着陈木修为的精进，他越发觉得古老的地球，埋藏了很多不为人所知的秘密。小小的地球仍然有许多他现如今境界都无法看透的地方，凭他元婴境的修为，随便一拳就能砸碎整个地球，但如今他却感受到来自地球一种不可抗拒的排斥之力。是时候告别地球了，想到这，陈木直接运转五行之力，飞入太空，站在太空俯瞰整个世界。总有一天，我陈木还会回来的，随后陈木直接打破了空间的束缚，离开了地球这个狭小的世界向着更加广阔无垠的宇宙深处进发。对于陈木而言修仙之路并未结束而只是刚刚开始，他深知在浩瀚的宇宙中还有无数的奥秘等待着他去探索和发现。陈木发誓，待他修成传说中的仙尊之境，他一定会再回地球一趟。PS：如果你对这个故事感兴趣，并想改进作者这粗糙的文笔，可以到https://semipieces.fun/zh-CN/articles/8 进行编辑，修改成你想要的版本。

In previous the article Mathjax loads slow. How to load local JS file with Nuxt?, we explain how to render math equations in Nuxt.js using local CDN. With local CDN, we could easily load many Javascript libraries without slowing down our page loading.PrismJS is a light Javascript library that highlights the code blocks on your pages. In the ordinary way that setup PrismJS (see How to do syntax highlighting for code blocks in Nuxt?), you need to import the file of almost every language that you need to highlight. For example, to highlight Typescript, you need to add import "prismjs/components/prism-typescript".  This is absolutely troublesome. However, PrismJS has multiple plugins that extends its functionality. The Autoloader plugin can automatically load the language you need, so you don't have to import the files of all your needed languages one by one. The easiest way to load Autoloader plugin is to use CDN. In this article, we will use the same method, which we used to load Mathjax, to setup PrismJS together with its autoloader plugin in Nuxt 3. 1. First, download the source code of PrismJS from Github:https://github.com/PrismJS/prism/archive/refs/tags/v1.29.0.zip. Or you can download it with npm:npm install prismjs2. Next, move it into the public directory in your Nuxt project. The directory structure will look like the following:Nuxt-app/ node_modules pages ... public/ prismjs3. Finally, configure your nuxt.config.ts, which will look like the following: css: [ "prismjs/themes/prism-tomorrow.css" ], app: { head: { link: [{ rel: "icon", type: "image/png", href: "/favicon.svg" }], script: [ { src: "/prismjs/components/prism-core.min.js", tagPosition: "bodyClose", }, { src: "/prismjs/plugins/autoloader/prism-autoloader.min.js", tagPosition: "bodyClose", }, ] }Note that we must add tagPosition: "bodyClose", because the script tag is placed in body not head. Now, PrismJS will automatically load every language you need, and all you need to do is just calling highlightAll().

Introduction In the recent past, the healthcare industry has dramatically transformed following the invention and integration of more advanced technologies used in data generation with the aim of enhancing healthcare practices (Shrotriya et al., 2023). These technological advancements have effectively propelled the healthcare industry forward especially in generating and analyzing healthcare outcomes. This substantial achievement can be attributed to the continuing development of technological devices in medicine, electronic health records, and the use of wearable gadgets that have recently boosted its trend. Patel and Sharma (2014) noted that Big Data has become a potent instrument with the potential to revolutionize healthcare and stimulate innovation in several industries. Apache Spark has emerged as one of the critical solutions to these challenges by supporting services and service components equipped with data processing and analysis features. Salloum et al. (2016) defined Apache Spark as an open-source distributed computing system that primarily manages big data processing tasks. It works based on in-memory computations and efficient query processing to quickly process large volumes of information. This paper analyses all the facets of the AI Spark Big model for enhancing healthcare industries' services, primarily through analyzing Apache Spark. 1. HealthCare Digitalization through the Spark ModelElectronic medical records (EMRs) and electronic health records (EHRs) are the primary frameworks for building up the patients' necessary clinical and medical-related information as they work towards enhancing the quality of the health care service, promoting the efficiency of the service delivery, enhancing control and reduction of costs and more importantly, reducing medical mistakes (Han & Zhang, 2015). The patients' EMR genomics include the genotyping and gene expression details, payer-provider affiliation information collected through genomics, prescription medications, insurance data, and related IoT devices, which contribute to enhancing the healthcare industry (Bedeley, 2017). On a positive note, it is also significant to acknowledge advancements in developing and deploying well-being monitoring solutions. Such systems consist of software and hardware with predictive capabilities. These tools can recognize that a certain patient is at risk, setting off audible alarms in that context and notifying the appropriate caregivers. The tools produce tremendous information, providing the principal clinical or medical response. the following are some measures whereby Spark has enhanced service delivery in the healthcare sector;a. EHRS ManagementAccording to Ansari (2019), knowing the significance of EHRs in present healthcare facilities, physicians and nurses can enter, retrieve, view, or share patients' records and information with other clinicians through the EHR systems. There are several challenges in data processing and storing data within EHRs because the volume and the types of data are also huge, and the access to these data must be timely. This remains a significant challenge when managing EHR for large patient populations, populations, and requires an appropriate solution. Apache Spark has played fundamental roles aimed at addressing these challenges following its ability to efficiently manage data and analytics. For example, Apache Spark has played a significant role in enhancing healthcare organizations by strengthening the connection of several players and to improving exchange between them (Ansari, 2019). Notably, Apache Spark has addressed EHR through different real-life situations Apache Spark.b. Disease outbreak forecastingAwareness and early intervention of diseases translate into an appreciation of the magnitude and extent of protection required to safeguard the populace and the efficient use of limited resources in the health sector (Shrotriya et al., 2023). The accuracy of an expected disease outbreak in a given population can be enhanced by assembling information from other sources, including public health, social networks, and the environment. When used with the current machine learning techniques, Spark helps healthcare organizations refine the prediction of an epidemic depending on the forecast's time and accuracy. c. Genetics and Personalized Medicine In the concept of personalized medicine, the objectives set are the enhancement of the interventions' precision and efficacy as a result of the use of the patient's genotype. However, managing, reviewing, and using data in genomics research have relatively become cumbersome due to the large volumes of mess and complex data streams produced (Shrotriya et al., 2023). Apache Spark has a deeper achievement in a genomic study where extensive analysis and variance exploration have been done. Specific case studies provided in the literature reviewed have pointed out that Apache Spark has a highly impactful role in promoting personalized medicine and patient care.d. Analysis of Medical ImagingImaging diagnosis is one of the critical areas of diagnosis in healthcare and involves methods such as radiography, magnetic resonance imaging, and computed axial tomography scans. Help is needed to manage all this medical imaging data, and it is a relatively necessary and valuable analysis by healthcare workers and professionals. Incorporating Apache Spark in photo processing and deep learning frameworks may bring significant changes in medical image analysis that can significantly improve the picture recognition approach (Shrotriya et al., 2023). These are why these abilities contribute to faster decisions on treatment regimens and improved diagnostics. Some real-life examples include the following: The application of Apache Spark in medical image processing has improved patient treatment and increased medical productivity.e. Telemedicine and Remote Patient MonitoringRecently, telemedicine and remote patient monitoring have been discovered as trends, enabling healthcare staff to provide treatments and oversee the patient's physiological data from a distance. Issues within this domain include the issue of large volumes of data as probed by remote monitoring devices and the need to implement the findings in real time. Spark is also helpful in identifying potential threats to health, improving the quantity and quality of healthcare services, and providing immediate data processing and analysis necessary in telemedicine practice. Case studies revealed research proving that Apache Spark can enhance telemedicine services and RPM systems. This could improve the quality of patients' care and healthcare delivery systems.2. Apache Spark in Medical Imaging AnalysisThe application of Apache Spark in medical imaging analysis can be deemed a revolution in the healthcare industry. Apache Spark has been argued to be the most suitable distributed computation engine for processing big image datasets (Tang et al., 2020). Diagnostic imaging evaluation entails considering many imaging methods, including X-ray, CT scan, and MRI scan, CT, and MRI, to discover features that point toward certain disease conditions or health complications. The enhancement of specialized processors and analysis capabilities for managing big data has become even more important in the latest and advancing developments in medical imaging data (Shrotriya et al., 2023). Analyzing medical pictures is relatively trivial as long as the topic concerns Apache Spark, as it incorporates features I have mentioned above, including in-memory processing, fault tolerance, and scalability. For academic and healthcare professionals, Spark can fundamentally support the excessive number of image sets to enhance diagnosis and tools. Also, implementing Spark will not affect the existing chains of respective processes in healthcare institutions since it is fully interoperable. Here, it shall be illustrated how, through Spark, the strength of the picture analysis and its consequences are supported. a. Reduction in Hospital Readmission Apache Spark has focused on delivering a premium reduction in readmission volume in the healthcare sector. Hence, increased readmission rates contribute to improved costs in the health system with that despaired outcome for the patient. Components such as electronic health records, demographic data, and other pertinent facets have been incorporated into healthcare organizations by analyzing the data to identify factors indicative of or may lead to hospital readmissions using Apache Spark. In detail, medical providers can harness extensive data analysis powered by Spark and machine learning algorithms to have a clearer picture of the patients most suitable for readmission and an improved approach that can minimize such cases across healthcare facilities (Shrotriya et al., 2023). The application of the analytics based on Apache Spark positively affects the high readmission rates of hospitals, and the outcomes are profitable for both patients and the scientific-healing complex.b. Early Detection of SepsiSepsis, if not detected at an early stage, should be treated immediately to avoid organ failure or even death. Sepsis, which the authors claim might result in death, starts with an infection and initiates an inflammatory response in the body (Lelubre & Vincent, 2018). Apache Spark is instrumental in rapid sepsis identification since it can analyze clinically relevant real-time data, including the patient's temperature, heart rate, blood pressure, glucose levels, and most routine laboratory results. Spark is intended to utilize machine learning to assist clinicians in discerning the indications of sepsis and prescribing the appropriate treatment immediately (Shrotriya et al., 2023). Apache Spark has been tested in various studies to be efficient in diagnosing sepsis early, enhancing t, treatment with minimal lethality.c. Cancer Studies and Treatment Optimization Apache Spark has transformed the analysis of cancer and the management of treatment procedures. Using available knowledge of the nature of cancer, namely the fact that it is a multifactorial disease associated with a vast amount of genomic, proteomic, and clinical data, one can state that cancer poses severe challenges and limitations to both scholars and clinicians. Apache Spark assists in improving the time taken for searching biomarkers, subtypes, and probable treatment solutions for cancer using big data analyzing lenses and speedy processing (Shrotriya et al., 2023). Moreover, by integrating AI and machine learning, Spark has eased the ability to formulate concrete strategies that enhance cancer therapy prospects without worsening side effects. d.  Accelerating Drug DiscoveryAs with conventional drug molecules, developing new drug molecules in the chemical and pharmaceutical industry can be time-consuming, expensive, and labor-intensive. It is incredibly vital in finding new drugs since it helps in understanding a wide of information in the form of chemicals and genomic and proteomic databases. The function and power of advanced Analysis in Spark allow researchers to discover new drugs that can cure diseases or even predict the side effects of particular medications. Machine learning and AI have benefited drug discovery because they offer more accurate estimations about how a given drug interacts with a target. In multiple cases, it has been described how the application of Apache Spark increases the speed of drug discovery operations to a great extent, thus improving the continuing development of new drugs and the treatment of patients.3. The Impacts of Apache Spark in the Management of Health PopulationIt has been crucial to have the Apache Spark in population health application. Spark, an advanced big data distributed computing framework, can address the challenges of big data about population health. Shrotriya and colleagues (2023) argues that Apache Spark as a possible solution for enhancing research findings and the utility of data in decision-making on population health intervention. Specifically, the new objectives of population health management will require the analysis of large volumes of data to detect patterns, developments in, and degrees of 'healthiness' within groups. Therefore, this data is applicable in establishing measures to counter various public health issues, allocating funds, and executing early interventions.Population data is rich and complex, so advanced functions for working with populations and large data sums are necessary (Gopalani & Arora, 2019). Apache Spark is a prime illustrated example characterized by several features, including its scalability, fault tolerance, and the ability to process big data in memory; these qualities make it possible to manage the entire population's health. Namely, by manipulating and analyzing Big Data in a health context, PH practitioners and researchers can discover relationships and patterns for evidence-based decision-making with the help of Spark. As a component of population health management in today's environment, Spark is easy to integrate into current practices due to its ability to handle many languages and data sets (Shrotriya et al., 2023). In addition to enhancing our knowledge of the factors that define the results concerning public health, machine learning technologies are suitable for creating high-level processing models for evaluating the population's condition. Below is a description of the ways by which the implementation of Apache Spark in the population health management process can make a significant impact on public health.4. Technological Advancements and IntegrationTogether with other advancements in the technology and compatibility of Apache Spark with the Industry standards, their applicability in improving the operations of healthcare structures has been enhanced to a greater extent. Several significant developments that have achieved regarding a substantial role in the spread of Spark in healthcare. One od these developments include machine learning libraries. This has boosted the work of the organization in improving innovations that are data-driven and patient service delivery. MLlib is one of the ten important machine learning libraries in Spark, and it has played crucial roles in contributing to the latter's growth in the healthcare sector (Nazari et al., 2019). These libraries provide ample coverage for dimensionality reduction, grouping, regression, and classification tasks. As illustrated in Figure 1, these resources are employed by academics working alongside healthcare professionals in building sophisticated models for the prognosis of the results for patients, for following ailment trends, and for modeling the interdependencies between the numerous health indicators. Some programming languages Spark supports are interoperability-friendly, particularly regarding integration into existing health functions. Some of them are R language, Python language, Java language, and Scala language. Such flexibility means that Spark can work concurrently with the other structures within healthcare organizations without disrupting the existing system. Thus, the problems that emanate from integration are considerably eased (Shrotriya et al., 2023). Figure 1 Illustration of the proposed ML framework for Spark.Ethical ConsiderationsThere are many universal topics that will continuously become important to adopting big data technologies like Apache Spark, especially now that healthcare organizations are embracing such technology; these are data privacy security and fairness issues. It becomes imperative that there are ethical principles that govern the use, analysis, and reporting of data, which enables the correct application of new technological tools in the health sector. While there is potential for using healthcare data analytics in the future, there are always ethical issues involved, which could be solved if there are guidelines and regulations to govern moral concerns via transparency and acceptance of responsibilities, positive results could be achieved (Zaharia et al., 2016). Hence, the healthcare business has to consider the ethical issues and use big data technologies, such as Apache Spark, properly and sustainably to enhance trust with the specific patient and stakeholder. Based on the description of Apache Spark, this could easily mean that the healthcare business could greatly benefit from Apache Spark by having extra processing and analysis functions. However, there are certain constraints within the platform that we need to address, as the utilization of this platform is rare at the moment. The significant challenges experienced by healthcare businesses incorporating Spark include the security and privacy of the information processed and analyzed and the requirement for skilled personnel to invest in it. Since the information content is processed in the context of the healthcare sector, it is nearly imperative to maintain the data's security, especially regarding the industry's strict compliance with data privacy when integrating Spark. Particular attention should be paid to traditional data in Spark to avoid violating the norms of critical current legislation, such as HIPAA. Because the data in the raw form and transit and transform within applications can contain susceptible information, businesses should embrace such measures as encryption, access control, and auditing trails. Nonetheless, Spark has brought other challenges to privacy practice within the healthcare sector, even as it continues to improve efficiency by processing data in real-time. ConclusionApache server Spark affects innovations in the healthcare business to a significant extent, enhances the quality of patient care, and optimizes decision-making based on big data analytics. Due to Spark's consistent extensibility, Spark incorporates sophisticated machine learning frameworks to address these issues, which elaborate the handling of comprehensive and intricate data in healthcare organizations that could benefit healthcare firms. Of course, Spark can be helpful in other fields like medical image processing and analysis, genomic research, disease surveillance, and population health management. Before going into the successful implementation of Spark in the healthcare field, three major concerns should be solved, and these are: However, for firms to gain full benefits of what Spark can offer, some guidelines must be laid down to ensure that sensitive data is well protected and meets the set laws regulating healthcare firms. The skills gap could be addressed by concentrating on the qualities of investment in practices of getting and developing a healthcare workforce that could employ Spark by promoting the culture of lifelong learning of employees.References Bedeley, R. T. (2017). An Investigation of Analytics and Business Intelligence Applications in Improving Healthcare Organization Performance: A Mixed Methods Research. The University of North Carolina at Greensboro. Gopalani, S., & Arora, R. (2019). You are comparing Apache spark and map-reduce with performance analysis using k-means—International Journal of Computer Applications, 113(1). Han, Z., & Zhang, Y. (2015, December). Spark: A big data processing platform based on memory computing. In 2015 Seventh International Symposium on Parallel Architectures, Algorithms and Programming (PAAP) (pp. 172-176). IEEE. Patel, J. A., & Sharma, P. (2014, August). Big data for better health planning. In 2014 International Conference on Advances in Engineering & technology research (ICAETR-2014) (pp. 1-5). IEEE. Salloum, S., Dautov, R., Chen, X., Peng, P. X., & Huang, J. Z. (2016). Big data analytics on Apache Spark. International Journal of Data Science and Analytics, 1, 145-164. Shrotriya, L., Sharma, K., Parashar, D., Mishra, K., Rawat, S. S., & Pagare, H. (2023). Apache Spark in healthcare: Advancing data-driven innovations and better patient care. International Journal of Advanced Computer Science and Applications, 14(6). Zaharia, M., Xin, R. S., Wendell, P., Das, T., Armbrust, M., Dave, A., ... & Stoica, I. (2016). Apache spark: a unified engine for big data processing. Communications of the ACM, 59(11), 56-65.

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