AI For Young Learners Pdf

AI For Young Learners Pdf — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

The Business Cloud

The Business Cloud is an API enabled self-service platform, developed by Domo, that provides an array of services like data connection and data visualization. == History == Domo, Inc. was founded in 2010 by Josh James who also co-founded the web analytics software company Omniture in 1996, which he took public in 2006. Domo launched the Domo Appstore, with 1,000 apps with social and mobile capabilities, in 2016. This appstore creates a network of business apps and an ecosystem of companies into a single, integrated business cloud. This decision came after Domo announced a $131 million round of funding from BlackRock. According to the company, the concept behind The Business Cloud is to connect smaller clouds relating to apps or other functional areas of a business into a single business cloud that allows self-service and other social features to customers. == Services == The Business Cloud is offered as a free service, claimed to be the world's first business cloud with Domo appstore as one of its core services. This free package includes all of the Domo's features and functionality including Domo platform, Domo Apps, visualizations, alerts, company directories, org charts, profiles, tasks and Domo Mobile. The Business Cloud allows customers to leverage their preferred cloud as well as on-premises software and monitor all aspects of their business in routine. The company is supported by a $500 million fund from investors all over the world.
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Common Image Generator Interface

The Common Image Generator Interface (CIGI) (pronounced sig-ee), is an on-the-wire data protocol that allows communication between an Image Generator and its host simulation. The interface is designed to promote a standard way for a host device to communicate with an image generator (IG) within the industry. CIGI enables plug-and-play by standard-compliant image generator vendors and reduces integration costs when upgrading visual systems. == Background == Most high-end simulators do not have everything running on a single machine the way popular home software flight simulators are currently implemented. The airplane model is run on one machine, normally referred to as the host, and the out the window visuals or scene graph program is run on another, usually referred to as an Image Generator (IG). Frequently there are multiple IGs required to display the surrounding environment created by a host. CIGI is the interface between the 'host' and the IGs. The main goal of CIGI is to capitalize on previous investments through the use of a common interface. CIGI is designed to assist suppliers and integrators of IG systems with ease of integration, code reuse, and overall cost reduction. In the past most image generators provided their own proprietary interface; every host had to implement that interface making changing image generators a costly ordeal. CIGI was created to standardize the interface between the host and the image generator so that little modification would be needed to switch image generators. The CIGI initiative was largely spearheaded by The Boeing Company during the early 21st century. The latest version of CIGI (CIGI 4.0) was developed by the Simulation Interoperability Standards Organization (SISO) in the form of SISO-STD-013-2014, Standard for Common Image Generator Interface (CIGI), Version 4.0, dated 22 August 2014. SISO-STD-013-2014 is freely available from SISO. == Definitions == Image generator – In this context an image generator consists of one or more rendering channels that produce an image that can be used to visualize an “Out-The-Window” scene, or images produced by various sensor simulations such as Infra-red, Day TV, electro-optical, and night vision. Host simulation – In this context a “Host” is the computational system that provides information about the device being simulated so that the image generator can portray the correct scenery to the user. This information is passed via CIGI to the image generator. == Maturation == CIGI 4 is the latest version of the standard as was approved by the Simulation Interoperability Standards Organization on August 22, 2014. CIGI became an international SISO standard known as SISO-STD-013-2014; which contains the CIGI version 4.0 Interface Control Document (ICD). CIGI 4.0 is the official standard, published by SISO. Previous versions of CIGI were spearheaded by Boeing include CIGI v3.3, in November 2008, v3.2 April 2006, v3.1 June 2004, v3 November 2003, v2 in March 2002, and the original (v1) in March 2001 == Protocol dependencies == Typically, CIGI uses UDP as its transport protocol, but CIGI does not require a specific transport mechanism, only packet definition conformance. CIGI traffic does not have a well known port; however, the use of ports 8004-8005 has been widely adopted by commercial image generator vendors implementations. == Development tools == === Host Emulator === The Host Emulator can be used as a surrogate to manipulate the interface when a simulation Host is not available. It is a Windows-based image generator Host application used to develop, integrate and test image generators that use the CIGI protocol. It provides a graphical user interface (GUI) for the creation, modification and deletion of entities; manipulation of views; control of environmental attributes and phenomena; and other host functions. The Host Emulator has several features that are useful for integration and testing. A free-flight mode allows for fixed-wing and rotorcraft flight, movement along entity axes and free rotation using a joystick or a joystick-like widget. Scripting and record/playback features support regression testing, demonstrations and other tasks needing exact reproduction of certain sequences of events. A packet-level snoop feature allows the user to examine the contents of CIGI messages, image generator response times and latencies. A Heartbeat Monitor Window shows a graphical timing history of the Image Generator's data frame rate. Other features include explicit packet creation, animation control, missile flyouts and a situation display window (Host Emulator 3.x only). === Multi-Purpose Viewer === The Multi-Purpose Viewer (MPV) provides the basic functionality expected of an Image Generator, such as loading and displaying a terrain database, displaying entities and so forth. The Multi-Purpose Viewer can be used as a surrogate to manipulate the interface when a real Image Generator is not available. The MPV is capable of operating with both the Windows and Linux operating systems. === CIGI Class Library === The CCL is an object-oriented software interface that automatically handles message composition and decomposition (i.e. packing, unpacking and byte swapping to the ICD specification) on both the Host and Image Generator sides of the interface. The CCL interprets Host or Image Generator messages based on compile time parameters. It also performs error handling and translation between different versions of CIGI. Each packet type has its own class. The individual packet members are accessed through packet class accessors. Outgoing messages are constructed by placing each packet into the outgoing buffer using a streaming operator. Incoming messages are parsed using callback or event-based mechanisms that supply the using program with fully populated packet objects. === Current tool suite === A set of CIGI development tools are managed and maintained by the SISO CIGI Product Support Group. The latest packages are available on SourceForge. Comments/Suggestions to the package can be directed to the SISO discussion board at: https://discussions.sisostds.org/index.htm?A0=SAC-PSG-CIGI Archived 2017-09-13 at the Wayback Machine === Wireshark === Wireshark is a free and open source packet analyzer. It is used for network troubleshooting, analysis, software and communications protocol development, and education. Wireshark provides a dissector for CIGI packets. As of October 2016, “The CIGI dissector is fully functional for CIGI version 2 and 3. Version 1 is not yet implemented.” === Older versions of CIGI === A CIGI Interface Control Document (ICD) and development suite is available in open source format. The tools, ICD, and accompanying user documentation can be found and downloaded from the CIGI sourceforge web site. The SourceForge version of the MPV is limited in its support of CIGI data packets and is intended to grow as needs arise. The MPV uses CIGI 3 as its interface, but the MPV is backward-compatible with earlier CIGI versions through the use of the CCL. The MPV uses the Open Scene Graph library to render a scene. The scene graph is manipulated according to the CIGI commands received from the Host via the CCL. The MPV itself is an application layer that consists of a small kernel leveraging heavily on a plug-in architecture for ease of maintainability and flexibility. An implementer can implement the interface from scratch, however a full suite of integration tools is available. These tools consist of three elements. The Host Emulator (HE), the Multi-Purpose Viewer (MPV), and the CIGI Class Library (CCL).
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Starlight Information Visualization System

Starlight is a software product originally developed at Pacific Northwest National Laboratory and now by Future Point Systems. It is an advanced visual analysis environment. In addition to using information visualization to show the importance of individual pieces of data by showing how they relate to one another, it also contains a small suite of tools useful for collaboration and data sharing, as well as data conversion, processing, augmentation and loading. The software, originally developed for the intelligence community, allows users to load data from XML files, databases, RSS feeds, web services, HTML files, Microsoft Word, PowerPoint, Excel, CSV, Adobe PDF, TXT files, etc. and analyze it with a variety of visualizations and tools. The system integrates structured, unstructured, geospatial, and multimedia data, offering comparisons of information at multiple levels of abstraction, simultaneously and in near real-time. In addition Starlight allows users to build their own named entity-extractors using a combination of algorithms, targeted normalization lists and regular expressions in the Starlight Data Engineer (SDE). As an example, Starlight might be used to look for correlations in a database containing records about chemical spills. An analyst could begin by grouping records according to the cause of the spill to reveal general trends. Sorting the data a second time, they could apply different colors based on related details such as the company responsible, age of equipment or geographic location. Maps and photographs could be integrated into the display, making it even easier to recognize connections among multiple variables. Starlight has been deployed to both the Iraq and Afghanistan wars and used on a number of large-scale projects. PNNL began developing Starlight in the mid-1990s, with funding from the Land Information Warfare Agency, a part of the Army Intelligence and Security Command and continued developed at the laboratory with funding from the NSA and the CIA. Starlight integrates visual representations of reports, radio transcripts, radar signals, maps and other information. The software system was recently honored with an R&D 100 Award for technical innovation. In 2006 Future Point Systems, a Silicon Valley startup, acquired rights to jointly develop and distribute the Starlight product in cooperation with the Pacific Northwest National Laboratory. The software is now also used outside of the military/intelligence communities in a number of commercial environments.
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Vinted

Vinted Group UAB is a Lithuanian technology company best known for its online marketplace Vinted. Vinted is the leading second-hand fashion marketplace in Europe and a go-to destination for all kinds of second-hand items. According to the company, its mission is to make second-hand the first choice worldwide. The company operates as an ecosystem of businesses, including the Vinted Marketplace (its peer-to-peer resale platform), Vinted Go (logistics and shipping services), Vinted Pay (in-app payment solutions), and Vinted Ventures (an investment arm supporting the circular economy). Headquartered in Vilnius, Lithuania, it also has offices in Germany and the Netherlands and employs more than 2,200 people. == History == Vinted was co-founded in 2008 by Milda Mitkute and Justas Janauskas in Vilnius, Lithuania. The idea originated when Mitkute was moving house and wanted a way to sell clothes she no longer needed. Janauskas helped her create a website where users could trade clothing items. In 2016, Dutch entrepreneur Thomas Plantenga joined Vinted as a strategy consultant and later became Chief Executive Officer, leading the company through a period of international growth. In 2019, Vinted became Lithuania’s first technology unicorn after raising €128 million at a €1 billion valuation in a funding round led by Lightspeed Venture Partners. In October 2020, it acquired United Wardrobe, a Dutch competitor, and in November 2020 German Kleiderkreisel and Mamikreisel were officially merged into the Vinted platform. In 2024 it acquired Trendsales, a Danish resale platform. According to Vogue Business, Vinted’s revenue grew 61% between 2022 and 2023 and the company posted a net profit of €17.8 million in 2023. Usage of Vinted in the UK has grown from 1.2 million users in 2021, to 8 million in 2023. In 2024, the group reported consolidated revenue of €813.4 million (up 36% from 2023) and a net profit of €76.7 million, up 330% from 2023. As of 2024, Vinted was valued at approximately €5 billion, operating in more than 26 markets worldwide and announcing plans to launch in Ireland, Greece, Latvia, Slovenia, and Estonia in 2025. As of 2025 the company employed more than 2,200 people. In April 2026, Vinted completed a secondary share transaction of €880m, valuing the company at €8bn. == Products and operations == Vinted primarily resells clothing but now supports multiple categories including homeware, kidswear, electronics, books, collectibles, and high-value fashion. Vinted has worked with public figures such as Paul Mescal and Alexa Chung on exclusive wardrobe sales and has also partnered directly with charities including Oxfam on initiatives which promote the social and environmental value of second-hand fashion, such as the Style for Change fashion show at London Fashion Week. In 2025, Vinted produced its first television format, the second-hand fashion competition series RE/Style, hosted by Emma Willis. The show features emerging fashion designers from across Europe creating runway-ready looks from second-hand garments and aired on Prime Video UK. In 2025, Vinted was reported as France’s top clothing retailer by sales volume. == Criticism == Vinted has faced scrutiny from European data protection authorities in France, Lithuania, and Poland following complaints regarding GDPR compliance and account blocking practices. In July 2024, the Lithuanian authority fined the company €2,375,276. The case was coordinated by a dedicated Vinted Working Group under the European Data Protection Board. In early 2024, Swedish police reported around 300 fraud cases linked to the platform, in which users’ bank accounts were targeted by scammers. In October 2024, Channel 4 in the United Kingdom aired a documentary examining safety and privacy concerns related to the platform, including the sexualisation of underage users’ images and risks associated with second-hand baby products lacking safety certification. In November 2025, BBC News reported that Vinted’s update to its sizing system in the United Kingdom led to widespread user criticism. Vinted said the update was intended to standardise sizing across international brands.
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Lose It!

Lose It! is an American health and wellness mobile app developed by FitNow, Inc. The app generates calorie budgets for users by tracking weight, exercise, food and calorie intake, and personal goals, primarily to assist them in achieving weight loss. == History == Lose It! was developed in Boston and debuted in 2008. The app and its associated company were founded by J.J. Allaire, Charles Teague and Paul Dicristina. Prior to founding Lose It!, Teague and Allaire had founded the online research tool Onfolio, which was acquired by Microsoft in 2006. The Lose It! app was originally released as an iOS app before being released as a website in 2010 and an Android app in 2011. In 2015, Lose It! announced plans to release the app internationally. Lose It! was also available as an app for Apple Watch at its launch in 2015. The app’s “Snap It” feature, which allows users to approximate calorie counts by taking pictures of their daily meals and snacks, was released in beta in 2016. Snap It was named an Innovation Awards Honoree at the 2017 Consumer Electronics Show in Las Vegas. In 2020, Patrick Wetherille, one of the company’s earliest employees, was appointed chief executive officer. == App == Lose It! is weight loss app. The app allows users to set goals such as increasing strength, overall health/maintenance, and weight loss. It provides users recommended calorie budgets based on data such as their current weight and their desired weight. Lose It! also tracks data such as exercise/activity level and food consumption and allows users to track calories consumed by scanning barcodes for food products then retrieving calorie information for products. The app can also estimate the amount of calories in a food products. Lose It! has integration features connecting it to other apps such as Fitbit and Runkeeper. It also has social features such as joining groups and sharing progress with friends. The Premium version of the app allows users to track foods according to specific diets like keto, heart healthy or Mediterranean.
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Commitment ordering

Commitment ordering (CO) is a class of interoperable serializability techniques in concurrency control of databases, transaction processing, and related applications. It allows optimistic (non-blocking) implementations. With the proliferation of multi-core processors, CO has also been increasingly utilized in concurrent programming, transactional memory, and software transactional memory (STM) to achieve serializability optimistically. CO is also the name of the resulting transaction schedule (history) property, defined in 1988 with the name dynamic atomicity. In a CO compliant schedule, the chronological order of commitment events of transactions is compatible with the precedence order of the respective transactions. CO is a broad special case of conflict serializability and effective means (reliable, high-performance, distributed, and scalable) to achieve global serializability (modular serializability) across any collection of database systems that possibly use different concurrency control mechanisms (CO also makes each system serializability compliant, if not already). Each not-CO-compliant database system is augmented with a CO component (the commitment order coordinator—COCO) which orders the commitment events for CO compliance, with neither data-access nor any other transaction operation interference. As such, CO provides a low overhead, general solution for global serializability (and distributed serializability), instrumental for global concurrency control (and distributed concurrency control) of multi-database systems and other transactional objects, possibly highly distributed (e.g., within cloud computing, grid computing, and networks of smartphones). An atomic commitment protocol (ACP; of any type) is a fundamental part of the solution, utilized to break global cycles in the conflict (precedence, serializability) graph. CO is the most general property (a necessary condition) that guarantees global serializability, if the database systems involved do not share concurrency control information beyond atomic commitment protocol (unmodified) messages and have no knowledge of whether transactions are global or local (the database systems are autonomous). Thus CO (with its variants) is the only general technique that does not require the typically costly distribution of local concurrency control information (e.g., local precedence relations, locks, timestamps, or tickets). It generalizes the popular strong strict two-phase locking (SS2PL) property, which in conjunction with the two-phase commit protocol (2PC), is the de facto standard to achieve global serializability across (SS2PL based) database systems. As a result, CO compliant database systems (with any different concurrency control types) can transparently join such SS2PL based solutions for global serializability. In addition, locking based global deadlocks are resolved automatically in a CO based multi-database environment, a vital side-benefit (including the special case of a completely SS2PL based environment; a previously unnoticed fact for SS2PL). Furthermore, strict commitment ordering (SCO; Raz 1991c), the intersection of Strictness and CO, provides better performance (shorter average transaction completion time and resulting in better transaction throughput) than SS2PL whenever read-write conflicts are present (identical blocking behavior for write-read and write-write conflicts; comparable locking overhead). The advantage of SCO is especially during lock contention. Strictness allows both SS2PL and SCO to use the same effective database recovery mechanisms. Two major generalizing variants of CO exist, extended CO (ECO; Raz 1993a) and multi-version CO (MVCO; Raz 1993b). They also provide global serializability without local concurrency control information distribution, can be combined with any relevant concurrency control, and allow optimistic (non-blocking) implementations. Both use additional information for relaxing CO constraints and achieving better concurrency and performance. Vote ordering (VO or Generalized CO (GCO); Raz 2009) is a container schedule set (property) and technique for CO and all its variants. Local VO is necessary for guaranteeing global serializability if the atomic commitment protocol (ACP) participants do not share concurrency control information (have the generalized autonomy property). CO and its variants inter-operate transparently, guaranteeing global serializability and automatic global deadlock resolution together in a mixed, heterogeneous environment with different variants. == Overview == The Commitment ordering (CO; Raz 1990, 1992, 1994, 2009) schedule property has been referred to also as Dynamic atomicity (since 1988), commit ordering, commit order serializability, and strong recoverability (since 1991). The latter is a misleading name since CO is incomparable with recoverability, and the term "strong" implies a special case. This means that a substantial recoverability property does not necessarily have the CO property and vice versa. In 2009 CO has been characterized as a major concurrency control method, together with the previously known (since the 1980s) three major methods: Locking, Time-stamp ordering, and Serialization graph testing, and as an enabler for the interoperability of systems using different concurrency control mechanisms. In a federated database system or any other more loosely defined multidatabase system, which are typically distributed in a communication network, transactions span multiple and possibly Distributed databases. Enforcing global serializability in such system is problematic. Even if every local schedule of a single database is still serializable, the global schedule of a whole system is not necessarily serializable. The massive communication exchanges of conflict information needed between databases to reach conflict serializability would lead to unacceptable performance, primarily due to computer and communication latency. The problem of achieving global serializability effectively had been characterized as open until the public disclosure of CO in 1991 by its inventor Yoav Raz (Raz 1991a; see also Global serializability). Enforcing CO is an effective way to enforce conflict serializability globally in a distributed system since enforcing CO locally in each database (or other transactional objects) also enforces it globally. Each database may use any, possibly different, type of concurrency control mechanism. With a local mechanism that already provides conflict serializability, enforcing CO locally does not cause any other aborts, since enforcing CO locally does not affect the data access scheduling strategy of the mechanism (this scheduling determines the serializability related aborts; such a mechanism typically does not consider the commitment events or their order). The CO solution requires no communication overhead since it uses (unmodified) atomic commitment protocol messages only, already needed by each distributed transaction to reach atomicity. An atomic commitment protocol plays a central role in the distributed CO algorithm, which enforces CO globally by breaking global cycles (cycles that span two or more databases) in the global conflict graph. CO, its special cases, and its generalizations are interoperable and achieve global serializability while transparently being utilized together in a single heterogeneous distributed environment comprising objects with possibly different concurrency control mechanisms. As such, Commitment ordering, including its special cases, and together with its generalizations (see CO variants below), provides a general, high performance, fully distributed solution (no central processing component or central data structure are needed) for guaranteeing global serializability in heterogeneous environments of multidatabase systems and other multiple transactional objects (objects with states accessed and modified only by transactions; e.g., in the framework of transactional processes, and within Cloud computing and Grid computing). The CO solution scales up with network size and the number of databases without any negative impact on performance (assuming the statistics of a single distributed transaction, e.g., the average number of databases involved with a single transaction, are unchanged). With the proliferation of Multi-core processors, Optimistic CO (OCO) has also been increasingly utilized to achieve serializability in software transactional memory, and numerous STM articles and patents utilizing "commit order" have already been published (e.g., Zhang et al. 2006). == The commitment ordering solution for global serializability == === General characterization of CO === Commitment ordering (CO) is a special case of conflict serializability. CO can be enforced with non-blocking mechanisms (each transaction can complete its task without having its data-access blocked, which allows optimistic concurrency control; however, commitment could be blo
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Webull

Webull Corporation, often stylized as simply Webull, is a U.S.-based financial services holding company headquartered in St. Petersburg, Florida. It owns and operates the Webull electronic trading platform for self-directed retail investors. Depending on jurisdiction, the Webull platform offers trading in stocks, exchange-traded funds (ETFs), options, margin, bonds, cryptocurrency and futures, as well as market-data tools. Webull began operations in 2016 under Hunan Fumi Information Technology, a China-based financial technology company founded by Wang Anquan. It launched U.S. brokerage services through Webull Financial LLC in 2018 and expanded during the retail-trading boom of 2020 and 2021. In April 2025, Webull became a publicly traded company on the Nasdaq through a merger with special-purpose acquisition company SK Growth Opportunities Corporation. The company's U.S. brokerage revenue relies substantially on payment for order flow, with options trading accounting for the larger share of its order-flow rebates in 2025. Webull has faced regulatory actions related to options customer approvals, complaint handling, suspicious activity reporting, social-media marketing and customer disclosures. It has also faced scrutiny from U.S. lawmakers and state officials over its historical and operational ties to China and the handling of U.S. customer data. == History == === Founding === Webull was founded in 2016 under Hunan Fumi Information Technology, a China-based financial technology company, by Wang Anquan, a former employee of Alibaba Group and Xiaomi. Hunan Fumi Information Technology received backing from Xiaomi, Shunwei Capital, and other investors in China. Fumi Technology was a Hunan-based fintech start-up incubated by Xiaomi and raised about CNY200 million (approximately US$30 million) in a Series B financing round in 2018. On May 24, 2017, Webull Financial LLC was established as a Delaware limited liability company. It began offering brokerage services in the United States in May 2018. Wang hired Anthony Denier as CEO of the U.S. brokerage that year and the two mapped out their strategy on napkins at a Mexican restaurant in New York City. Webull Corporation was incorporated in the Cayman Islands in September 2019 as the group's holding company. === Retail trading boom === In May 2020, the company received SEC approval to launch a robo-advisor on its platform. By August 2020, the platform had over 11 million registered users, and in October 2020, it had 750,000 daily active users. Webull introduced options trading in 2020 and later added cryptocurrency trading through a separate digital-asset business. In November 2020, Webull began supporting cryptocurrency transactions. In December 2020, Webull launched trading services in Hong Kong. During the GameStop short squeeze in January 2021, Webull gained attention as some retail traders looked for alternatives to Robinhood. On January 27, 2021, Webull recorded its highest-ever number of active daily users, at 952,000, and the Webull app was downloaded across the Apple App and Google Play stores an estimated 100,000 times. That week, approximately 1.2 million people downloaded the Webull mobile app, which the company reported as a 1,548% week-over-week increase. On January 28, 2021, Webull was directed by its clearing house to temporarily halt buy orders for stocks affected by the GameStop short squeeze. In June 2021, Webull was reported to be considering a U.S. initial public offering that could raise up to $400 million. === Restructuring and expansion === Webull restructured its China-related corporate arrangements in 2022 and later stated that Hunan Fumi was no longer affiliated with the group. In 2022 and 2023, Webull expanded in several non-U.S. markets, including Singapore, Australia, South Africa, Japan, the United Kingdom and Indonesia. In June 2023, Webull moved cryptocurrency trading to a separate app called Webull Pay. By the end of 2023, Webull had 4.3 million funded accounts and US$8.2 billion in customer assets. In January 2024, Anthony Denier was promoted to group president of Webull Corporation. In November 2024, Webull launched overnight, or extended-hours, trading, expanding the trading window of U.S. stocks for users inside and outside the United States. === SPAC merger and Nasdaq listing === On February 28, 2024, Webull agreed to go public through a business combination with SK Growth Opportunities Corporation (NASDAQ: SKGR), a special-purpose acquisition company, in a deal that valued the company at approximately US$7.3 billion. The proposed valuation drew scrutiny because of Webull's limited financial disclosure at announcement, reliance on payment for order flow and small expected public float. SK Growth shareholders approved the business combination on March 30, 2025, and the transaction closed on April 10, 2025. Webull's Class A ordinary shares and warrants began trading on the Nasdaq on April 11, 2025 under the ticker symbols BULL and BULLW (incentive warrants traded under BULLZ until their redemption in June 2025). The merger brought Webull to the public market but generated little cash for the company: after shareholder redemptions, Webull disclosed net proceeds of US$430,066 from the transaction. After the listing, Webull's shares experienced extreme volatility, rising as much as 500% to US$79.56 on April 14, 2025, after closing at US$13.25 on the prior trading day. The initial post-listing surge increased the value of Webull holdings owned by earlier investors, including RIT Capital Partners, which had first invested in Webull in 2021. In April 2026, after Webull's shares had fallen about 70% over the previous year, the company authorized a US$100 million share repurchase program. == Business model and financials == Webull provides a self-directed electronic trading platform available through mobile, desktop and web applications. Depending on jurisdiction, the platform offers trading in stocks, exchange-traded funds, options, margin, futures, fixed income products, cryptocurrency, cash management features and market data tools. In the United States, Webull Financial LLC is a registered broker-dealer and member of FINRA and the Securities Investor Protection Corporation, while Webull operates in other markets through locally licensed brokerage subsidiaries. Webull operates a commission-free or low-cost brokerage model for self-directed retail investors. In the United States, a substantial part of its trading-related revenue comes from payment for order flow, while in some non-U.S. markets the company more commonly charges commissions directly to customers. The platform is aimed at more active retail investors, including users seeking options tools, extended-hours trading and real-time market data. For 2025, Webull reported total revenue of US$571.0 million, up from US$390.2 million in 2024. Equity and option order-flow rebates accounted for US$304.1 million, or 53.3% of revenue, making order-flow rebates the company's largest reported revenue category. Interest-related income accounted for US$154.3 million, handling charge income for US$87.3 million and other revenue for US$25.3 million. Options were the larger component of the company's order-flow rebates in 2025, generating US$210.0 million compared with US$94.2 million from equities. Webull also generates revenue from interest-related activities, including margin financing, customer bank deposits, stock lending and corporate bank deposits. The company has stated that its interest-related income is affected by interest rates, customer cash balances, margin balances and demand for stock lending. The company had approximately 20 million registered users worldwide as of February 2024. As of December 31, 2025, it reported 26.8 million registered users, 5.0 million funded accounts and US$24.6 billion in customer assets. As of March 2025, Webull operated in Hong Kong, Singapore, Australia, South Africa, Japan, the United Kingdom, the United States, Indonesia, Canada, Brazil, Thailand, Malaysia and Mexico. == Marketing and sponsorships == Webull has used paid digital advertising, referral incentives, free-stock promotions, affiliate marketing and sports sponsorships to acquire customers and promote its brand. In its 2025 annual filing, the company reported marketing and branding expenses of US$152.3 million in 2023, US$138.7 million in 2024 and US$135.9 million in 2025. Webull said most of its advertising and promotion costs were related to paid search and paid social advertising, and that it had reduced free-stock promotions while shifting toward deposit- and asset-transfer-based incentives. In September 2021, BSE Global, the parent company of the Brooklyn Nets and New York Liberty, entered into a global multi-year agreement with Webull. Under the agreement, Webull became an official sponsor and online brokerage partner of the teams, with branding that included a jersey patch on Brooklyn Nets uniforms. Spo
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Swap chain

In computer graphics, a swap chain (also swapchain) is a series of virtual framebuffers used by the graphics card and graphics API for frame rate stabilization, stutter reduction, and several other purposes. Because of these benefits, many graphics APIs require the use of a swap chain. The swap chain usually exists in graphics memory, but it can exist in system memory as well. A swap chain with two buffers is a kind of double buffer. == Function == In every swap chain there are at least two buffers. The first framebuffer, the screenbuffer, is the buffer that is rendered to the output of the video card. The remaining buffers are known as backbuffers. Each time a new frame is displayed, the first backbuffer in the swap chain takes the place of the screenbuffer, this is called presentation or swapping. A variety of other actions may be taken on the previous screenbuffer and other backbuffers (if they exist). The screenbuffer may be simply overwritten or returned to the back of the swap chain for further processing. The action taken is decided by the client application and is API dependent. == Direct3D == Microsoft Direct3D implements a SwapChain class. Each host device has at least one swap chain assigned to it, and others may be created by the client application. The API provides three methods of swapping: copy, discard, and flip. When the SwapChain is set to flip, the screenbuffer is copied onto the last backbuffer, then all the existing backbuffers are copied forward in the chain. When copy is set, each backbuffer is copied forward, but the screenbuffer is not wrapped to the last buffer, leaving it unchanged. Flip does not work when there is only one backbuffer, as the screenbuffer is copied over the only backbuffer before it can be presented. In discard mode, the driver selects the best method. == Comparison with triple buffering == Outside the context of Direct3D, triple buffering refers to the technique of allowing an application to draw to whichever back buffer was least recently updated. This allows the application to always proceed with rendering, regardless of the pace at which frames are being drawn by the application or the pace at which frames are being sent to the display. Triple buffering may result in a frame being discarded without being displayed if two or more newer frames are completely rendered in the time it takes for one frame to be sent to the display. By contrast, Direct3D swap chains are a strict first-in, first-out queue, so every frame that is drawn by the application will be displayed even if newer frames are available. Direct3D does not implement a most-recent buffer swapping strategy, and Microsoft's documentation calls a Direct3D swap chain of three buffers "triple buffering". Triple buffering as described above is superior for interactive purposes such as gaming, but Direct3D swap chains of more than three buffers can be better for tasks such as presenting frames of a video where the time taken to decode each frame may be highly variable.
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Learning curve (machine learning)

In machine learning (ML), a learning curve (or training curve) is a graphical representation that shows how a model's performance on a training set (and usually a validation set) changes with the number of training iterations (epochs) or the amount of training data. Typically, the number of training epochs or training set size is plotted on the x-axis, and the value of the loss function (and possibly some other metric such as the cross-validation score) on the y-axis. Synonyms include error curve, experience curve, improvement curve and generalization curve. More abstractly, learning curves plot the difference between learning effort and predictive performance, where "learning effort" usually means the number of training samples, and "predictive performance" means accuracy on testing samples. Learning curves have many useful purposes in ML, including: choosing model parameters during design, adjusting optimization to improve convergence, and diagnosing problems such as overfitting (or underfitting). Learning curves can also be tools for determining how much a model benefits from adding more training data, and whether the model suffers more from a variance error or a bias error. If both the validation score and the training score converge to a certain value, then the model will no longer significantly benefit from more training data. == Formal definition == When creating a function to approximate the distribution of some data, it is necessary to define a loss function L ( f θ ( X ) , Y ) {\displaystyle L(f_{\theta }(X),Y)} to measure how good the model output is (e.g., accuracy for classification tasks or mean squared error for regression). We then define an optimization process which finds model parameters θ {\displaystyle \theta } such that L ( f θ ( X ) , Y ) {\displaystyle L(f_{\theta }(X),Y)} is minimized, referred to as θ ∗ {\displaystyle \theta ^{}} . === Training curve for amount of data === If the training data is { x 1 , x 2 , … , x n } , { y 1 , y 2 , … y n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\},\{y_{1},y_{2},\dots y_{n}\}} and the validation data is { x 1 ′ , x 2 ′ , … x m ′ } , { y 1 ′ , y 2 ′ , … y m ′ } {\displaystyle \{x_{1}',x_{2}',\dots x_{m}'\},\{y_{1}',y_{2}',\dots y_{m}'\}} , a learning curve is the plot of the two curves i ↦ L ( f θ ∗ ( X i , Y i ) ( X i ) , Y i ) {\displaystyle i\mapsto L(f_{\theta ^{}(X_{i},Y_{i})}(X_{i}),Y_{i})} i ↦ L ( f θ ∗ ( X i , Y i ) ( X i ′ ) , Y i ′ ) {\displaystyle i\mapsto L(f_{\theta ^{}(X_{i},Y_{i})}(X_{i}'),Y_{i}')} where X i = { x 1 , x 2 , … x i } {\displaystyle X_{i}=\{x_{1},x_{2},\dots x_{i}\}} === Training curve for number of iterations === Many optimization algorithms are iterative, repeating the same step (such as backpropagation) until the process converges to an optimal value. Gradient descent is one such algorithm. If θ i ∗ {\displaystyle \theta _{i}^{}} is the approximation of the optimal θ {\displaystyle \theta } after i {\displaystyle i} steps, a learning curve is the plot of i ↦ L ( f θ i ∗ ( X , Y ) ( X ) , Y ) {\displaystyle i\mapsto L(f_{\theta _{i}^{}(X,Y)}(X),Y)} i ↦ L ( f θ i ∗ ( X , Y ) ( X ′ ) , Y ′ ) {\displaystyle i\mapsto L(f_{\theta _{i}^{}(X,Y)}(X'),Y')}
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Mixed raster content

Mixed raster content (MRC) is a method for compressing images that contain both binary-compressible text and continuous-tone components, using image segmentation methods to improve the level of compression and the quality of the rendered image. By separating the image into components with different compressibility characteristics, the most efficient and accurate compression algorithm for each component can be applied. MRC-compressed images are typically packaged into a hybrid file format such as DjVu and sometimes PDF. This allows for multiple images, and the instructions to properly render and reassemble them, to be stored within a single file. Some image scanners optionally support MRC when scanning to PDF. A typical manual states that without MRC, the image is generated in a single process, with text and graphics not distinguished. With MRC, separate processes are used for text, graphics, and other elements, producing clearer graphics and sharper text, at the price of slightly slower processing. MRC is recommended to optimise the scanning of documents with harder-to-read text or lower-quality graphics. MRC can also reduce the size of the scanned file, though higher compression using JBIG2 can sometimes lead to character substitution errors in scanned documents. == File format == A form of MRC is defined by international standard bodies as ISO/IEC 16485, or ITU recommendation T.44 (accessible free of charge). It defines a file format with bilevel masks and two data layers in each "stripe" of the image. The mask can be encoded in ITU T.4, JBIG1, or JBIG2, while the images can be JPEG, JBIG1, or run-length encoded color. The format is loosely based on JPEG, with a APP13 segment registered for this purpose. It is not known whether this file format is actually used, as formats like DjVu and PDF have their own ways of defining layers and masks.
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Pocketbook (application)

Pocketbook was a Sydney-based free budget planner and personal finance app launched in 2012. The app helped users setup and manage budgets, track spending and manage bills. As of 2016 Pocketbook claimed to support over 250,000 Australians, in January 2018 that number was 435,000. After being acquired by Zip Co Ltd in 2016, it was announced in 2022 that the app was to be shut down and all user accounts deleted. == History == Pocketbook was founded by Alvin Singh and Bosco Tan in 2012. It was conceived in 2011 in a Wolli Creek apartment as a tool for Alvin and Bosco to take control of their money. In 2013, Pocketbook raised $500,000 from technology fund Tank Stream Ventures, and a group of investors including TV personality David Koch, Geoff Levy, David Shein and Peter Cooper. In September 2016 Digital retail finance and payment industry player zipMoney (now trading as Zip Co Limited) acquired Pocketbook in a $7.5m deal == Features == The app synced with the bank account of users and would organize spending into different categories. Users could also be reminded of bill payments, analyse spending and set spending limits. They can also be alerted of fraudulent transactions and deductions. The app employs security measures like end to end encryption, CloudFlare protection, fraud detection, identity protection etc. Pocketbook was available via web and mobile version. == Awards == Personal Finance Innovator of the Year by Fintech Business Awards 2017 Innovator of the Year by OPTUS MyBusiness Awards 2017 Best Finance App of 2016 by Australian Fintech Best Personal Finance App: Pocketbook won the 2016 Finder Innovation Awards, presented at a gala dinner hosted by media personality and The New Inventors presenter James O'Loghlin. Best Mobile App of the Year Winner: StartCon hosted the first annual Australasian Startup Awards. Over 200 nominations in 14 categories and an overall winner were reviewed, and winners were determined by public voting, with over 63,000 votes in total. Best New Startup 2014 by StartupSmart. Finalist in the SWIFT Innotribe startup competition in Dubai in 2013.
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Vulnerability assessment (computing)

Vulnerability assessment is a process of defining, identifying and classifying the security holes in information technology systems. An attacker can exploit a vulnerability to violate the security of a system. Some known vulnerabilities are Authentication Vulnerability, Authorization Vulnerability and Input Validation Vulnerability. == Purpose == Before deploying a system, it first must go through from a series of vulnerability assessments that will ensure that the build system is secure from all the known security risks. When a new vulnerability is discovered, the system administrator can again perform an assessment, discover which modules are vulnerable, and start the patch process. After the fixes are in place, another assessment can be run to verify that the vulnerabilities were actually resolved. This cycle of assess, patch, and re-assess has become the standard method for many organizations to manage their security issues. The primary purpose of the assessment is to find the vulnerabilities in the system, but the assessment report conveys to stakeholders that the system is secured from these vulnerabilities. If an intruder gained access to a network consisting of vulnerable Web servers, it is safe to assume that he gained access to those systems as well. Because of assessment report, the security administrator will be able to determine how intrusion occurred, identify compromised assets and take appropriate security measures to prevent critical damage to the system. == Assessment types == Depending on the system a vulnerability assessment can have many types and level. === Host assessment === A host assessment looks for system-level vulnerabilities such as insecure file permissions, application level bugs, backdoor and Trojan horse installations. It requires specialized tools for the operating system and software packages being used, in addition to administrative access to each system that should be tested. Host assessment is often very costly in term of time, and thus is only used in the assessment of critical systems. Tools like COPS and Tiger are popular in host assessment. === Network assessment === In a network assessment one assess the network for known vulnerabilities. It locates all systems on a network, determines what network services are in use, and then analyzes those services for potential vulnerabilities. This process does not require any configuration changes on the systems being assessed. Unlike host assessment, network assessment requires little computational cost and effort. == Vulnerability assessment vs penetration testing == Vulnerability assessment and penetration testing are two different testing methods. They are differentiated on the basis of certain specific parameters. == Regulatory requirements == Vulnerability assessments are mandated or strongly recommended by several regulatory frameworks. In the United States healthcare sector, the Health Insurance Portability and Accountability Act (HIPAA) Security Rule requires covered entities to conduct periodic evaluations of their security posture, and a December 2024 Notice of Proposed Rulemaking would explicitly require vulnerability scanning at least every six months for systems containing electronic protected health information. The Payment Card Industry Data Security Standard (PCI DSS) requires quarterly vulnerability scans for organizations that process credit card transactions, and the NIST Cybersecurity Framework includes vulnerability assessment as a core component of its Identify function.
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Metadata repository

A metadata repository is a database created to store metadata. Metadata is information about the structures that contain the actual data. Metadata is often said to be "data about data", but this is misleading. Data profiles are an example of actual "data about data". Metadata adds one layer of abstraction to this definition– it is data about the structures that contain data. Metadata may describe the structure of any data, of any subject, stored in any format. A well-designed metadata repository typically contains data far beyond simple definitions of the various data structures. Typical repositories store dozens to hundreds of separate pieces of information about each data structure. Comparing the metadata of a couple data items - one digital and one physical - clarify what metadata is: First, digital: For data stored in a database one may have a table called "Patient" with many columns, each containing data which describes a different attribute of each patient. One of these columns may be named "Patient_Last_Name". What is some of the metadata about the column that contains the actual surnames of patients in the database? We have already used two items: the name of the column that contains the data (Patient_Last_Name) and the name of the table that contains the column (Patient). Other metadata might include the maximum length of last name that may be entered, whether or not last name is required (can we have a patient without Patient_Last_Name?), and whether the database converts any surnames entered in lower case to upper case. Metadata of a security nature may show the restrictions which limit who may view these names. Second, physical: For data stored in a brick and mortar library, one have many volumes and may have various media, including books. Metadata about books would include ISBN, Binding_Type, Page_Count, Author, etc. Within Binding_Type, metadata would include possible bindings, material, etc. This contextual information of business data include meaning and content, policies that govern, technical attributes, specifications that transform, and programs that manipulate. == Definition == The metadata repository is responsible for physically storing and cataloging metadata. Data in a metadata repository should be generic, integrated, current, and historical: Generic Meta model should store the metadata by generic terms instead of storing it by an applications-specific defined way, so that if your data base standard changes from one product to another the physical meta model of the metadata repository would not need to change. Integration of the metadata repository allows all business areas' metadata to be in an integrated fashion: Covering all domains and subject areas of the organization. current and historical The metadata repository should have accessible current and historical metadata. Metadata repositories used to be referred to as a data dictionary. With the transition of needs for the metadata usage for business intelligence has increased so is the scope of the metadata repository increased. Earlier data dictionaries are the closest place to interact technology with business. Data dictionaries are the universe of metadata repository in the initial stages but as the scope increased Business glossary and their tags to variety of status flags emerged in the business side while consumption of the technology metadata, their lineage and linkages made the repository, the source for valuable reports to bring business and technology together and helped data management decisions easier as well as assess the cost of the changes. Metadata repository explores the enterprise wide data governance, data quality and master data management (includes master data and reference data) and integrates this wealth of information with integrated metadata across the organization to provide decision support system for data structures, even though it only reflects the structures consumed from various systems. == Repository vs. registry == Repository has additional functionalities compared with registry. Metadata repository not only stores metadata like Metadata registry but also adds relationships with related metadata types. Metadata when related in a flow from its point of entry into organization up to the deliverables is considered as the lineage of that data point. Metadata when related across other related metadata types is called linkages. By providing the relationships to all the metadata points across the organization and maintaining its integrity with an architecture to handle the changes, metadata repository provides the basic material for understanding the complete data flow and their definitions and their impact. Also the important feature is to maintain the version control though this statement for contrasting is open for discussion. These definitions are still evolving, so the accuracy of the definitions needs refinement. The purpose of registry is to define the metadata element and maintained across the organization. And data models and other data management teams refer to the registry for any changes to follow. While Metadata repository sources metadata from various metadata systems in the organizations and reflects what is in the upstream. Repository never acts as an upstream while registry is used as an upstream for metadata changes. == Reason for use == Metadata repository enables all the structure of the organizations data containers to one integrated place. This opens plethora of resourceful information for making calculated business decisions. This tool uses one generic form of data model to integrate all the models thus brings all the applications and programs of the organization into one format. And on top of it applying the business definitions and business processes brings the business and technology closer that will help organizations make reliable roadmaps with definite goals. With one stop information, business will have more control on the changes, and can do impact analysis of the tool. Usually business spends much time and money to make decisions based on discovery and research on impacts to make changes or to add new data structures or remove structures in data management of the organization. With a structured and well maintained repository, moving the product from ideation to delivery takes the least amount of time (considering other variables are constant). To sum it up: Integration of the metadata across the organization Build relationship between various metadata types Build relationship between various disparate systems Define business golden copy of definitions Version control of the changes at structure level Interaction with Reference data Link view to master data Automatic synchronization with various authorized metadata source systems More control to business decisions Validate the structures by overlapping the models Discovering discrepancies, gaps, lineage, metrics at data structure level Each database management system (DBMS) and database tools have their own language for the metadata components within. Database applications already have their own repositories or registries that are expected to provide all of the necessary functionality to access the data stored within. Vendors do not want other companies to be capable of easily migrating data away from their products and into competitors products, so they are proprietary with the way they handle metadata. CASE tools, DBMS dictionaries, ETL tools, data cleansing tools, OLAP tools, and data mining tools all handle and store metadata differently. Only a metadata repository can be designed to store the metadata components from all of these tools. == Design == Metadata repositories should store metadata in four classifications: ownership, descriptive characteristics, rules and policies, and physical characteristics. Ownership, showing the data owner and the application owner. The descriptive characteristics, define the names, types and lengths, and definitions describing business data or business processes. Rules and policies, will define security, data cleanliness, timelines for data, and relationships. Physical characteristics define the origin or source, and physical location. Like building a logical data model for creating a database, a logical meta model can help identify the metadata requirements for business data. The metadata repository will be centralized, decentralized, or distributed. A centralized design means that there is one database for the metadata repository that stores metadata for all applications business wide. A centralized metadata repository has the same advantages and disadvantages of a centralized database. Easier to manage because all the data is in one database, but the disadvantage is that bottlenecks may occur. A decentralized metadata repository stores metadata in multiple databases, either separated by location and or departments of the business. This makes management of the repository more involved than a centraliz
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Clipmap

In computer graphics, clipmapping is a method of clipping a mipmap to a subset of data pertinent to the geometry being displayed. This is useful for loading as little data as possible when memory is limited, such as on a graphics processing unit. The technique is used for LODing in NVIDIA’s implementation of voxel cone tracing. The high-resolution levels of the mipmapped scene representation are clipped to a region near the camera, while lower resolution levels are clipped further away. == MegaTexture == MegaTexture is a clipmap implementation developed by id Software. It was introduced in their id Tech 4 engine and also appeared in id Tech 5 and id Tech 6 before being removed in id Tech 7. MegaTexture is a texture allocation technique that uses a single, extremely large texture rather than repeating multiple smaller textures. It is also featured in Splash Damage's game Enemy Territory: Quake Wars, and was developed by id Software former technical director John Carmack. MegaTexture employs a single large texture space for static terrain. The texture is stored on removable media or a computer's hard drive and streamed as needed, allowing large amounts of detail and variation over a large area with comparatively little RAM usage. Depending on the pixel resolution per square meter, covering a large area could require several gigabytes of memory. However, RAM is also filled by the rest of the game and the underlying operating system, limiting the amount available for texturing. As the player moves around the game, different sections of the MegaTexture are loaded into memory. They are then scaled to the correct size and applied to the 3D models of the terrain. Id has presented a more advanced technique that builds upon the MegaTexture idea and virtualizes both the geometry and the textures to obtain unique geometry down to the equivalent of the texel: the sparse voxel octree (SVO). It works by raycasting the geometry represented by voxels (instead of triangles) stored in an octree. The goal is to stream parts of the octree into video memory, going further down along the tree for nearby objects to give them more details, and to use higher level, larger voxels for farther objects, which give an automatic level of detail (LOD) system for both geometry and textures at the same time. The geometric detail that can be obtained using this method is nearly infinite, which removes the need for faking 3-dimensional details with techniques such as normal mapping. Despite that most voxel rendering tests use very large amounts of memory (up to several GB), Jon Olick of id Software claimed the technology is able to compress such SVO to 1.15 bits per voxel of position data. == Virtual texturing == Unlike clipmaps, which clip each mip level around a viewpoint-dependent clipcenter and therefore work best for terrain, virtual texturing preprocesses texture data into equally sized tiles that can be streamed for arbitrary textured geometry. Rage, powered by the id Tech 5 engine, uses a more advanced technique called virtual texturing. Textures can measure up to 128000×128000 pixels and are also used for in-game models and sprites, etc. and not just the terrain. Wolfenstein: The New Order and the 2016 version of Doom also use these. Carmageddon: Reincarnation also uses virtual texturing, though unlike id's virtual texturing system, which is designed for unique texture-mapping everywhere, their system is designed to use storage space sparingly while still offering good blend of texture variation and resolution.
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Lossless join decomposition

In database design, a lossless join decomposition is a decomposition of a relation r {\displaystyle r} into relations r 1 , r 2 {\displaystyle r_{1},r_{2}} such that a natural join of the two smaller relations yields back the original relation. This is central in removing redundancy safely from databases while preserving the original data. Lossless join can also be called non-additive. == Definition == A relation r {\displaystyle r} on schema R {\displaystyle R} decomposes losslessly onto schemas R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} if π R 1 ( r ) ⋈ π R 2 ( r ) = r {\displaystyle \pi _{R_{1}}(r)\bowtie \pi _{R_{2}}(r)=r} , that is r {\displaystyle r} is the natural join of its projections onto the smaller schemas. A pair ( R 1 , R 2 ) {\displaystyle (R_{1},R_{2})} is a lossless-join decomposition of R {\displaystyle R} or said to have a lossless join with respect to a set of functional dependencies F {\displaystyle F} if any relation r ( R ) {\displaystyle r(R)} that satisfies F {\displaystyle F} decomposes losslessly onto R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} . Decompositions into more than two schemas can be defined in the same way. == Criteria == A decomposition R = R 1 ∪ R 2 {\displaystyle R=R_{1}\cup R_{2}} has a lossless join with respect to F {\displaystyle F} if and only if the closure of R 1 ∩ R 2 {\displaystyle R_{1}\cap R_{2}} includes R 1 ∖ R 2 {\displaystyle R_{1}\setminus R_{2}} or R 2 ∖ R 1 {\displaystyle R_{2}\setminus R_{1}} . In other words, one of the following must hold: ( R 1 ∩ R 2 ) → ( R 1 ∖ R 2 ) ∈ F + {\displaystyle (R_{1}\cap R_{2})\to (R_{1}\setminus R_{2})\in F^{+}} ( R 1 ∩ R 2 ) → ( R 2 ∖ R 1 ) ∈ F + {\displaystyle (R_{1}\cap R_{2})\to (R_{2}\setminus R_{1})\in F^{+}} === Criteria for multiple sub-schemas === Multiple sub-schemas R 1 , R 2 , . . . , R n {\displaystyle R_{1},R_{2},...,R_{n}} have a lossless join if there is some way in which we can repeatedly perform lossless joins until all the schemas have been joined into a single schema. Once we have a new sub-schema made from a lossless join, we are not allowed to use any of its isolated sub-schema to join with any of the other schemas. For example, if we can do a lossless join on a pair of schemas R i , R j {\displaystyle R_{i},R_{j}} to form a new schema R i , j {\displaystyle R_{i,j}} , we use this new schema (rather than R i {\displaystyle R_{i}} or R j {\displaystyle R_{j}} ) to form a lossless join with another schema R k {\displaystyle R_{k}} (which may already be joined (e.g., R k , l {\displaystyle R_{k,l}} )). == Example == Let R = { A , B , C , D } {\displaystyle R=\{A,B,C,D\}} be the relation schema, with attributes A, B, C and D. Let F = { A → B C } {\displaystyle F=\{A\rightarrow BC\}} be the set of functional dependencies. Decomposition into R 1 = { A , B , C } {\displaystyle R_{1}=\{A,B,C\}} and R 2 = { A , D } {\displaystyle R_{2}=\{A,D\}} is lossless under F because R 1 ∩ R 2 = A {\displaystyle R_{1}\cap R_{2}=A} and we have a functional dependency A → B C {\displaystyle A\rightarrow BC} . In other words, we have proven that ( R 1 ∩ R 2 → R 1 ∖ R 2 ) ∈ F + {\displaystyle (R_{1}\cap R_{2}\rightarrow R_{1}\setminus R_{2})\in F^{+}} .
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