Kubity

Kubity is a cloud-based 3D communication tool that works on desktop computers, the web, smartphones, tablets, augmented reality gear, and virtual reality glasses. Kubity is powered by several proprietary 3D processing engines including "Paragone" and "Etna" that prepare the 3D file for transfer over mobile devices. Kubity has practical applications for architecture, interior design, engineering, product design, film, and video games among others. The majority of its users create 3D models using SketchUp or Autodesk Revit software. Kubity products include the Kubity web app and Kubity Go (a mobile application for iOS and Android). Kubity is compatible across many platforms, devices and operating systems including: iOS, Android, Firefox, Chrome, Windows, MacOS, and Linux. == History == Kubity was created by SPK Technology (ex Kubity S.A.S.), a Paris-based software company specializing in automatic 3D data optimization and visualization. Founded in 2012 by a group of software engineers and an urban projects developer, they united around a simple idea: create a way for anyone, anywhere to simply and intuitively explore 3D models on smartphones and computers. In order to bring architects, engineers and designers together with their clients around a 3D model, it was essential to develop an interactive platform that supported multiple desktop and mobile devices for instantaneous and fluid 3D navigation. With specifications in place, 15 engineers fused together several technologies: 3D design, data compression, decimation and rendering optimization, web and mobile transfer, and virtual reality headset integration. In January 2014, the first public Kubity prototype (1.0 Amethyst) was launched to a small group of beta testers with a plug-in that allowed users to import 3D models from SketchUp to their browser. A global release was announced in April 2014 at the SketchUp Basecamp in Vail, Colorado. In May 2015, Kubity launched a web application that worked using WebGL technology (2.0 Citrine). For the first time, users were able to drag and drop any SketchUp file in a web browser without having to install a plug-in. In December 2015, Kubity launched a mobile application on the App Store for iPhone, iPod, and iPad as well as on Google Play for Android devices (3.0 Druzy). In November 2016, Kubity launched support for Oculus Rift and HTC Vive (4.0 Emerald). Beginning in November 2017, Kubity launched a full suite rollout of mobile applications over six months that included Kubity AR for augmented reality, Kubity VR for virtual reality, and Kubity Mirror for remote presentations and screen mirroring (5.0 Feldspar). In September 2018, a one-click plugin for SketchUp and Revit (Kubity PRO), along with a mobile-first revamp of Kubity Go was launched, allowing PRO-to-Go device pairing for automatic mobile sync (6.0 Gypsum). In early 2019, the Kubity Go application was updated to include fully integrated AR, VR, and screen mirroring functionalities, killing off the dedicated companion apps Kubity AR, Kubity VR and Kubity Mirror in the process (7.0 Heliotrope). In January 2020, support for the Kubity PRO plugin for SketchUp and Revit was migrated to a SketchUp-only web app. == Technology == Kubity is powered by a proprietary 3D crystallization engine known as "Paragone"; a technology developed by SPK Technology. Paragone takes constrained information from a 3D file and runs it through the "BlockWave" algorithm (US Patent 10,482.629), also developed by SPK Technology. BlockWave is a multiphase optimization algorithm that combines 3D design, data compression, decimation and rendering optimization, web and mobile transfer, and mixed reality headset integration to create a crystallized universal format of the original file. One phase of the BlockWave algorithm is based on the quadric-based polygonal surface simplification algorithm, performed using predefined heuristics, and is associated with a plurality of simplified versions of the 3D model, each version being associated with a predefined level of detail adapted to the user specific end device. BlockWave extracts data content, geometry and textures, then sets quadrics for each top of the original 3D model, and identifies pairs of adjacent tops linked by vertices. The algorithm uses a local collapsing operator and a top-plan error metric to obtain a fixed number of faces or a maximum defined error; 3D meshing is simplified by replacing two points with one, then deleting the degrading faces and updating adjacent relations. Once decimation is completed, texture optimization is set using texture target parameters allowing maximized GPU memory to improve computing time. With texture encoding completed, the crystallized universal 3D file can now be easily opened on any user-specific end device and played across most digital devices with real-time rendering. == Features == === 3D Crystallization === A user converts (or crystallizes) a 3D file by exporting it with the Kubity web app. Crystallization adds features like AR/VR and cinematic fly-through tour as well as assigns the model a dedicated QR code. === Automatic Mobile Sync === When a 3D model is exported, it is automatically synced to Kubity Go on the user's mobile device. From there, it can be accessed, explored, and shared with others with or without an internet connection. === Security and Management === User models can be managed all in one place on Kubity Go or in a browser from their account. Models can be renamed, password-protected, shared, and played. === Augmented Reality === Developed using Apple ARKit and Google ARCore technology, Kubity Go's augmented reality feature maps the environment in a room detecting horizontal planes like tables and floors to track and place 3D objects. By blending digital objects and information with the environment, Kubity allows users to interact with 3D models in true augmented reality. Built-in communication features allows users to instantly share 3D models with anyone over text, email, social media, or direct device-to-device with a QR Code. Platform Support AR supports devices running iOS11 including: iPhone SE, iPhone 6s, iPhone 6s Plus, iPhone 7, iPhone 7 Plus, iPhone 8, iPhone X, all iPad Pro models, and iPad (2017). AR for Android requires Android 7.0 or later and access to the Google Play Store. === Virtual Reality === VR allows users to explore SketchUp models and Revit projects on-the-go right from a mobile device using Oculus Go, Google Cardboard, Samsung Gear VR, or the glasses-free Magic Window feature. Kubity's virtual reality feature is compatible with Oculus Go, Google Cardboard viewers and other cardboard compatible devices including clip-on style VR glasses like Homido Mini, as well as the mobile virtual reality headset, Samsung Gear VR. Samsung Gear VR supports: Galaxy S6, Galaxy S6 Edge, Galaxy S6 Edge+, Samsung Galaxy Note 5, Galaxy S7, Galaxy S7 Edge, Galaxy S8, Galaxy S8+, Samsung Galaxy Note Fan Edition, Samsung Galaxy Note 8, Samsung Galaxy A8/A8+ (2018), and Samsung Galaxy S9/Galaxy S9+. === Screen Mirroring === Screen mirroring allows a user to sync the sender device to a receiver on a webpage, then control from the sender device to give a remote presentation of the 3D model. Devices are easily synced by entering a six-digit number displayed on the receiving computer. == Platform support == On iOS, the Kubity application is compatible with devices running on the version 9.0 or higher. On Android, Kubity is compatible with devices running on the version 4.4 “Kit Kat” or higher. The web version of Kubity applications currently support web browsers compatible with WebGL2 : Mozilla Firefox and Google Chrome. AR is compatible with devices running iOS11 including: iPhone SE, iPhone 6s, iPhone 6s Plus, iPhone 7, iPhone 7 Plus, iPhone 8, iPhone X, all iPad Pro models, and iPad (2017), and Android devices. Requires Android 7.0 or later and access to the Google Play Store. VR is compatible with Google Cardboard viewers and other cardboard compatible devices including clip-on style VR glasses like Homido Mini, as well as the Samsung Gear VR and Oculus Go. Samsung Gear VR supports: Galaxy S6, Galaxy S6 Edge, Galaxy S6 Edge+, Samsung Galaxy Note 5, Galaxy S7, Galaxy S7 Edge, Galaxy S8, Galaxy S8+, Samsung Galaxy Note Fan Edition, Samsung Galaxy Note 8, Samsung Galaxy A8/A8+ (2018) and Samsung Galaxy S9/Galaxy S9+.

CrewAI

CrewAI is an open-source software framework and platform for building AI agents and multi-agent systems. Written primarily in Python, it is used to define artificial-intelligence agents, assign tasks to them, and coordinate their work through agent teams and workflows. The framework is associated with CrewAI Inc., a startup developing enterprise tools for automating business workflows with large language model-based agents. == History == CrewAI was first released on the Python Package Index in December 2023. The project was created by João Moura and later developed by CrewAI Inc. and open-source contributors. In October 2024, TechCrunch reported that CrewAI had raised $18 million across seed and Series A funding rounds from investors including Boldstart Ventures, Craft Ventures, Earl Grey Capital, and Insight Partners. The report also stated that Andrew Ng and HubSpot co-founder Dharmesh Shah had invested in the company. SiliconANGLE described the company as the developer of an open-source framework for building artificial-intelligence agents and reported that the funding consisted of a seed round led by Boldstart Ventures and a Series A led by Insight Partners. By late 2024, CrewAI had introduced commercial enterprise products built on top of its open-source components. TechCrunch reported that the company's enterprise offering added access controls, analytics, support, and templates for workflow automation. == Features == CrewAI is designed around groups of agents, sometimes called "crews", that can be assigned roles, goals, and tasks. The framework supports agent collaboration, task delegation, tool use, memory, and knowledge sources for retrieval-augmented generation workflows. The project describes two main building blocks: "Crews", which are used for autonomous agent collaboration, and "Flows", which are used for more controlled event-driven workflows. The framework is independent of LangChain and is released under the MIT License. It can be installed as a Python package and is commonly used with external large language model APIs or local models, depending on the developer's configuration. == Business model == CrewAI combines an open-source framework with commercial enterprise products. Its enterprise products are intended for organizations that need to build, monitor, and manage agent-based automations with additional security, observability, and administrative controls.

NLWeb

Natural Language Web or NLWeb was introduced by Microsoft in 2025. It is an open Python project designed to simplify the creation of natural language interfaces for websites. It enables users to query website contents using natural language, similar to interacting with an AI assistant. Every instance functions as a Model Context Protocol (MCP) server allowing websites to make their content discoverable and accessible to AI agents and other participants. NLWeb leverages existing web standards like Schema.org and RSS to build conversational capabilities of processing user queries through language models, performing semantic searches against website content and generating natural responses. It is platform-agnostic, running on all major systems and connecting to any vector database. Content to be indexed by NLWeb works best when it is organized in an AI friendly way. This means short, interlinked and semantically annotated articles work best. Initial adopters of NLWeb include TripAdvisor, Shopify, Eventbrite, and Hearst.

AlphaGo

AlphaGo is a computer program that plays the board game Go. It was developed by the London-based DeepMind Technologies, an acquired subsidiary of Google. Subsequent versions of AlphaGo became increasingly powerful, including a version that competed under the name Master. After retiring from competitive play, AlphaGo Master was succeeded by an even more powerful version known as AlphaGo Zero, which was completely self-taught without learning from human games. AlphaGo Zero was then generalized into a program known as AlphaZero, which played additional games, including chess and shogi. AlphaZero has in turn been succeeded by a program known as MuZero which learns without being taught the rules. AlphaGo and its successors use a Monte Carlo tree search algorithm to find its moves based on knowledge previously acquired by machine learning, specifically by an artificial neural network (a deep learning method) by extensive training, both from human and computer play. A neural network is trained to identify the best moves and the winning percentages of these moves. This neural network improves the strength of the tree search, resulting in stronger move selection in the next iteration. In October 2015, in a match against Fan Hui, the original AlphaGo became the first computer Go program to beat a human professional Go player without handicap on a full-sized 19×19 board. In March 2016, it beat Lee Sedol in a five-game match, the first time a computer Go program has beaten a 9-dan professional without handicap. Although it lost to Lee Sedol in the fourth game, Lee resigned in the final game, giving a final score of 4 games to 1 in favour of AlphaGo. In recognition of the victory, AlphaGo was awarded an honorary 9-dan by the Korea Baduk Association. The lead up and the challenge match with Lee Sedol were documented in a documentary film also titled AlphaGo, directed by Greg Kohs. The win by AlphaGo was chosen by Science as one of the Breakthrough of the Year runners-up on 22 December 2016. At the 2017 Future of Go Summit, the Master version of AlphaGo beat Ke Jie, the number one ranked player in the world at the time, in a three-game match, after which AlphaGo was awarded professional 9-dan by the Chinese Weiqi Association. After the match between AlphaGo and Ke Jie, DeepMind retired AlphaGo, while continuing AI research in other areas. The self-taught AlphaGo Zero achieved a 100–0 victory against the early competitive version of AlphaGo, and its successor AlphaZero was perceived as the world's top player in Go by the end of the 2010s. == History == Go is considered much more difficult for computers to win than other games such as chess, because its strategic and aesthetic nature makes it hard to directly construct an evaluation function, and its much larger branching factor makes it prohibitively difficult to use traditional AI methods such as alpha–beta pruning, tree traversal and heuristic search. Almost two decades after IBM's computer Deep Blue beat world chess champion Garry Kasparov in the 1997 match, the strongest Go programs using artificial intelligence techniques only reached about amateur 5-dan level, and still could not beat a professional Go player without a handicap. In 2012, the software program Zen, running on a four PC cluster, beat Masaki Takemiya (9p) twice at five- and four-stone handicaps. In 2013, Crazy Stone beat Yoshio Ishida (9p) at a four-stone handicap. According to DeepMind's David Silver, the AlphaGo research project was formed around 2014 to test how well a neural network using deep learning can compete at Go. AlphaGo represents a significant improvement over previous Go programs. In 500 games against other available Go programs, including Crazy Stone and Zen, AlphaGo running on a single computer won all but one. In a similar matchup, AlphaGo running on multiple computers won all 500 games played against other Go programs, and 77% of games played against AlphaGo running on a single computer. The distributed version in October 2015 was using 1,202 CPUs and 176 GPUs. === Match against Fan Hui === In October 2015, the distributed version of AlphaGo defeated the European Go champion Fan Hui, a 2-dan (out of 9 dan possible) professional, five to zero. This was the first time a computer Go program had beaten a professional human player on a full-sized board without handicap. The announcement of the news was delayed until 27 January 2016 to coincide with the publication of a paper in the journal Nature describing the algorithms used. === Match against Lee Sedol === AlphaGo played South Korean professional Go player Lee Sedol, ranked 9-dan, one of the best players at Go, with five games taking place at the Four Seasons Hotel in Seoul, South Korea on 9, 10, 12, 13, and 15 March 2016, which were video-streamed live. Out of five games, AlphaGo won four games and Lee won the fourth game which made him recorded as the only human player who beat AlphaGo in all of its 74 official games. AlphaGo ran on Google's cloud computing with its servers located in the United States. The match used Chinese rules with a 7.5-point komi, and each side had two hours of thinking time plus three 60-second byoyomi periods. The version of AlphaGo playing against Lee used a similar amount of computing power as was used in the Fan Hui match. The Economist reported that it used 1,920 CPUs and 280 GPUs. At the time of play, Lee Sedol had the second-highest number of Go international championship victories in the world after South Korean player Lee Chang-ho who kept the world championship title for 16 years. Since there is no single official method of ranking in international Go, the rankings may vary among the sources. While he was ranked top sometimes, some sources ranked Lee Sedol as the fourth-best player in the world at the time. AlphaGo was not specifically trained to face Lee nor was designed to compete with any specific human players. The first three games were won by AlphaGo following resignations by Lee. However, Lee beat AlphaGo in the fourth game, winning by resignation at move 180. AlphaGo then continued to achieve a fourth win, winning the fifth game by resignation. The prize was US$1 million. Since AlphaGo won four out of five and thus the series, the prize will be donated to charities, including UNICEF. Lee Sedol received $150,000 for participating in all five games and an additional $20,000 for his win in Game 4. In June 2016, at a presentation held at a university in the Netherlands, Aja Huang, one of the Deep Mind team, revealed that they had patched the logical weakness that occurred during the 4th game of the match between AlphaGo and Lee, and that after move 78 (which was dubbed the "divine move" by many professionals), it would play as intended and maintain Black's advantage. Before move 78, AlphaGo was leading throughout the game, but Lee's move caused the program's computing powers to be diverted and confused. Huang explained that AlphaGo's policy network of finding the most accurate move order and continuation did not precisely guide AlphaGo to make the correct continuation after move 78, since its value network did not determine Lee's 78th move as being the most likely, and therefore when the move was made AlphaGo could not make the right adjustment to the logical continuation. === Sixty online games === On 29 December 2016, a new account on the Tygem server named "Magister" (shown as 'Magist' at the server's Chinese version) from South Korea began to play games with professional players. It changed its account name to "Master" on 30 December, then moved to the FoxGo server on 1 January 2017. On 4 January, DeepMind confirmed that the "Magister" and the "Master" were both played by an updated version of AlphaGo, called AlphaGo Master. As of 5 January 2017, AlphaGo Master's online record was 60 wins and 0 losses, including three victories over Go's top-ranked player, Ke Jie, who had been quietly briefed in advance that Master was a version of AlphaGo. After losing to Master, Gu Li offered a bounty of 100,000 yuan (US$14,400) to the first human player who could defeat Master. Master played at the pace of 10 games per day. Many quickly suspected it to be an AI player due to little or no resting between games. Its adversaries included many world champions such as Ke Jie, Park Jeong-hwan, Yuta Iyama, Tuo Jiaxi, Mi Yuting, Shi Yue, Chen Yaoye, Li Qincheng, Gu Li, Chang Hao, Tang Weixing, Fan Tingyu, Zhou Ruiyang, Jiang Weijie, Chou Chun-hsun, Kim Ji-seok, Kang Dong-yun, Park Yeong-hun, and Won Seong-jin; national champions or world championship runners-up such as Lian Xiao, Tan Xiao, Meng Tailing, Dang Yifei, Huang Yunsong, Yang Dingxin, Gu Zihao, Shin Jinseo, Cho Han-seung, and An Sungjoon. All 60 games except one were fast-paced games with three 20 or 30 seconds byo-yomi. Master offered to extend the byo-yomi to one minute when playing with Nie Weiping in consideration of his age. After winning its 59th game Master revealed itse

The Machine Question

The Machine Question: Critical Perspectives on AI, Robots, and Ethics is a 2012 nonfiction book by David J. Gunkel that discusses the evolution of the theory of human ethical responsibilities toward non-human things and to what extent intelligent, autonomous machines can be considered to have legitimate moral responsibilities and what legitimate claims to moral consideration they can hold. The book was awarded as the 2012 Best Single Authored Book by the Communication Ethics Division of the National Communication Association. == Content == The book is spread across three chapters, with the first two chapters focusing on an overall review of the history of philosophy and its discussion of moral agency, moral rights, human rights, and animal rights and the third chapter focusing on what defines "thingness" and why machines have been excluded from moral and ethical consideration due to a misuse of the patient/agent binary. The first chapter, titled Moral Agency, breaks down the history of said agency based on what it included and excluded in various parts of history. Gunkel also raises the conflict between discussing the morality of humans toward objects and the theory of the philosophy of technology that "technology is merely a tool: a means to an end". The main issue, he explains, in defining what constitutes an appropriate moral agent is that there will be things left outside of what is included, as the definition is based on a set of characteristics that will inherently not be all-encompassing. The subject of consciousness is broached and subsequently derided by Gunkel because of it being one of the main arguments against machine rights, while Gunkel points out that no "settled definition" of the term exists and that he considers it no better than a synonym used for "the occultish soul". In addition, the issue of the other minds problem entails that no proper understanding of consciousness can come to pass due to the inability to properly understand the mind of a being that is not oneself. The second chapter, titled Moral Patiency, focuses on the patient end of the topic and discusses the expansion of the fields of animal studies and environmental studies. Gunkel describes moral patients as the ones that are to be the object of moral consideration and deserve such consideration even if they lack their own agency, such as animals, thus allowing moral consideration itself to be broader and more inclusive. The topic of other minds is discussed again when examining the question of whether animals can suffer, a question that Gunkel ultimately abandons because it encounters the same problems that the topic of consciousness does. Especially because the subject of animal rights is often only afforded for the animals deemed to be "cute", but often not including "reptiles, insects, or microbes". Gunkel continues on to examine environmental ethics and information ethics, but finds them to be too anthropocentric, just as all the other examined models have been. The third chapter, titled Thinking Otherwise, proposes a combination of Heideggerian ontology and Levinasian ethics to properly discuss the otherness of technology and machines, but finds that the patient/agent binary is unable to be properly extended to confine the extent of "the machine question". In discussing the land ethic philosophy espoused by Aldo Leopold, Gunkel proposes that it is the entire relationship between agent and patient that should have moral consideration and not a specific definition based on either side, as each part contributes to the relationship as a whole and cannot be removed without breaking that relationship. == Critical reception == Choice: Current Reviews for Academic Libraries writer R. S. Stansbury explained that the book is able to use simple examples to discuss difficult topics and separate ideas and that it would be "useful for philosophy students, and for engineering students interested in exploring the ethical implications of their work". Dominika Dzwonkowska, writing for International Philosophical Quarterly, stated that the "unprecedented value of the book is that Gunkel not only analyzes important aspects of the immediate problem but also that he places his discussion in the context of philosophical discussions on such related issues as rights discourse." Mark Coeckelbergh in Ethics and Information Technology noted that focusing on the question itself of the machine question allows further exploration of machine ethics and the expansion of general ethics and that the book's questions point out that "good, critical philosophical reflection on machines is not only about how we should cope with machines, but also about how we (should) think and what role technology plays (and should play) in this thinking." A review in Notre Dame Philosophical Reviews by Colin Allen criticized some of Gunkel's methodology and the indecisiveness of his ultimate answer to the machine question, but also acknowledged that the book "succeeded in connecting the ethics of robots and AI to a much broader ethical discussion than has been represented in the literature on machine ethics to date". Blay Whitby, in a review for AISB Quarterly, lauded The Machine Question for its "clear exposition" and wide range of references to other works, concluding that the book is "essential reading for philosophers interested in AI, robot ethics, or animal ethics". In a twin review of The Machine Question and Robot Ethics: The Ethical and Social Implications of Robots by Patrick Lin, Keith Abney, and George A. Bekey, Techné: Research in Philosophy and Technology reviewer Jeff Shaw called Gunkel's book a good introduction to the "complex field of robot ethics" and that both books are "highly recommended to both the general reader as well as to experts in the field of robotics, philosophy, and ethics." In a 2017 paper for Ethics and Information Technology, Katharyn Hogan investigated whether the machine question presented by Gunkel in the book is any different from the longstanding animal question. She concludes that the real question that is revealed from this discussion is whether humans deserve any moral preference over artificial life in the first place.

SGT STAR

SGT STAR, also known as Sgt. Star or Sergeant Star, was a chatbot operated by the United States Army to answer questions about recruitment. == Background == After the September 11 attacks, traffic increased significantly to chatrooms on the U.S. Army's website, goarmy.com, increasing costs of staffing the live chatrooms. As a cost-cutting measure, the SGT STAR project was initiated as a partnership between the United States Army Accessions Command and Spectre AI, a wholly owned subsidiary of Next IT. Next IT, a Spokane, Washington-based company deploys "intelligent virtual assistants," using its software dubbed "ActiveAgent" which is a framework for functional presence engines. Testing began in 2003, and SGT STAR launched to the public in 2006. "STAR" is an acronym for "strong, trained and ready." SGT STAR was launched as a chat interface on goarmy.com, but has since been developed as a mobile application, as well as a life-size animated projection that has appeared live at public events. SGT STAR can also interact with users on Facebook. == FOIA request == In 2013, the Electronic Frontier Foundation filed a Freedom of Information Act request to learn more about SGT STAR, including input and output patterns (questions and answers), usage statistics, contracts, and privacy policies. They received these records in April 2014, after coverage from various media outlets and a tongue-in-cheek campaign to "Free Sgt. Star."

Linear belief function

Linear belief functions are an extension of the Dempster–Shafer theory of belief functions to the case when variables of interest are continuous. Examples of such variables include financial asset prices, portfolio performance, and other antecedent and consequent variables. The theory was originally proposed by Arthur P. Dempster in the context of Kalman Filters and later was elaborated, refined, and applied to knowledge representation in artificial intelligence and decision making in finance and accounting by Liping Liu. == Concept == A linear belief function intends to represent our belief regarding the location of the true value as follows: We are certain that the truth is on a so-called certainty hyperplane but we do not know its exact location; along some dimensions of the certainty hyperplane, we believe the true value could be anywhere from –∞ to +∞ and the probability of being at a particular location is described by a normal distribution; along other dimensions, our knowledge is vacuous, i.e., the true value is somewhere from –∞ to +∞ but the associated probability is unknown. A belief function in general is defined by a mass function over a class of focal elements, which may have nonempty intersections. A linear belief function is a special type of belief function in the sense that its focal elements are exclusive, parallel sub-hyperplanes over the certainty hyperplane and its mass function is a normal distribution across the sub-hyperplanes. Based on the above geometrical description, Shafer and Liu propose two mathematical representations of a LBF: a wide-sense inner product and a linear functional in the variable space, and as their duals over a hyperplane in the sample space. Monney proposes still another structure called Gaussian hints. Although these representations are mathematically neat, they tend to be unsuitable for knowledge representation in expert systems. == Knowledge representation == A linear belief function can represent both logical and probabilistic knowledge for three types of variables: deterministic such as an observable or controllable, random whose distribution is normal, and vacuous on which no knowledge bears. Logical knowledge is represented by linear equations, or geometrically, a certainty hyperplane. Probabilistic knowledge is represented by a normal distribution across all parallel focal elements. In general, assume X is a vector of multiple normal variables with mean μ and covariance Σ. Then, the multivariate normal distribution can be equivalently represented as a moment matrix: M ( X ) = ( μ Σ ) . {\displaystyle M(X)=\left({\begin{array}{{20}c}\mu \\\Sigma \end{array}}\right).} If the distribution is non-degenerate, i.e., Σ has a full rank and its inverse exists, the moment matrix can be fully swept: M ( X → ) = ( μ Σ − 1 − Σ − 1 ) {\displaystyle M({\vec {X}})=\left({\begin{array}{{20}c}\mu \Sigma ^{-1}\\-\Sigma ^{-1}\end{array}}\right)} Except for normalization constant, the above equation completely determines the normal density function for X. Therefore, M ( X → ) {\displaystyle M({\vec {X}})} represents the probability distribution of X in the potential form. These two simple matrices allow us to represent three special cases of linear belief functions. First, for an ordinary normal probability distribution M(X) represents it. Second, suppose one makes a direct observation on X and obtains a value μ. In this case, since there is no uncertainty, both variance and covariance vanish, i.e., Σ = 0. Thus, a direct observation can be represented as: M ( X ) = ( μ 0 ) {\displaystyle M(X)=\left({\begin{array}{{20}c}\mu \\0\end{array}}\right)} Third, suppose one is completely ignorant about X. This is a very thorny case in Bayesian statistics since the density function does not exist. By using the fully swept moment matrix, we represent the vacuous linear belief functions as a zero matrix in the swept form follows: M ( X → ) = [ 0 0 ] {\displaystyle M({\vec {X}})=\left[{\begin{array}{{20}c}0\\0\end{array}}\right]} One way to understand the representation is to imagine complete ignorance as the limiting case when the variance of X approaches to ∞, where one can show that Σ−1 = 0 and hence M ( X → ) {\displaystyle M({\vec {X}})} vanishes. However, the above equation is not the same as an improper prior or normal distribution with infinite variance. In fact, it does not correspond to any unique probability distribution. For this reason, a better way is to understand the vacuous linear belief functions as the neutral element for combination (see later). To represent the remaining three special cases, we need the concept of partial sweeping. Unlike a full sweeping, a partial sweeping is a transformation on a subset of variables. Suppose X and Y are two vectors of normal variables with the joint moment matrix: M ( X , Y ) = [ μ 1 Σ 11 Σ 21 μ 2 Σ 12 Σ 22 ] {\displaystyle M(X,Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}\mu _{1}\\\Sigma _{11}\\\Sigma _{21}\end{array}}&{\begin{array}{{20}c}\mu _{2}\\\Sigma _{12}\\\Sigma _{22}\end{array}}\end{array}}\right]} Then M(X, Y) may be partially swept. For example, we can define the partial sweeping on X as follows: M ( X → , Y ) = [ μ 1 ( Σ 11 ) − 1 − ( Σ 11 ) − 1 Σ 21 ( Σ 11 ) − 1 μ 2 − μ 1 ( Σ 11 ) − 1 Σ 12 ( Σ 11 ) − 1 Σ 12 Σ 22 − Σ 21 ( Σ 11 ) − 1 Σ 12 ] {\displaystyle M({\vec {X}},Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}\mu _{1}(\Sigma _{11})^{-1}\\-(\Sigma _{11})^{-1}\\\Sigma _{21}(\Sigma _{11})^{-1}\end{array}}&{\begin{array}{{20}c}\mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}\\(\Sigma _{11})^{-1}\Sigma _{12}\\\Sigma _{22}-\Sigma _{21}(\Sigma _{11})^{-1}\Sigma _{12}\end{array}}\end{array}}\right]} If X is one-dimensional, a partial sweeping replaces the variance of X by its negative inverse and multiplies the inverse with other elements. If X is multidimensional, the operation involves the inverse of the covariance matrix of X and other multiplications. A swept matrix obtained from a partial sweeping on a subset of variables can be equivalently obtained by a sequence of partial sweepings on each individual variable in the subset and the order of the sequence does not matter. Similarly, a fully swept matrix is the result of partial sweepings on all variables. We can make two observations. First, after the partial sweeping on X, the mean vector and covariance matrix of X are respectively μ 1 ( Σ 11 ) − 1 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}} and − ( Σ 11 ) − 1 {\displaystyle -(\Sigma _{11})^{-1}} , which are the same as that of a full sweeping of the marginal moment matrix of X. Thus, the elements corresponding to X in the above partial sweeping equation represent the marginal distribution of X in potential form. Second, according to statistics, μ 2 − μ 1 ( Σ 11 ) − 1 Σ 12 {\displaystyle \mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional mean of Y given X = 0; Σ 22 − Σ 21 ( Σ 11 ) − 1 Σ 12 {\displaystyle \Sigma _{22}-\Sigma _{21}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional covariance matrix of Y given X = 0; and ( Σ 11 ) − 1 Σ 12 {\displaystyle (\Sigma _{11})^{-1}\Sigma _{12}} is the slope of the regression model of Y on X. Therefore, the elements corresponding to Y indices and the intersection of X and Y in M ( X → , Y ) {\displaystyle M({\vec {X}},Y)} represents the conditional distribution of Y given X = 0. These semantics render the partial sweeping operation a useful method for manipulating multivariate normal distributions. They also form the basis of the moment matrix representations for the three remaining important cases of linear belief functions, including proper belief functions, linear equations, and linear regression models. === Proper linear belief functions === For variables X and Y, assume there exists a piece of evidence justifying a normal distribution for variables Y while bearing no opinions for variables X. Also, assume that X and Y are not perfectly linearly related, i.e., their correlation is less than 1. This case involves a mix of an ordinary normal distribution for Y and a vacuous belief function for X. Thus, we represent it using a partially swept matrix as follows: M ( X → , Y ) = [ 0 0 0 μ 2 0 Σ 22 ] {\displaystyle M({\vec {X}},Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}0\\0\\0\end{array}}&{\begin{array}{{20}c}\mu _{2}\\0\\\Sigma _{22}\\\end{array}}\end{array}}\right]} This is how we could understand the representation. Since we are ignorant on X, we use its swept form and set μ 1 ( Σ 11 ) − 1 = 0 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}=0} and − ( Σ 11 ) − 1 = 0 {\displaystyle -(\Sigma _{11})^{-1}=0} . Since the correlation between X and Y is less than 1, the regression coefficient of X on Y approaches to 0 when the variance of X approaches to ∞. Therefore, ( Σ 11 ) − 1 Σ 12 = 0 {\displaystyle (\Sigma _{11})^{-1}\Sigma _{12}=0} . Similarly, one can prove that μ 1 ( Σ 11 ) − 1 Σ 12 = 0 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}=0} and Σ 21 ( Σ 11 ) −