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  Confused deputy problem

  In information security, a confused deputy is a computer program that is tricked by another program (with fewer privileges or less rights) into misusing its authority on the system. It is a specific type of privilege escalation. The confused deputy problem is often cited as an example of why capability-based security is important. Capability systems protect against the confused deputy problem, whereas access-control list–based systems do not. Such systems can mitigate the confused deputy problem by eliminating ambient authority, allowing programs to act only on resources for which they hold explicit capabilities, whereas access-control list–based systems are more susceptible to it. However, this protection depends on correct implementation; in formally verified capability systems such as seL4, it can be shown that the kernel enforces capability constraints correctly, preventing such behavior at the system level. == Example == In the original example of a confused deputy, there was a compiler program provided on a commercial timesharing service. Users could run the compiler and optionally specify a filename where it would write debugging output, and the compiler would be able to write to that file if the user had permission to write there. The compiler also collected statistics about language feature usage. Those statistics were stored in a file called "(SYSX)STAT", in the directory "SYSX". To make this possible, the compiler program was given permission to write to files in SYSX. But there were other files in SYSX: in particular, the system's billing information was stored in a file "(SYSX)BILL". A user ran the compiler and named "(SYSX)BILL" as the desired debugging output file. This produced a confused deputy problem. The compiler made a request to the operating system to open (SYSX)BILL. Even though the user did not have access to that file, the compiler did, so the open succeeded. The compiler wrote the compilation output to the file (here "(SYSX)BILL") as normal, overwriting it, and the billing information was destroyed. === The confused deputy === In this example, the compiler program is the deputy because it is acting at the request of the user. The program is seen as 'confused' because it was tricked into overwriting the system's billing file. Whenever a program tries to access a file, the operating system needs to know two things: which file the program is asking for, and whether the program has permission to access the file. In the example, the file is designated by its name, “(SYSX)BILL”. The program receives the file name from the user, but does not know whether the user had permission to write the file. When the program opens the file, the system uses the program's permission, not the user's. When the file name was passed from the user to the program, the permission did not go along with it; the permission was increased by the system silently and automatically. It is not essential to the attack that the billing file be designated by a name represented as a string. The essential points are that: the designator for the file does not carry the full authority needed to access the file; the program's own permission to access the file is used implicitly. == Other examples == A cross-site request forgery (CSRF) is an example of a confused deputy attack that uses the web browser to perform sensitive actions against a web application. A common form of this attack occurs when a web application uses a cookie to authenticate all requests transmitted by a browser. Using JavaScript, an attacker can force a browser into transmitting authenticated HTTP requests. The Samy computer worm used cross-site scripting (XSS) to turn the browser's authenticated MySpace session into a confused deputy. Using XSS the worm forced the browser into posting an executable copy of the worm as a MySpace message which was then viewed and executed by friends of the infected user. Clickjacking is an attack where the user acts as the confused deputy. In this attack a user thinks they are harmlessly browsing a website (an attacker-controlled website) but they are in fact tricked into performing sensitive actions on another website. An FTP bounce attack can allow an attacker to connect indirectly to TCP ports to which the attacker's machine has no access, using a remote FTP server as the confused deputy. Another example relates to personal firewall software. It can restrict Internet access for specific applications. Some applications circumvent this by starting a browser with instructions to access a specific URL. The browser has authority to open a network connection, even though the application does not. Firewall software can attempt to address this by prompting the user in cases where one program starts another which then accesses the network. However, the user frequently does not have sufficient information to determine whether such an access is legitimate—false positives are common, and there is a substantial risk that even sophisticated users will become habituated to clicking "OK" to these prompts. Not every program that misuses authority is a confused deputy. Sometimes misuse of authority is simply a result of a program error. The confused deputy problem occurs when the designation of an object is passed from one program to another, and the associated permission changes unintentionally, without any explicit action by either party. It is insidious because neither party did anything explicit to change the authority. Another example is when an administrator authorizes an AI agent to act on their behalf, and that AI subsequently delegates authority to another AI agent neither vetted nor authorized by the original administrator. The unvetted AI can then act without permissions or oversight from the original developer. == Solutions == In some systems it is possible to ask the operating system to open a file using the permissions of another client. This solution has some drawbacks: It requires explicit attention to security by the server. A naive or careless server might not take this extra step. It becomes more difficult to identify the correct permission if the server is in turn the client of another service and wants to pass along access to the file. It requires the client to trust the server to not abuse the borrowed permissions. Note that intersecting the server and client's permissions does not solve the problem either, because the server may then have to be given very wide permissions (all of the time, rather than those needed for a given request) in order to act for arbitrary clients. The simplest way to solve the confused deputy problem is to bundle together the designation of an object and the permission to access that object. This is exactly what a capability is. Using capability security in the compiler example, the client would pass to the server a capability to the output file, such as a file descriptor, rather than the name of the file. Since it lacks a capability to the billing file, it cannot designate that file for output. In the cross-site request forgery example, a URL supplied "cross"-site would include its own authority independent of that of the client of the web browser.

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  Zesta

  Zesta is an online food ordering and delivery platform operating across the African region. Formerly known as Square Eats, the company rebranded to Zesta in 2025. Zesta connects customers with restaurants and stores, offering delivery services for food, groceries, parcel delivery and other essentials. == History == Zesta was originally founded as Square Eats in 2020 by twin brothers Henry Newman and Randall Newman when they were 21 years old. It was launched in Gaborone, Botswana, and quickly gained traction as a leading food delivery service in the country. The company halted operations and took a strategic decision to reinvent the business in 2022. In 2025, the company announced its rebranding to Zesta, highlighting its commitment to evolving beyond food delivery to become a super app. === COVID-19 initiative === During the COVID-19 pandemic, Zesta (then Square Eats) implemented measures to ensure safety and hygiene, including providing free gloves and hand sanitizer to drivers and introducing contactless delivery options. These efforts positioned the platform as a trusted service during the pandemic. == Service == Zesta facilitates delivery from a wide range of merchant partners via a smartphone app, available on iOS and Android platforms, or through its website. Customers can browse their favorite restaurants, place orders, and have meals delivered to their doorstep efficiently.

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  Swizzling (computer graphics)

  In computer graphics, swizzles are a class of operations that transform vectors by rearranging components. Swizzles can also project from a vector of one dimensionality to a vector of another dimensionality, such as taking a three-dimensional vector and creating a two-dimensional or five-dimensional vector using components from the original vector. For example, if A = {1,2,3,4}, where the components are x, y, z, and w respectively, one could compute B = A.wwxy, whereupon B would equal {4,4,1,2}. Additionally, one could create a two-dimensional vector with A.wx or a five-dimensional vector with A.xyzwx. Combining vectors and swizzling can be employed in various ways. This is common in GPGPU applications. In terms of linear algebra, this is equivalent to multiplying by a matrix whose rows are standard basis vectors. If A = ( 1 , 2 , 3 , 4 ) T {\displaystyle A=(1,2,3,4)^{T}} , then swizzling A {\displaystyle A} as above looks like A . w w x y = [ 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 ] [ 1 2 3 4 ] = [ 4 4 1 2 ] . {\displaystyle A.\!wwxy={\begin{bmatrix}0&0&0&1\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{bmatrix}}{\begin{bmatrix}1\\2\\3\\4\end{bmatrix}}={\begin{bmatrix}4\\4\\1\\2\end{bmatrix}}.}

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  Morphing

  Morphing is a special effect in motion pictures and animations that changes (or morphs) one image or shape into another through a seamless transition. Traditionally such a depiction would be achieved through dissolving techniques on film. Since the early 1990s, this has been replaced by computer software to create more realistic transitions. A similar method is applied to audio recordings, for example, by changing voices or vocal lines. == Early transformation techniques == Long before digital morphing, several techniques were used for similar image transformations. Some of those techniques are closer to a matched dissolve – a gradual change between two pictures without warping the shapes in the images – while others did change the shapes in between the start and end phases of the transformation. === Tabula scalata === Known since at least the end of the 16th century, Tabula scalata is a type of painting with two images divided over a corrugated surface. Each image is only correctly visible from a certain angle. If the pictures are matched properly, a primitive type of morphing effect occurs when changing from one viewing angle to the other. === Mechanical transformations === Around 1790 French shadow play showman François Dominique Séraphin used a metal shadow figure with jointed parts to have the face of a young woman changing into that of a witch. Some 19th century mechanical magic lantern slides produced changes to the appearance of figures. For instance a nose could grow to enormous size, simply by slowly sliding away a piece of glass with black paint that masked part of another glass plate with the picture. === Matched dissolves === In the first half of the 19th century "dissolving views" were a popular type of magic lantern show, mostly showing landscapes gradually dissolving from a day to night version or from summer to winter. Other uses are known, for instance Henry Langdon Childe showed groves transforming into cathedrals. The 1910 short film Narren-grappen shows a dissolve transformation of the clothing of a female character. Maurice Tourneur's 1915 film Alias Jimmy Valentine featured a subtle dissolve transformation of the main character from respected citizen Lee Randall into his criminal alter ego Jimmy Valentine. The Peter Tchaikovsky Story in a 1959 TV-series episode of Disneyland features a swan automaton transforming into a real ballet dancer. In 1985, Godley & Creme created a "morph" effect using analogue cross-fades on parts of different faces in the video for "Cry". === Animation === In animation, the morphing effect was created long before the introduction of cinema. A phenakistiscope designed by its inventor Joseph Plateau was printed around 1835 and shows the head of a woman changing into a witch and then into a monster. Émile Cohl's 1908 animated film Fantasmagorie featured much morphing of characters and objects drawn in simple outlines. == Digital morphing == In the early 1990s, computer techniques capable of more convincing results saw increasing use. These involved distorting one image at the same time that it faded into another through marking corresponding points and vectors on the "before" and "after" images used in the morph. For example, one would morph one face into another by marking key points on the first face, such as the contour of the nose or location of an eye, and mark where these same points existed on the second face. The computer would then distort the first face to have the shape of the second face at the same time that it faded the two faces. To compute the transformation of image coordinates required for the distortion, the algorithm of Beier and Neely can be used. === Concerns === In 1993 concerns were raised about the authenticity of digitally altered images arising from morphing. Images of fake "tween" people found half way between two morphed people created a skeptical media long before AI. === Early examples === In or before 1986, computer graphics company Omnibus created a digital animation for a Tide commercial with a Tide detergent bottle smoothly morphing into the shape of the United States. The effect was programmed by Bob Hoffman. Omnibus re-used the technique in the movie Flight of the Navigator (1986). It featured scenes with a computer generated spaceship that appeared to change shape. The plaster cast of a model of the spaceship was scanned and digitally modified with techniques that included a reflection mapping technique that was also developed by programmer Bob Hoffman. The 1986 movie The Golden Child implemented early digital morphing effects from animal to human and back. Willow (1988) featured a more detailed digital morphing sequence with a person changing into different animals. A similar process was used a year later in Indiana Jones and the Last Crusade to create Walter Donovan's gruesome demise. Both effects were created by Industrial Light & Magic, using software developed by Tom Brigham and Doug Smythe (AMPAS). In 1991, morphing appeared notably in the Michael Jackson music video "Black or White" and in the movies Terminator 2: Judgment Day and Star Trek VI: The Undiscovered Country. The first application for personal computers to offer morphing was Gryphon Software Morph on the Macintosh. Other early morphing systems included ImageMaster, MorphPlus and CineMorph, all of which premiered for the Amiga in 1992. Other programs became widely available within a year, and for a time the effect became common to the point of cliché. For high-end use, Elastic Reality (based on MorphPlus) saw its first feature film use in In The Line of Fire (1993) and was used in Quantum Leap (work performed by the Post Group). At VisionArt Ted Fay used Elastic Reality to morph Odo for Star Trek: Deep Space Nine. The Snoop Dogg music video "Who Am I? (What's My Name?)", where Snoop Dogg and the others morph into dogs. Elastic Reality was later purchased by Avid, having already become the de facto system of choice, used in many hundreds of films. The technology behind Elastic Reality earned two Academy Awards in 1996 for Scientific and Technical Achievement going to Garth Dickie and Perry Kivolowitz. The effect is technically called a "spatially warped cross-dissolve". The first social network designed for user-generated morph examples to be posted online was Galleries by Morpheus. In late 1991 Yeti Productions employed a young Stephen Regelous to run it's 486 computer graphics system in Wellington New Zealand. After producer Barry Thomas showed him Michael Jackson's "Black or White", Regelous wrote 10,000 lines of C++ code of triangle-based digital morphing software. Together they created morphing based TV commercials for The NZ Cancer Society, Fit food, Salvation Army and others. The Fit food commercial employed morphing with 35mm, pin registered, digitally controlled motion control designed and made by Russell Collins with software by Stephen Regelous. In Taiwan, Aderans, a hair loss solutions provider, did a TV commercial featuring a morphing sequence in which people with lush, thick hair morph into one another, reminiscent of the end sequence of the "Black or White" video. === Present use === Morphing algorithms continue to advance and programs can automatically morph images that correspond closely enough with relatively little instruction from the user. This has led to the use of morphing techniques to create convincing slow-motion effects where none existed in the original film or video footage by morphing between each individual frame using optical flow technology. Morphing has also appeared as a transition technique between one scene and another in television shows, even if the contents of the two images are entirely unrelated. The algorithm in this case attempts to find corresponding points between the images and distort one into the other as they crossfade. While perhaps less obvious than in the past, morphing is used heavily today. Whereas the effect was initially a novelty, today, morphing effects are most often designed to be seamless and invisible to the eye. A particular use for morphing effects is modern digital font design. Using morphing technology, called interpolation or multiple master tech, a designer can create an intermediate between two styles, for example generating a semibold font by compromising between a bold and regular style, or extend a trend to create an ultra-light or ultra-bold. The technique is commonly used by font design studios. == Software == After Effects Animate Elastic Reality FantaMorph Gryphon Software Morph Morph Age Morpheus Nuke SilhouetteFX

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  VLLM

  vLLM is an open-source software framework for inference and serving of large language models and related multimodal models. Originally developed at the University of California, Berkeley's Sky Computing Lab, the project is centered on PagedAttention, a memory-management method for transformer key–value caches, and supports features such as continuous batching, distributed inference, quantization, and OpenAI-compatible APIs. According to a project maintainer, the "v" in vLLM originally referred to "virtual", inspired by virtual memory. == History == vLLM was introduced in 2023 by researchers affiliated with the Sky Computing Lab at UC Berkeley. Its core ideas were described in the 2023 paper Efficient Memory Management for Large Language Model Serving with PagedAttention, which presented the system as a high-throughput and memory-efficient serving engine for large language models. In 2025, the PyTorch Foundation announced that vLLM had become a Foundation-hosted project. PyTorch's project page states that the University of California, Berkeley contributed vLLM to the Linux Foundation in July 2024. In January 2026, TechCrunch reported that the creators of vLLM had launched the startup Inferact to commercialize the project, raising $150 million in seed funding. == Architecture == According to its 2023 paper, vLLM was designed to improve the efficiency of large language model serving by reducing memory waste in the key–value cache used during transformer inference. The paper introduced PagedAttention, an algorithm inspired by virtual memory and paging techniques in operating systems, and described vLLM as using block-level memory management and request scheduling to increase throughput while maintaining similar latency. The project documentation and repository describe support for continuous batching, chunked prefill, speculative decoding, prefix caching, quantization, and multiple forms of distributed inference and serving. PyTorch has described vLLM as a high-throughput, memory-efficient inference and serving engine that supports a range of hardware back ends, including NVIDIA and AMD GPUs, Google TPUs, AWS Trainium, and Intel processors.

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  Fillrate

  In computer graphics, a video card's pixel fillrate refers to the number of pixels that can be rendered on the screen and written to video memory in one second. Pixel fillrates are given in megapixels per second or in gigapixels per second (in the case of newer cards), and are obtained by multiplying the number of render output units (ROPs) by the clock frequency of the graphics processing unit (GPU) of a video card. A similar concept, texture fillrate, refers to the number of texture map elements (texels) the GPU can map to pixels in one second. Texture fillrate is obtained by multiplying the number of texture mapping units (TMUs) by the clock frequency of the GPU. Texture fillrates are given in mega or gigatexels per second. However, there is no full agreement on how to calculate and report fillrates. Another possible method is to multiply the number of pixel pipelines by the GPU's clock frequency. The results of these multiplications correspond to a theoretical number. The actual fillrate depends on many other factors. In the past, the fillrate has been used as an indicator of performance by video card manufacturers such as ATI and NVIDIA, however, the importance of the fillrate as a measurement of performance has declined as the bottleneck in graphics applications has shifted. For example, today, the number and speed of unified shader processing units has gained attention. Although fillrate doesn't provide a substantial bottleneck in games, it can still provide a bottleneck for certain parts of the game, for example applying a gaussian blur can be bottlenecked by fillrate. Scene complexity can be increased by overdrawing, which happens when an object is drawn to the frame buffer, and another object (such as a wall) is then drawn on top of it, covering it up. The time spent drawing the first object is thus wasted because it is not visible. When a sequence of scenes is extremely complex (many pixels have to be drawn for each scene), the frame rate for the sequence may drop. When designing graphics intensive applications, one can determine whether the application is fillrate-limited (or shader limited) by seeing if the frame rate increases dramatically when the application runs at a lower resolution or in a smaller window. Although this is not a full-proof method, modern videogame engines can dynamically reduce the level-of-detail required and thereby reducing fillrate-limited applications. The best way to find fillrate bottlenecks is to use GPU vendor software like NVIDIA Nsight Graphics, AMD Radeon GPU Profile and the Intel Graphics Performance Analyzers.

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  Vatican News App

  The Vatican News App is an official mobile application software issued by the Vatican's Dicastery for Communication. Formerly titled The Pope App, the app was launched on January 23, 2013, under the auspices of the Pontifical Council for Social Communications, a now-defunct dicastery that was merged into the Secretariat (now Dicastery) for Communication in March 2016. Initially, The Pope App was available only on iOS devices, but became available for Android phones at the end of February 2013. The app is available for download on iOS and Android in five languages: English, French, Italian, Portuguese and Spanish. It was originally promoted as an application with focus on the figure of the Pope which made it possible to follow the Pope's events while they are taking place. Alerts notified the followers by informing and offering access to "official papal-related content in a variety of formats". The app also enabled its users to see areas of the Vatican through webcams allocated throughout St. Peter's Square in Rome that broadcast images. In early 2018, The Pope App was relaunched as the Vatican News App, accompanied by a redesign that eliminated many of the previous version's features, reducing the app to a more conventional news service, with increased emphasis on news from the Vatican and the worldwide Catholic Church and less focus on the day-to-day activities of the Pope.

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  Graphical Kernel System

  The Graphical Kernel System (GKS) is a 2D computer graphics system using vector graphics, introduced in 1977. It was suitable for making line and bar charts and similar tasks. A key concept was cross-system portability, based on an underlying coordinate system that could be represented on almost any hardware. GKS is best known as the basis for the graphics in the GEM GUI system used on the Atari ST and as part of Ventura Publisher. A draft international standard was circulated for review in September 1983. Final ratification of the standard was achieved in 1985, making it the first ISO graphics standard. A 3D system modelled on GKS was introduced as PHIGS, which saw some use in the 1980s and early 1990s. == Overview == GKS provides a set of drawing features for two-dimensional vector graphics suitable for charting and similar duties. The calls are designed to be portable across different programming languages, graphics devices and hardware, so that applications written to use GKS will be readily portable to many platforms and devices. GKS was fairly common on computer workstations in the 1980s and early 1990s. GKS formed the basis of Digital Research's GSX which evolved into VDI, one of the core components of GEM. GEM was the native GUI on the Atari ST and was occasionally seen on PCs, particularly in conjunction with Ventura Publisher. GKS was little used commercially outside these markets, but remains in use in some scientific visualization packages. It is also the underlying API defining the Computer Graphics Metafile. One popular application based on an implementation of GKS is the GR Framework, a C library for high-performance scientific visualization that has become a common plotting backend among Julia users. A main developer and promoter of the GKS was José Luis Encarnação, formerly director of the Fraunhofer Institute for Computer Graphics (IGD) in Darmstadt, Germany. GKS has been standardized in the following documents: ANSI standard ANSI X3.124 of 1985. ISO 7942:1985 standard, revised as ISO 7942:1985/Amd 1:1991 and ISO/IEC 7942-1:1994, as well as ISO/IEC 7942-2:1997, ISO/IEC 7942-3:1999 and ISO/IEC 7942-4:1998 The language bindings are ISO standard ISO 8651. GKS-3D (Graphical Kernel System for Three Dimensions) functional definition is ISO standard ISO 8805, and the corresponding C bindings are ISO/IEC 8806. The functionality of GKS is wrapped up as a data model standard in the STEP standard, section ISO 10303-46.

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  Lexical substitution

  Lexical substitution is the task of identifying a substitute for a word in the context of a clause. For instance, given the following text: "After the match, replace any remaining fluid deficit to prevent chronic dehydration throughout the tournament", a substitute of game might be given. Lexical substitution is strictly related to word sense disambiguation (WSD), in that both aim to determine the meaning of a word. However, while WSD consists of automatically assigning the appropriate sense from a fixed sense inventory, lexical substitution does not impose any constraint on which substitute to choose as the best representative for the word in context. By not prescribing the inventory, lexical substitution overcomes the issue of the granularity of sense distinctions and provides a level playing field for automatic systems that automatically acquire word senses (a task referred to as Word Sense Induction). == Evaluation == In order to evaluate automatic systems on lexical substitution, a task was organized at the Semeval-2007 evaluation competition held in Prague in 2007. A Semeval-2010 task on cross-lingual lexical substitution has also taken place. == Skip-gram model == The skip-gram model takes words with similar meanings into a vector space (collection of objects that can be added together and multiplied by numbers) that are found close to each other in N-dimensions (list of items). A variety of neural networks (computer system modeled after a human brain) are formed together as a result of the vectors and networks that are related together. This all occurs in the dimensions of the vocabulary that has been generated in a network. The model has been used in lexical substitution automation and prediction algorithms. One such algorithm developed by Oren Melamud, Omer Levy, and Ido Dagan uses the skip-gram model to find a vector for each word and its synonyms. Then, it calculates the cosine distance between vectors to determine which words will be the best substitutes. === Example === In a sentence like "The dog walked at a quick pace" each word has a specific vector in relation to the other. The vector for "The" would be [1,0,0,0,0,0,0] because the 1 is the word vocabulary and the 0s are the words surrounding that vocabulary, which create a vector.

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  Reflection lines

  Engineers use reflection lines to judge a surface's quality. Reflection lines reveal surface flaws, particularly discontinuities in normals indicating that the surface is not C 2 {\displaystyle C^{2}} . Reflection lines may be created and examined on physical surfaces or virtual surfaces with the help of computer graphics. For example, the shiny surface of an automobile body is illuminated with reflection lines by surrounding the car with parallel light sources. Virtually, a surface can be rendered with reflection lines by modulating the surfaces point-wise color according to a simple calculation involving the surface normal, viewing direction and a square wave environment map. == Mathematical definition == Consider a point p {\displaystyle p} on a surface M {\displaystyle M} with (normalized) normal n {\displaystyle n} . If an observer views this point from infinity at view direction v {\displaystyle v} then the reflected view direction r {\displaystyle r} is: r = v − 2 ( n ⋅ v ) n . {\displaystyle r=v-2(n\cdot v)n.} (The vector v {\displaystyle v} is decomposed into its normal part v n = ( n ⋅ v ) v {\displaystyle v_{n}=(n\cdot v)v} and tangential part v t = v − v n {\displaystyle v_{t}=v-v_{n}} . Upon reflection, the tangential part is kept and the normal part is negated.) For reflection lines we consider the surface M {\displaystyle M} surrounded by parallel lines with direction a {\displaystyle a} , representing infinite, non-dispersive light sources. For each point p {\displaystyle p} on M {\displaystyle M} we determine which line is seen from direction v {\displaystyle v} . The position on each line is of no interest. Define the vector r p {\displaystyle r_{p}} to be the reflection direction r {\displaystyle r} projected onto a plane P {\displaystyle P} that is orthogonal to a {\displaystyle a} : r p = r − ( r ⋅ a ) a {\displaystyle r_{p}=r-(r\cdot a)a} and similarly let v p {\displaystyle v_{p}} be the viewing direction projected onto P {\displaystyle P} : v p = v − ( v ⋅ a ) a {\displaystyle v_{p}=v-(v\cdot a)a} Finally, define v o {\displaystyle v_{o}} to be the direction lying in P {\displaystyle P} perpendicular to a {\displaystyle a} and v p {\displaystyle v_{p}} : v o = a × v p {\displaystyle v_{o}=a\times v_{p}} Using these vectors, the reflection line function θ ( p ) : M → ( − π , π ] {\displaystyle \theta (p):M\rightarrow (-\pi ,\pi ]} is a scalar function mapping points p {\displaystyle p} on the surface to angles between v p {\displaystyle v_{p}} and r p {\displaystyle r_{p}} : θ = arctan ⁡ ( r p ⋅ v o , r p ⋅ v p ) {\displaystyle \theta =\arctan {(r_{p}\cdot v_{o},r_{p}\cdot v_{p})}} where a r c t a n ( y , x ) {\displaystyle arctan(y,x)} is the atan2 function producing a number in the range ( − π , π ] {\displaystyle (-\pi ,\pi ]} . ( v p {\displaystyle v_{p}} and v o {\displaystyle v_{o}} can be viewed as a local coordinate system in P {\displaystyle P} with x {\displaystyle x} -axis in direction v p {\displaystyle v_{p}} and y {\displaystyle y} -axis in direction v o {\displaystyle v_{o}} .) Finally, to render the reflection lines positive values θ > 0 {\displaystyle \theta >0} are mapped to a light color and non-positive values to a dark color. == Highlight lines == Highlight lines are a view-independent alternative to reflection lines. Here the projected normal is directly compared against some arbitrary vector x {\displaystyle x} perpendicular to the light source: θ = arctan ⁡ ( n a ⋅ a ⊥ , n a ⋅ x ) {\displaystyle \theta =\arctan {(n_{a}\cdot a^{\perp },n_{a}\cdot x)}} where n a {\displaystyle n_{a}} is the surface normal projected on the light source plane P {\displaystyle P} : n a ^ / | n a ^ | , n a ^ = n − ( n ⋅ a ) a {\displaystyle {\hat {n_{a}}}/|{\hat {n_{a}}}|,{\hat {n_{a}}}=n-(n\cdot a)a} The relationship between reflection lines and highlight lines is likened to that between specular and diffuse shading.

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  Couchbase Server

  Couchbase Server, originally known as Membase, is a source-available, distributed (shared-nothing architecture) multi-model NoSQL document-oriented database software package optimized for interactive applications. These applications may serve many concurrent users by creating, storing, retrieving, aggregating, manipulating and presenting data. In support of these kinds of application needs, Couchbase Server is designed to provide easy-to-scale key-value, or JSON document access, with low latency and high sustainability throughput. It is designed to be clustered from a single machine to very large-scale deployments spanning many machines. Couchbase Server provided client protocol compatibility with memcached, but added disk persistence, data replication, live cluster reconfiguration, rebalancing and multitenancy with data partitioning. == Product history == Membase was developed by several leaders of the memcached project, who had founded a company, NorthScale, to develop a key-value store with the simplicity, speed, and scalability of memcached, but also the storage, persistence and querying capabilities of a database. The original membase source code was contributed by NorthScale, and project co-sponsors Zynga and Naver Corporation (then known as NHN) to a new project on membase.org in June 2010. On February 8, 2011, the Membase project founders and Membase, Inc. announced a merger with CouchOne (a company with many of the principal players behind CouchDB) with an associated project merger. The merged company was called Couchbase, Inc. In January 2012, Couchbase released Couchbase Server 1.8. In September of 2012, Orbitz said it had changed some of its systems to use Couchbase. In December of 2012, Couchbase Server 2.0 (announced in July 2011) was released and included a new JSON document store, indexing and querying, incremental MapReduce and replication across data centers. == Architecture == Every Couchbase node consists of a data service, index service, query service, and cluster manager component. Starting with the 4.0 release, the three services can be distributed to run on separate nodes of the cluster if needed. In the parlance of Eric Brewer's CAP theorem, Couchbase is normally a CP type system meaning it provides consistency and partition tolerance, or it can be set up as an AP system with multiple clusters. === Cluster manager === The cluster manager supervises the configuration and behavior of all the servers in a Couchbase cluster. It configures and supervises inter-node behavior like managing replication streams and re-balancing operations. It also provides metric aggregation and consensus functions for the cluster, and a RESTful cluster management interface. The cluster manager uses the Erlang programming language and the Open Telecom Platform. ==== Replication and fail-over ==== Data replication within the nodes of a cluster can be controlled with several parameters. In December of 2012, support was added for replication between different data centers. === Data manager === The data manager stores and retrieves documents in response to data operations from applications. It asynchronously writes data to disk after acknowledging to the client. In version 1.7 and later, applications can optionally ensure data is written to more than one server or to disk before acknowledging a write to the client. Parameters define item ages that affect when data is persisted, and how max memory and migration from main-memory to disk is handled. It supports working sets greater than a memory quota per "node" or "bucket". External systems can subscribe to filtered data streams, supporting, for example, full text search indexing, data analytics or archiving. ==== Data format ==== A document is the most basic unit of data manipulation in Couchbase Server. Documents are stored in JSON document format with no predefined schemas. Non-JSON documents can also be stored in Couchbase Server (binary, serialized values, XML, etc.) ==== Object-managed cache ==== Couchbase Server includes a built-in multi-threaded object-managed cache that implements memcached compatible APIs such as get, set, delete, append, prepend etc. ==== Storage engine ==== Couchbase Server has a tail-append storage design that is immune to data corruption, OOM killers or sudden loss of power. Data is written to the data file in an append-only manner, which enables Couchbase to do mostly sequential writes for update, and provide an optimized access patterns for disk I/O. === Performance === A performance benchmark done by Altoros in 2012, compared Couchbase Server with other technologies. Cisco Systems published a benchmark that measured the latency and throughput of Couchbase Server with a mixed workload in 2012. == Licensing and support == Couchbase Server is a packaged version of Couchbase's open source software technology and is available in a community edition without recent bug fixes with an Apache 2.0 license and an edition for commercial use. Couchbase Server builds are available for Ubuntu, Debian, Red Hat, SUSE, Oracle Linux, Microsoft Windows and macOS operating systems. Couchbase has supported software developers' kits for the programming languages .NET, PHP, Ruby, Python, C, Node.js, Java, Go, and Scala. == SQL++ == A query language called SQL++ (formerly called N1QL), is used for manipulating the JSON data in Couchbase, just like SQL manipulates data in RDBMS. It has SELECT, INSERT, UPDATE, DELETE, MERGE statements to operate on JSON data. It was initially announced in March 2015 as "SQL for documents". The SQL++ data model is non-first normal form (N1NF) with support for nested attributes and domain-oriented normalization. The SQL++ data model is also a proper superset and generalization of the relational model. === Example === Like query SELECT FROM `bucket` WHERE email LIKE "%@example.org"; Array query SELECT FROM `bucket` WHERE ANY x IN friends SATISFIES x.name = "Pavan" END; == Couchbase Mobile == Couchbase Mobile / Couchbase Lite is a mobile database providing data replication. Couchbase Lite (originally TouchDB) provides native libraries for offline-first NoSQL databases with built-in peer-to-peer or client-server replication mechanisms. Sync Gateway manages secure access and synchronization of data between Couchbase Lite and Couchbase Server. Couchbase Lite added support for Vector Search in version 3.2, allowing cloud to edge support for vector search in mobile applications. == Uses == Couchbase began as an evolution of Memcached, a high-speed data cache, and can be used as a drop-in replacement for Memcached, providing high availability for memcached application without code changes. Couchbase is used to support applications where a flexible data model, easy scalability, and consistent high performance are required, such as tracking real-time user activity or providing a store of user preferences or online applications. Couchbase Mobile, which stores data locally on devices (usually mobile devices) is used to create “offline-first” applications that can operate when a device is not connected to a network and synchronize with Couchbase Server once a network connection is re-established. The Catalyst Lab at Northwestern University uses Couchbase Mobile to support the Evo application, a healthy lifestyle research program where data is used to help participants improve dietary quality, physical activity, stress, or sleep. Amadeus uses Couchbase with Apache Kafka to support their “open, simple, and agile” strategy to consume and integrate data on loyalty programs for airline and other travel partners. High scalability is needed when disruptive travel events create a need to recognize and compensate high value customers. Starting in 2012, it played a role in LinkedIn's caching systems, including backend caching for recruiter and jobs products, counters for security defense mechanisms, for internal applications. == Alternatives == For caching, Couchbase competes with Memcached and Redis. For document databases, Couchbase competes with other document-oriented database systems. It is commonly compared with MongoDB, Amazon DynamoDB, Oracle RDBMS, DataStax, Google Bigtable, MariaDB, IBM Cloudant, Redis Enterprise, SingleStore, and MarkLogic.

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  Human image synthesis

  Human image synthesis is technology that can be applied to make believable and even photorealistic renditions of human-likenesses, moving or still. It has effectively existed since the early 2000s. Many films using computer generated imagery have featured synthetic images of human-like characters digitally composited onto the real or other simulated film material. Towards the end of the 2010s deep learning artificial intelligence has been applied to synthesize images and video that look like humans, without need for human assistance, once the training phase has been completed, whereas the old school 7D-route required massive amounts of human work. == Timeline of human image synthesis == In 1971 Henri Gouraud made the first CG geometry capture and representation of a human face. Modeling was his wife Sylvie Gouraud. The 3D model was a simple wire-frame model and he applied the Gouraud shader he is most known for to produce the first known representation of human-likeness on computer. The 1972 short film A Computer Animated Hand by Edwin Catmull and Fred Parke was the first time that computer-generated imagery was used in film to simulate moving human appearance. The film featured a computer simulated hand and face (watch film here). The 1976 film Futureworld reused parts of A Computer Animated Hand on the big screen. The 1983 music video for song Musique Non-Stop by German band Kraftwerk aired in 1986. Created by the artist Rebecca Allen, it features non-realistic looking, but clearly recognizable computer simulations of the band members. The 1994 film The Crow was the first film production to make use of digital compositing of a computer simulated representation of a face onto scenes filmed using a body double. Necessity was the muse as the actor Brandon Lee portraying the protagonist was tragically killed accidentally on-stage. In 1999 Paul Debevec et al. of USC captured the reflectance field of a human face with their first version of a light stage. They presented their method at the SIGGRAPH 2000 In 2003 audience debut of photo realistic human-likenesses in the 2003 films The Matrix Reloaded in the burly brawl sequence where up-to-100 Agent Smiths fight Neo and in The Matrix Revolutions where at the start of the end showdown Agent Smith's cheekbone gets punched in by Neo leaving the digital look-alike unnaturally unhurt. The Matrix Revolutions bonus DVD documents and depicts the process in some detail and the techniques used, including facial motion capture and limbal motion capture, and projection onto models. In 2003 The Animatrix: Final Flight of the Osiris a state-of-the-art want-to-be human likenesses not quite fooling the watcher made by Square Pictures. In 2003 digital likeness of Tobey Maguire was made for movies Spider-man 2 and Spider-man 3 by Sony Pictures Imageworks. In 2005 the Face of the Future project was an established. by the University of St Andrews and Perception Lab, funded by the EPSRC. The website contains a "Face Transformer", which enables users to transform their face into any ethnicity and age as well as the ability to transform their face into a painting (in the style of either Sandro Botticelli or Amedeo Modigliani). This process is achieved by combining the user's photograph with an average face. In 2009 Debevec et al. presented new digital likenesses, made by Image Metrics, this time of actress Emily O'Brien whose reflectance was captured with the USC light stage 5 Motion looks fairly convincing contrasted to the clunky run in the Animatrix: Final Flight of the Osiris which was state-of-the-art in 2003 if photorealism was the intention of the animators. In 2009 a digital look-alike of a younger Arnold Schwarzenegger was made for the movie Terminator Salvation though the end result was critiqued as unconvincing. Facial geometry was acquired from a 1984 mold of Schwarzenegger. In 2010 Walt Disney Pictures released a sci-fi sequel entitled Tron: Legacy with a digitally rejuvenated digital look-alike of actor Jeff Bridges playing the antagonist CLU. In SIGGGRAPH 2013 Activision and USC presented a real-time "Digital Ira" a digital face look-alike of Ari Shapiro, an ICT USC research scientist, utilizing the USC light stage X by Ghosh et al. for both reflectance field and motion capture. The end result both precomputed and real-time rendering with the modernest game GPU shown here and looks fairly realistic. In 2014 The Presidential Portrait by USC Institute for Creative Technologies in conjunction with the Smithsonian Institution was made using the latest USC mobile light stage wherein President Barack Obama had his geometry, textures and reflectance captured. In 2014 Ian Goodfellow et al. presented the principles of a generative adversarial network. GANs made the headlines in early 2018 with the deepfakes controversies. For the 2015 film Furious 7 a digital look-alike of actor Paul Walker who died in an accident during the filming was done by Weta Digital to enable the completion of the film. In 2016 techniques which allow near real-time counterfeiting of facial expressions in existing 2D video have been believably demonstrated. In 2016 a digital look-alike of Peter Cushing was made for the Rogue One film where its appearance would appear to be of same age as the actor was during the filming of the original 1977 Star Wars film. In SIGGRAPH 2017 an audio driven digital look-alike of upper torso of Barack Obama was presented by researchers from University of Washington. It was driven only by a voice track as source data for the animation after the training phase to acquire lip sync and wider facial information from training material consisting 2D videos with audio had been completed. Late 2017 and early 2018 saw the surfacing of the deepfakes controversy where porn videos were doctored using deep machine learning so that the face of the actress was replaced by the software's opinion of what another persons face would look like in the same pose and lighting. In 2018 Game Developers Conference Epic Games and Tencent Games demonstrated "Siren", a digital look-alike of the actress Bingjie Jiang. It was made possible with the following technologies: CubicMotion's computer vision system, 3Lateral's facial rigging system and Vicon's motion capture system. The demonstration ran in near real time at 60 frames per second in the Unreal Engine 4. In 2018 at the World Internet Conference in Wuzhen the Xinhua News Agency presented two digital look-alikes made to the resemblance of its real news anchors Qiu Hao (Chinese language) and Zhang Zhao (English language). The digital look-alikes were made in conjunction with Sogou. Neither the speech synthesis used nor the gesturing of the digital look-alike anchors were good enough to deceive the watcher to mistake them for real humans imaged with a TV camera. In September 2018 Google added "involuntary synthetic pornographic imagery" to its ban list, allowing anyone to request the search engine block results that falsely depict them as "nude or in a sexually explicit situation." In February 2019 Nvidia open sources StyleGAN, a novel generative adversarial network. Right after this Phillip Wang made the website ThisPersonDoesNotExist.com with StyleGAN to demonstrate that unlimited amounts of often photo-realistic looking facial portraits of no-one can be made automatically using a GAN. Nvidia's StyleGAN was presented in a not yet peer reviewed paper in late 2018. At the June 2019 CVPR the MIT CSAIL presented a system titled "Speech2Face: Learning the Face Behind a Voice" that synthesizes likely faces based on just a recording of a voice. It was trained with massive amounts of video of people speaking. Since 1 July 2019 Virginia has criminalized the sale and dissemination of unauthorized synthetic pornography, but not the manufacture., as § 18.2–386.2 titled 'Unlawful dissemination or sale of images of another; penalty.' became part of the Code of Virginia. The law text states: "Any person who, with the intent to coerce, harass, or intimidate, maliciously disseminates or sells any videographic or still image created by any means whatsoever that depicts another person who is totally nude, or in a state of undress so as to expose the genitals, pubic area, buttocks, or female breast, where such person knows or has reason to know that he is not licensed or authorized to disseminate or sell such videographic or still image is guilty of a Class 1 misdemeanor.". The identical bills were House Bill 2678 presented by Delegate Marcus Simon to the Virginia House of Delegates on 14 January 2019 and three-day later an identical Senate bill 1736 was introduced to the Senate of Virginia by Senator Adam Ebbin. Since 1 September 2019 Texas senate bill SB 751 amendments to the election code came into effect, giving candidates in elections a 30-day protection period to the elections during which making and distributing digital look-alikes or synthetic fakes of the candidates is an offense. Th

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  Pronunciation assessment

  Automatic pronunciation assessment uses computer speech recognition to determine how accurately speech has been pronounced, instead of relying on a human instructor or proctor. It is also called speech verification, pronunciation evaluation, and pronunciation scoring. This technology is used to grade speech quality, for language testing, for computer-aided pronunciation teaching (CAPT) in computer-assisted language learning (CALL), for speaking skill remediation, and for accent reduction. Pronunciation assessment is different from dictation or automatic transcription, because instead of determining unknown speech, it verifies learners' pronunciation of known word(s), often from prior transcription of the same utterance; ideally scoring the intelligibility of the learners' speech. Sometimes pronunciation assessment evaluates the prosody of the learners' speech, such as intonation, pitch, tempo, rhythm, and syllable and word stress, although those are usually not essential for being understood in most languages. Pronunciation assessment is also used in reading tutoring, for example in products from Google, Microsoft, and Amira Learning. Automatic pronunciation assessment can also be used to help diagnose and treat speech disorders such as apraxia. == Intelligibility == Intelligibility refers to how well a learner's utterance is understood by a listener, rather than how much it sounds like a native speaker. This is separate from measures of fluency, such as so-called "Goodness of Pronunciation" (GoP) scores, which estimate how closely an utterance aligns with those of native speakers. Intelligibility is widely regarded as the most important communicative goal in pronunciation teaching and assessment. For example, in the Common European Framework of Reference for Languages (CEFR) assessment criteria for "overall phonological control", intelligibility outweighs formally correct pronunciation at all levels. Studies in applied linguistics have shown that accent reduction does not always increase intelligibility because listeners can often comprehend heavily accented speech without difficulty. Pronunciation assessment systems often rely on acoustic methods such as GoP which compare learner speech to reference models to produce phoneme-level scores, which are in turn aggregated to produce word and phrase scores. While these methods are effective for identifying deviations from native speakers' utterances, they do not effectively measure how understandable speech is to human listeners. Intelligibility is influenced by broader linguistic and contextual factors such as stress placement, speech rate, and coarticulation, which are not represented in purely segmental scores. The earliest work on pronunciation assessment avoided measuring genuine listener intelligibility, a shortcoming corrected in 2011 at the Toyohashi University of Technology, and included in the Versant high-stakes English fluency assessment from Pearson and mobile apps from 17zuoye Education & Technology, but still missing in 2023 products from Google Search, Microsoft, Educational Testing Service, Speechace, and ELSA. Assessing authentic listener intelligibility is essential for avoiding inaccuracies from accent bias, especially in high-stakes assessments; from words with multiple correct pronunciations; and from phoneme coding errors in machine-readable pronunciation dictionaries. In 2022, researchers found that some newer speech-to-text systems, based on end-to-end reinforcement learning to map audio signals directly into words, produce word and phrase confidence scores (from 10-25ms audio frame logit aggregation) closely correlated with genuine listener intelligibility. Others have been able to assess intelligibility using Levenshtein or dynamic time warping distance measures from Wav2Vec2 representation of good speech. Further work through 2025 has focused specifically on measuring intelligibility. A 2025 study of 42 pronunciation and speech coaching apps (32 mobile and 10 web) found that none offered intelligibility assessment. Instead, most provided only segmental and accent-focused scoring. About two-thirds of the apps provided some form of specific pronunciation feedback, usually with phonetic transcriptions, but accompanied by visual cues (such as animations of the vocal tract or the lips and tongue from the front) in only about 5% of the apps. Less than a third provided feedback on learner perception of exemplar speech. == Evaluation == Although there are as yet no industry-standard benchmarks for evaluating pronunciation assessment accuracy, researchers occasionally release evaluation speech corpuses for others to use for improving assessment quality. Such evaluation databases often emphasize formally unaccented pronunciation to the exclusion of genuine intelligibility evident from blinded listener transcriptions. As of mid-2025, state of the art approaches for automatically transcribing phonemes typically achieve an error rate of about 10% from known good speech. The International Speech Communication Association (ISCA) 2025 Workshop on Speech and Language Technology in Education (SLaTE) administered a Speak & Improve Challenge: Spoken Language Assessment and Feedback, introducing benchmarks for evaluating pronunciation assessment and remediation systems across languages, accents, and learner populations. The challenge emphasized cross-lingual generalization and alignment with human intelligibility judgments, for more robust and interpretable assessment systems. Ethical issues in pronunciation assessment are present in both human and automatic methods. Authentic validity, fairness, and mitigating bias in evaluation are all crucial. Diverse speech data should be included in automatic pronunciation assessment models. Combining human judgments, especially blinded transcriptions from a wide diversity of listeners, with automated feedback can improve accuracy and fairness. Second language learners benefit substantially from their use of widely available speech recognition systems for dictation, virtual assistants, and AI chatbots. In such systems, users naturally try to correct their own errors evident in speech recognition results that they notice. Such use improves their grammar and vocabulary development along with their pronunciation skills. The extent to which explicit pronunciation assessment and remediation approaches improve on such self-directed interactions remains an open question. Similarly, automatic dictation results have been shown to reflect intelligibility about as well as human scorers. == Recent developments == During 2021–22, a smartphone-based CAPT system was used to sense articulation through both audible and inaudible signals, providing feedback at the phoneme level. Some promising areas for improvement which were being developed in 2024 include articulatory feature extraction and transfer learning to suppress unnecessary corrections. Other interesting advances under development include "augmented reality" interfaces for mobile devices using optical character recognition to provide pronunciation training on text found in user environments. In 2024, audio multimodal large language models were first described as assessing pronunciation. That work has been carried forward by other researchers in 2025 who report positive results. Subsequently, researchers demonstrated pronunciation scoring by providing a language model with textual descriptions of speech, including the speech-to-text transcript, phoneme sequences, pauses, and phoneme sequence matching; this approach can achieve performance similar to multimodal LLMs that analyze raw audio while avoiding their higher computational cost. In 2025, the Duolingo English Test authors published a description of their pronunciation assessment method, purportedly built to measure intelligibility rather than accent imitation. While achieving a correlation of 0.82 with expert human ratings, very close to inter-rater agreement and outperforming alternative methods, the method is nonetheless based on experts' scores along the six-point CEFR common reference levels scale, instead of actual blinded listener transcriptions. Further promising work in 2025 includes assessment feedback aligning learner speech to synthetic utterances using interpretable features, identifying continuous spans of words for remediation feedback; synthesizing corrected speech matching learners' self-perceived voices, which they prefer and imitate more accurately as corrections; and streaming such interactions. On January 21, 2026, Educational Testing Service's TOEFL iBT high-stakes English language test, required by US university admissions and employers from English as a foreign language applicants more often than all other internet-based tests combined, changed its speaking assessments. While official rubrics claim that the new scoring will be based primarily on intelligibility, the new test's technical description indicates that it ju

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  Graphics address remapping table

  The graphics address remapping table (GART), also known as the graphics aperture remapping table, or graphics translation table (GTT), is an I/O memory management unit (IOMMU) used by Accelerated Graphics Port (AGP) and PCI Express (PCIe) graphics cards. The GART allows the graphics card direct memory access (DMA) to the host system memory, through which buffers of textures, polygon meshes and other data are loaded. AMD later reused the same mechanism for I/O virtualization with other peripherals including disk controllers and network adapters. A GART is used as a means of data exchange between the main memory and video memory through which buffers (i.e. paging/swapping) of textures, polygon meshes and other data are loaded, but can also be used to expand the amount of video memory available for systems with only integrated or shared graphics (i.e. no discrete or inbuilt graphics processor), such as Intel HD Graphics processors. However, this type of memory (expansion) remapping has a caveat that affects the entire system: specifically, any GART, pre-allocated memory becomes pooled and cannot be utilised for any other purposes but graphics memory and display rendering. Since PCI Express, the GART is extended to the GTT (Graphics Translation Table), which act as a buffer or cache between system memory and graphics card, and in PCI Express, the GTT buffer size is changeable by the GPU driver. == Operating system support == === Windows === Support for AGP GART was added since Windows 95 OSR2. Later, support for GTT was added since Windows XP SP2 and Windows Vista. === Linux === Jeff Hartmann served as the primary maintainer of the Linux kernel's agpgart driver, which began as part of Brian Paul's Utah GLX accelerated Mesa 3D driver project. The developers primarily targeted Linux 2.4.x kernels, but made patches available against older 2.2.x kernels. Dave Jones heavily reworked agpgart for the Linux 2.6.x kernels, along with more contributions from Jeff Hartmann. === FreeBSD === In FreeBSD, the agpgart driver appeared in its 4.1 release. === Solaris === AGPgart support was introduced into Solaris Express Developer Edition as of its 7/05 release.

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  Supertoroid

  In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surfaces (technically, a topological torus) whose shape is defined by mathematical formulas similar to those that define the superellipsoids. The plural of "supertorus" is either supertori or supertoruses. The family was described and named by Alan Barr in 1994. Barr's supertoroids have been fairly popular in computer graphics as a convenient model for many objects, such as smooth frames for rectangular things. One quarter of a supertoroid can provide a smooth and seamless 90-degree joint between two superquadric cylinders. However, they are not algebraic surfaces (except in special cases). == Formulas == Alan Barr's supertoroids are defined by parametric equations similar to the trigonometric equations of the torus, except that the sine and cosine terms are raised to arbitrary powers. Namely, the generic point P(u, v) of the surface is given by P ( u , v ) = ( X ( u , v ) Y ( u , v ) Z ( u , v ) ) = ( ( a + C u s ) C v t ( b + C u s ) S v t S u s ) {\displaystyle P(u,v)=\left({\begin{array}{c}X(u,v)\\Y(u,v)\\Z(u,v)\end{array}}\right)=\left({\begin{array}{c}(a+C_{u}^{s})C_{v}^{t}\\(b+C_{u}^{s})S_{v}^{t}\\S_{u}^{s}\end{array}}\right)} where C θ ε = sgn ⁡ ( cos ⁡ θ ) | cos ⁡ θ | ε , S θ ε = sgn ⁡ ( sin ⁡ θ ) | sin ⁡ θ | ε , {\displaystyle {\begin{aligned}C_{\theta }^{\varepsilon }&=\operatorname {sgn} (\cos \theta )\,\left|\,\cos \theta \,\right|^{\varepsilon },\\S_{\theta }^{\varepsilon }&=\operatorname {sgn} (\sin \theta )\ \left|\,\sin \theta \ \right|^{\varepsilon },\end{aligned}}} sgn is the sign function, and the parameters u, v range from 0 to 360 degrees (0 to 2π radians). In these formulas, the parameter s > 0 controls the "squareness" of the vertical sections, t > 0 controls the squareness of the horizontal sections, and a, b ≥ 1 are the major radii in the x and y directions. With s = t = 1 and a = b = R one obtains the ordinary torus with major radius R and minor radius 1, with the center at the origin and rotational symmetry about the z-axis. In general, the supertorus defined as above spans the intervals: − ( a + 1 ) ≤ x ≤ + ( a + 1 ) − ( b + 1 ) ≤ y ≤ + ( b + 1 ) − 1 ≤ z ≤ + 1 {\displaystyle {\begin{array}{rcccl}-(a+1)&\leq &x&\leq &+(a+1)\\[4pt]-(b+1)&\leq &y&\leq &+(b+1)\\[4pt]-1&\leq &z&\leq &+1\end{array}}} The whole shape is symmetric about the planes x = 0, y = 0, and z = 0. The hole runs in the z direction and spans the intervals − ( a − 1 ) ≤ x ≤ + ( a − 1 ) − ( b − 1 ) ≤ y ≤ + ( b − 1 ) − ∞ ≤ z ≤ + ∞ {\displaystyle {\begin{array}{rcccl}-(a-1)&\leq &x&\leq &+(a-1)\\[4pt]-(b-1)&\leq &y&\leq &+(b-1)\\[4pt]-\infty &\leq &z&\leq &+\infty \end{array}}} A curve of constant u on this surface is a horizontal Lamé curve with exponent ⁠ 2 t , {\displaystyle {\tfrac {2}{t}},} ⁠ scaled in x and y and displaced in z. A curve of constant v, projected on the plane x = 0 or y = 0, is a Lamé curve with exponent ⁠ 2 s , {\displaystyle {\tfrac {2}{s}},} ⁠ scaled and horizontally shifted. If v = 0, the curve is planar and spans the intervals: a − 1 ≤ x ≤ a + 1 − 1 ≤ z ≤ + 1 {\displaystyle {\begin{array}{rcccl}a-1&\leq &x&\leq &a+1\\[4pt]-1&\leq &z&\leq &+1\end{array}}} and similarly if v = 90°, 180°, 270°. The curve is also planar if a = b. In general, if a ≠ b and v is not a multiple of 90 degrees, the curve of constant v will not be planar; and, conversely, a vertical plane section of the supertorus will not be a Lamé curve. The basic supertoroid shape defined above is often modified by non-uniform scaling to yield supertoroids of specific width, length, and vertical thickness. == Plotting code == The following GNU Octave code generates plots of a supertorus:

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