GitHub Codespaces is a cloud-based online integrated development environment developed by GitHub. It allows users to create and manage development environments directly within the browser or through Visual Studio Code desktop. Codespaces is tightly integrated with GitHub repositories and enables on-demand coding, debugging, and testing in a full-featured development container hosted in the cloud. == Features == Instant development environments integrated with GitHub Browser-based and desktop access via Visual Studio Code Configurable Dockerfile or devcontainer.json environments Built-in support for GitHub Copilot, extensions, snippets, and SSH. == Licensing == GitHub Codespaces is proprietary software and available to GitHub users under various subscription plans. Codespaces includes a monthly usage quota for free tier users of 120 hours, and expanded access for GitHub education, Pro, Team, and GitHub Enterprise plans. == GitHub Classroom == GitHub Classroom is an educational tool developed by GitHub to streamline the process of managing programming assignments and coursework. Integrated with GitHub repositories, it allows instructors to distribute starter code, automate grading workflows, and track student progress. GitHub Classroom is widely used in computer science education and supports integration with GitHub Codespaces for cloud-based development environments. == Programming languages supported == == Extensions == Some of the popular extensions include:
Object model
In computing, object model has two related but distinct meanings: The properties of objects in general in a specific computer programming language, technology, notation or methodology that uses them. Examples are the object models of Java, the Component Object Model (COM), or Object-Modeling Technique (OMT). Such object models are usually defined using concepts such as class, generic function, message, inheritance, polymorphism, and encapsulation. There is an extensive literature on formalized object models as a subset of the formal semantics of programming languages. A collection of objects or classes through which a program can examine and manipulate some specific parts of its world. In other words, the object-oriented interface to some service or system. Such an interface is said to be the object model of the represented service or system. For example, the Document Object Model (DOM) is a collection of objects that represent a page in a web browser, used by script programs to examine and dynamically change the page. There is a Microsoft Excel object model [1] for controlling Microsoft Excel from another program, and the ASCOM Telescope Driver is an object model for controlling an astronomical telescope. == Features == An object model consists of the following important features: === Object reference === Objects can be accessed via object references. To invoke a method in an object, the object reference and method name are given, together with any arguments. === Interfaces === An interface provides a definition of the signature of a set of methods without specifying their implementation. An object will provide a particular interface if its class contains code that implement the method of that interface. An interface also defines types that can be used to declare the type of variables or parameters and return values of methods. === Actions === An action in object-oriented programming (OOP) is initiated by an object invoking a method in another object. An invocation can include additional information needed to carry out the method. The receiver executes the appropriate method and then returns control to the invoking object, sometimes supplying a result. === Exceptions === Programs can encounter various errors and unexpected conditions of varying seriousness. During the execution of the method many different problems may be discovered. Exceptions provide a clean way to deal with error conditions without complicating the code. A block of code may be defined to throw an exception whenever particular unexpected conditions or errors arise. This means that control passes to another block of code that catches the exception.
Far-Play
Far-Play (stylized fAR-Play, from augmented reality) was a software platform developed at the University of Alberta, for creating location-based, scavenger-hunt style games which use the GPS and web-connectivity features of a player's smartphone. According to the development team, "our long-term objective is to develop a general framework that supports the implementation of AARGs that are fun to play and also educational". It utilizes Layar, an augmented reality smartphone application, QR codes located at particular real-world sites, or a phone's web browser, to facilitate games which require players to be in close physical proximity to predefined "nodes". A node, referred to by the developers as a Virtual Point of Interest (vPOI), is a point in space defined by a set of map coordinates; fAR-Play uses the GPS function of a player's smartphone — or, for indoor games, which are not easily tracked by GPS satellites, specially-created QR codes— to confirm that they are adequately near a given node. Once a player is within a node's proximity, Layar's various augmented reality features can be utilized to display a range of extra content overlaid upon the physical play-space or launch another application for extra functionality. == Development and features == fAR-Play began development in 2008, emerging from a collaborative project undertaken by a group of University of Alberta students from the Computer Science and Humanities Computing departments. fAR-Play is still under development, but a beta version is available for testing by request. fAR-Play's development is managed by a team of interdisciplinary professors and students at the University of Alberta. Currently, the developing team's roster includes Supervising Professors Geoffrey Rockwell and Eleni Stroulia, Developers Lucio Gutierrez and Matthew Delaney, and Website Developers Calen Henry and Garry Wong. === Technology === fAR-Play relies on a number of open- and closed-source web technologies as tools to create, and enhance the users' experience. Layar is the recommended client-side frontend for delivering game content to the player; it is available on Android and iOS, which covers over 91% of smartphones. While Layar is not a requirement to play fAR-Play games, the application does supply additional augmented reality functionality; Layar also includes a built-in QR scanner. Depending on the design of the particular game, the player may instead use a dedicated QR code scanner; the developers recommend BeeTagg, but any such application will do. Layar or a QR code scanner are the maximum software requirements to play a fAR-Play game, making implementation of games on a wide variety of platforms relatively straightforward. fAR-Play games can also be designed for play strictly within a mobile phone's web browser. On the server side, fAR-Play's engine is composed of an Apache server which manages the system's web interface, including the mobile and desktop versions of the fAR-Play website, and a Java-based REST framework for managing the database of nodes. === Features === As a platform for designing AR games, as opposed to an AR game itself, fAR-Play offers little in the way of explicit shapes or patterns for games to take; instead, these elements are left to the game designer or players to develop. However, the nonspecific nature of nodes, the many options they offer for content delivery, and the open design of the platform are such that these elements can be developed extensively. Functionally, fAR-Play is a tool for tracking arbitrary points in space and a given player's proximity to them; what it does beyond that is up to the developers' and players' discretion. However, the fAR-Play website contains a leaderboard which tracks registered user's total scores. Players are assigned levels based on their total score, ranging from Novice — Super Player. Player profiles will display nodes that the player has recently caught, and any achievements the player has gained. Additionally, players can share their adventure progress, achievements, and the capture of vPOIs on Facebook. == How to play == In order to participate in the locative aspects of fAR-Play games, users must have an Android or iOS mobile device and access to wireless internet. Players can participate in fAR-Play anonymously, or create and sign into a fAR-Play account. Those who choose to play anonymously will lose the ability to track their progress across multiple games. When signed in, the player is presented with a list of games that are currently available for play. Each game includes a brief description and the various "adventures" available to the player. Once the game has been started, the player has three different methods for capturing nodes: they may scan a QR in the physical space, discover a node through the Layar camera virtual view, or receive a link in their device's web browser. === QR codes and Layar === QR codes can only be used as a method for capturing nodes and initiating games when there is a physical code present. In order to scan a QR code, players are required to have an application which can capture and recognize QR codes. If the player is utilizing a QR scanning application that has a built in browser, they will be required to log into fAR-Play through the app. Layar is a free to download augmented reality app, containing a built in QR code scanner, which enables its users to participate in fAR-Play games. === Capturing nodes === Layar permits the player to see nodes on their mobile device, guiding the player to their goal. Using this application, the player is able to navigate to their objective with map provided by Google Maps' API or by using their camera — Layar overlays a virtual image onto the real-world scene presented by the camera. The representations on screen expand in size as the player approaches the node destination, simulating relative distance. If the player taps any of the nodes that are presented on the screen, they will be provided additional information about that node, including the node's name and a brief description. Nodes can be captured by tapping the "capture" button. === Playing on browsers === The player can also play fAR-Play games within their mobile device's browser. By visiting https://archive.today/20131123223038/http://farplay.ualberta.ca/far-play/ on a mobile device, players will be presented with a fully realized user interface, permitting full interaction with the games. The player can capture the in game vPOIs through their browser by tapping the "nodes" button. This will bring up a list of all the accessible nodes, complete with a brief description for each location. By clicking on one of the nodes, the player is shown to a screen with a mapped location of the vPOI, an in-depth description of it, and hints. At the top of the page, the player can tap "CAPTURE THIS NODE" and advance in the game. When attempting to capture a node, the developer may or may not associate a challenge with the node. For example, in the game "Zombies ate my Campus", when players are attempting to capture a node, they're presented with a multiple choice question associated with the current node. === Game types === Players complete an adventure when they have captured all of the nodes within it. fAR-Play provides two game modes: in a Virtual Scavenger Hunt, nodes must be captured in a specific order; in a Virtual Treasure Hunt, the order is unimportant. == Existing fAR-Play games == Games currently available through fAR-Play include: Giselle Ever After Thought Hub Comics Arts Capture Challenge Pioneering Edmonton The Intelliphone Challenge A Tour of Atwater Zombies ate my Campus == For developers == fAR-Play's ultimate goal is to provide a simple, effective platform for the creation of locative augmented reality games, but the developer tools are still under active development and not openly available to the public. Access can be granted on a case-by-case basis, however, and a developer's manual is available. Users with development privileges can create new games or edit their existing games, in addition to playing their own or others' games. === Adventures === Games that are developed with fAR-Play are segmented into components called "Adventures". To progress through each game adventure, the player must reach and capture virtual points of interest, referred to in the game as vPOIs. In order to capture a vPOI, the player must travel to a physical location that is set by the developer. It is the developer's choice to include a challenge question to capture the vPOI, though it is not mandatory. A deduction of points can be implemented if the player submits an incorrect answer to a challenge question. === Points and achievements === Each of the nodes will reward the player with a predetermined number of points once they have been captured by the player. These points are added to the player's total points. Each of the adventures that are created require a predetermined number of vPOIs
Information Age
The Information Age is a historical period that began in the mid-20th century. It is characterized by a rapid shift from traditional industries, as established during the Industrial Revolution, to an economy centered on information technology. The onset of the Information Age has been linked to the development of the transistor in 1947. Advances in computer miniaturization, internet communication, and semiconductor technology enabled the rapid expansion of digital systems and global information networks. The Information Age transformed industries such as education, healthcare, finance, entertainment, and communication through digital infrastructure and connected technologies. The rise of smartphones and cloud-based services further accelerated global internet accessibility and digital interaction. == Digital applications and mobile technology == The expansion of Android and iOS ecosystems during the 21st century contributed to the widespread use of utility applications and mobile productivity tools. Applications related to calculations, scheduling, digital organization, and educational support became increasingly common on smartphones and tablets. Mobile utility software demonstrates how modern digital platforms support accessibility and everyday online services. Independent developers have contributed to this technological ecosystem through lightweight applications focused on mobile usability and internet-based functionality. == Influence on modern society == The Information Age has reshaped the way individuals communicate, consume information, and interact with digital services. Social media platforms, artificial intelligence systems, cloud storage, and mobile computing continue to influence modern economies and online communities worldwide. Emerging technologies such as the Internet of things, machine learning, and advanced automation are often associated with the transition toward the Fourth Industrial Revolution. == History == The digital revolution converted technology from analog format to digital format. By doing this, it became possible to make copies that were identical to the original. In digital communications, for example, repeating hardware was able to amplify the digital signal and pass it on with no loss of information in the signal. Of equal importance to the revolution was the ability to easily move the digital information between media and to access or distribute it remotely. One turning point of the revolution was the change from analog to digitally recorded music. During the 1980s, the digital format of optical compact discs gradually replaced analog formats, such as vinyl records and cassette tapes, as the popular medium of choice. === Previous inventions === Humans have manufactured tools for counting and calculating since ancient times, such as the abacus, astrolabe, equatorium, and mechanical timekeeping devices. More complicated devices started appearing in the 1600s, including the slide rule and mechanical calculators. By the early 1800s, the Industrial Revolution had produced mass-market calculators like the arithmometer and the enabling technology of the punch card. Charles Babbage proposed a mechanical general-purpose computer called the Analytical Engine, but it was never successfully built, and was largely forgotten by the 20th century, and unknown to most of the inventors of modern computers. The Second Industrial Revolution, in the last quarter of the 19th century, developed useful electrical circuits and the telegraph. In the 1880s, Herman Hollerith developed electromechanical tabulating and calculating devices using punch cards and unit record equipment, which became widespread in business and government. Meanwhile, various analog computer systems used electrical, mechanical, or hydraulic systems to model problems and calculate answers. These included an 1872 tide-predicting machine, differential analysers, perpetual calendar machines, the Deltar for water management in the Netherlands, network analyzers for electrical systems, and various machines for aiming military guns and bombs. The construction of problem-specific analog computers continued in the late 1940s and beyond, with FERMIAC for neutron transport, Project Cyclone for various military applications, and the Phillips Machine for economic modeling. Building on the complexity of the Z1 and Z2, German inventor Konrad Zuse used electromechanical systems to complete in 1941 the Z3, the world's first working programmable, fully automatic digital computer. Also, during World War II, Allied engineers constructed electromechanical bombes to break the German Enigma machine encoding. The base-10 electromechanical Harvard Mark I was completed in 1944, and was to some degree improved with inspiration from Charles Babbage's designs. === 1947–1969: Origins === In 1947, the first working transistor, the germanium-based point-contact transistor, was invented by John Bardeen and Walter Houser Brattain while working under William Shockley at Bell Labs. This led the way to more advanced digital computers. From the late 1940s, universities, the military, and businesses developed computer systems to digitally replicate and automate previously manually performed mathematical calculations, with the LEO being the first commercially available general-purpose computer. Digital communication became economical for widespread adoption after the invention of the personal computer in the 1970s. Claude Shannon, a Bell Labs mathematician, is generally credited with laying the foundations of digitalization in his pioneering 1948 article, A Mathematical Theory of Communication. In 1948, Bardeen and Brattain patented an insulated-gate transistor (IGFET) with an inversion layer. Their concept forms the basis of CMOS and DRAM technology today. In 1957, at Bell Labs, Frosch and Derick were able to manufacture planar silicon dioxide transistors, later a team at Bell Labs demonstrated a working MOSFET. The first integrated circuit milestone was achieved by Jack Kilby in 1958. Other important technological developments included the invention of the monolithic integrated circuit chip by Robert Noyce at Fairchild Semiconductor in 1959, made possible by the planar process developed by Jean Hoerni. In 1963, complementary MOS (CMOS) was developed by Chih-Tang Sah and Frank Wanlass at Fairchild Semiconductor. The self-aligned gate transistor, which further facilitated mass production, was invented in 1966 by Robert Bower at Hughes Aircraft and independently by Robert Kerwin, Donald Klein, and John Sarace at Bell Labs. In 1962, AT&T deployed the T-carrier for long-haul pulse-code modulation (PCM) digital voice transmission. The T1 format carried 24 pulse-code modulated, time-division multiplexed speech signals, each encoded in 64 kbit/s streams, leaving 8 kbit/s of framing information, which facilitated the synchronization and demultiplexing at the receiver. Over the subsequent decades, the digitisation of voice became the norm for all but the last mile (where analogue continued to be the norm right into the late 1990s). Following the development of MOS integrated circuit chips in the early 1960s, MOS chips reached higher transistor density and lower manufacturing costs than bipolar integrated circuits by 1964. MOS chips further increased in complexity at a rate predicted by Moore's law, leading to large-scale integration (LSI) with hundreds of transistors on a single MOS chip by the late 1960s. The application of MOS LSI chips to computing was the basis for the first microprocessors, as engineers began recognizing that a complete computer processor could be contained on a single MOS LSI chip. In 1968, Fairchild engineer Federico Faggin improved MOS technology with his development of the silicon-gate MOS chip, which he later used to develop the Intel 4004, the first single-chip microprocessor. It was released by Intel in 1971 and laid the foundations for the microcomputer revolution that began in the 1970s. MOS technology also led to the development of semiconductor image sensors suitable for digital cameras. The first such image sensor was the charge-coupled device, developed by Willard S. Boyle and George E. Smith at Bell Labs in 1969, based on MOS capacitor technology. === 1969–1989: Invention of the internet, rise of home computers === The public was first introduced to the concepts that led to the Internet when a message was sent over the ARPANET in 1969. Packet switched networks such as ARPANET, Mark I, CYCLADES, Merit Network, Tymnet, and Telenet, were developed in the late 1960s and early 1970s using a variety of protocols. The ARPANET in particular led to the development of protocols for internetworking, in which multiple separate networks could be joined into a network of networks. The Whole Earth movement of the 1960s advocated the use of new technology. In the 1970s, the home computer was introduced, time-sharing computers, the video game console, the first coin-op vide
Frictionless sharing
Frictionless sharing refers to the transparent or automatic dissemination of user activity across social media platforms, typically without requiring explicit action from the user each time content is shared. The concept gained prominence in 2011 after Mark Zuckerberg announced a series of new features for Facebook at the F8 developers conference, framing the changes as enabling “real-time serendipity in a friction-less experience.” == History and concept == Before 2011, the term “frictionless sharing” was occasionally used in academic and technical contexts to describe sharing of resources with minimal effort, such as through social bookmarking or Creative Commons licensing to reduce barriers to reuse of research data. The concept took on a broader cultural meaning when Facebook introduced its Timeline interface and new “social apps” in 2011. These features enabled third-party applications to automatically publish user activity to the platform—effectively shifting sharing from a deliberate act to a passive process. For example, integrating music streaming service Spotify meant that any song a user listened to could automatically appear in a Facebook “Ticker,” allowing friends to see the activity and click through to play the song themselves. == Zuckerberg’s vision == Zuckerberg articulated a vision of a Web in which sharing occurs by default rather than by choice: “You read, you watch, you listen, you buy—and everyone you know will hear all about it on Facebook.” This “frictionless” model assumes ongoing consent after an initial opt-in. Once users connect an app to their profile, any future activity with that app may be automatically shared. This shift from intentional posting to ambient sharing represented a significant evolution in how personal data is distributed online. == Criticism and debate == Many commentators and users have raised concerns about frictionless sharing. While some criticism centers on online privacy, others focus on how automatic updates can flood news feeds and erode the social value of sharing. Critics argue that when sharing becomes automatic, it dilutes the personal curation that makes social media exchanges meaningful. According to Slate, this approach risks “killing taste,” because users typically choose to share only select content they find worth highlighting, rather than everything they consume. AL.com similarly observed that the frictionless model encourages over-sharing, overwhelming both users and their networks with minor or trivial activities. For example, integrating multiple platforms—such as Twitter, Foursquare, Pinterest, Spotify, and others—can create an incessant stream of updates that some users may find intrusive or irritating. This can lead to what critics describe as “narcissistic” or noisy timelines, potentially undermining the “social” nature of social media. == Business model and data implications == For Facebook, frictionless sharing offers clear business advantages. More frequent and detailed sharing provides valuable data that can be used to refine targeted advertising and personalize content delivery. The model also encourages users to spend more time on the platform, reinforcing its position as a central hub of online social activity. Other technology companies have experimented with similar approaches. Google has introduced forms of cross-platform integration that facilitate automatic activity sharing, though with a more explicit opt-in structure compared to Facebook. This approach has been described as “friction with consent,” allowing users to manually enable or disable integrations on a per-service basis.
Neural operators
Neural operators are a class of deep learning architectures designed to learn maps between infinite-dimensional function spaces. Neural operators represent an extension of traditional artificial neural networks, marking a departure from the typical focus on learning mappings between finite-dimensional Euclidean spaces or finite sets. Neural operators directly learn operators between function spaces; they can receive input functions, and the output function can be evaluated at any discretization. The primary application of neural operators is in learning surrogate maps for the solution operators of partial differential equations (PDEs), which are critical tools in modeling the natural environment. Standard PDE solvers can be time-consuming and computationally intensive, especially for complex systems. Neural operators have demonstrated improved performance in solving PDEs compared to existing machine learning methodologies while being significantly faster than numerical solvers. Neural operators have also been applied to various scientific and engineering disciplines such as turbulent flow modeling, computational mechanics, graph-structured data, and the geosciences. In particular, they have been applied to learning stress-strain fields in materials, classifying complex data like spatial transcriptomics, predicting multiphase flow in porous media, and carbon dioxide migration simulations. Finally, the operator learning paradigm allows learning maps between function spaces, and is different from parallel ideas of learning maps from finite-dimensional spaces to function spaces, and subsumes these settings as special cases when limited to a fixed input resolution. == Operator learning == Understanding and mapping relationships between function spaces has many applications in engineering and the sciences. In particular, one can cast the problem of solving partial differential equations as identifying a map between function spaces, such as from an initial condition to a time-evolved state. In other PDEs this map takes an input coefficient function and outputs a solution function. Operator learning is a machine learning paradigm to learn solution operators mapping the input function to the output function . Using traditional machine learning methods, addressing this problem would involve discretizing the infinite-dimensional input and output function spaces into finite-dimensional grids and applying standard learning models, such as neural networks. This approach reduces the operator learning to finite-dimensional function learning and has some limitations, such as generalizing to discretizations beyond the grid used in training. The primary properties of neural operators that differentiate them from traditional neural networks is discretization invariance and discretization convergence. Unlike conventional neural networks, which are fixed on the discretization of training data, neural operators can adapt to various discretizations without re-training. This property improves the robustness and applicability of neural operators in different scenarios, providing consistent performance across different resolutions and grids. == Definition and formulation == Architecturally, neural operators are similar to feed-forward neural networks in the sense that they are composed of alternating linear maps and non-linearities. Since neural operators act on and output functions, neural operators have been instead formulated as a sequence of alternating linear integral operators on function spaces and point-wise non-linearities. Using an analogous architecture to finite-dimensional neural networks, similar universal approximation theorems have been proven for neural operators. In particular, it has been shown that neural operators can approximate any continuous operator on a compact set. Neural operators seek to approximate some operator G : A → U {\displaystyle {\mathcal {G}}:{\mathcal {A}}\to {\mathcal {U}}} between function spaces A {\displaystyle {\mathcal {A}}} and U {\displaystyle {\mathcal {U}}} by building a parametric map G ϕ : A → U {\displaystyle {\mathcal {G}}_{\phi }:{\mathcal {A}}\to {\mathcal {U}}} . Such parametric maps G ϕ {\displaystyle {\mathcal {G}}_{\phi }} can generally be defined in the form G ϕ := Q ∘ σ ( W T + K T + b T ) ∘ ⋯ ∘ σ ( W 1 + K 1 + b 1 ) ∘ P , {\displaystyle {\mathcal {G}}_{\phi }:={\mathcal {Q}}\circ \sigma (W_{T}+{\mathcal {K}}_{T}+b_{T})\circ \cdots \circ \sigma (W_{1}+{\mathcal {K}}_{1}+b_{1})\circ {\mathcal {P}},} where P , Q {\displaystyle {\mathcal {P}},{\mathcal {Q}}} are the lifting (lifting the codomain of the input function to a higher dimensional space) and projection (projecting the codomain of the intermediate function to the output dimension) operators, respectively. These operators act pointwise on functions and are typically parametrized as multilayer perceptrons. σ {\displaystyle \sigma } is a pointwise nonlinearity, such as a rectified linear unit (ReLU), or a Gaussian error linear unit (GeLU). Each layer t = 1 , … , T {\displaystyle t=1,\dots ,T} has a respective local operator W t {\displaystyle W_{t}} (usually parameterized by a pointwise neural network), a kernel integral operator K t {\displaystyle {\mathcal {K}}_{t}} , and a bias function b t {\displaystyle b_{t}} . Given some intermediate functional representation v t {\displaystyle v_{t}} with domain D {\displaystyle D} in the t {\displaystyle t} -th hidden layer, a kernel integral operator K ϕ {\displaystyle {\mathcal {K}}_{\phi }} is defined as ( K ϕ v t ) ( x ) := ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x):=\int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy,} where the kernel κ ϕ {\displaystyle \kappa _{\phi }} is a learnable implicit neural network, parametrized by ϕ {\displaystyle \phi } . In practice, one is often given the input function to the neural operator at a specific resolution. For instance, consider the setting where one is given the evaluation of v t {\displaystyle v_{t}} at n {\displaystyle n} points { y j } j n {\displaystyle \{y_{j}\}_{j}^{n}} . Borrowing from Nyström integral approximation methods such as Riemann sum integration and Gaussian quadrature, the above integral operation can be computed as follows: ∫ D κ ϕ ( x , y , v t ( x ) , v t ( y ) ) v t ( y ) d y ≈ ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j , {\displaystyle \int _{D}\kappa _{\phi }(x,y,v_{t}(x),v_{t}(y))v_{t}(y)dy\approx \sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}},} where Δ y j {\displaystyle \Delta _{y_{j}}} is the sub-area volume or quadrature weight associated to the point y j {\displaystyle y_{j}} . Thus, a simplified layer can be computed as v t + 1 ( x ) ≈ σ ( ∑ j n κ ϕ ( x , y j , v t ( x ) , v t ( y j ) ) v t ( y j ) Δ y j + W t ( v t ( y j ) ) + b t ( x ) ) . {\displaystyle v_{t+1}(x)\approx \sigma \left(\sum _{j}^{n}\kappa _{\phi }(x,y_{j},v_{t}(x),v_{t}(y_{j}))v_{t}(y_{j})\Delta _{y_{j}}+W_{t}(v_{t}(y_{j}))+b_{t}(x)\right).} The above approximation, along with parametrizing κ ϕ {\displaystyle \kappa _{\phi }} as an implicit neural network, results in the graph neural operator (GNO). There have been various parameterizations of neural operators for different applications. These typically differ in their parameterization of κ {\displaystyle \kappa } . The most popular instantiation is the Fourier neural operator (FNO). FNO takes κ ϕ ( x , y , v t ( x ) , v t ( y ) ) := κ ϕ ( x − y ) {\displaystyle \kappa _{\phi }(x,y,v_{t}(x),v_{t}(y)):=\kappa _{\phi }(x-y)} and by applying the convolution theorem, arrives at the following parameterization of the kernel integral operator: ( K ϕ v t ) ( x ) = F − 1 ( R ϕ ⋅ ( F v t ) ) ( x ) , {\displaystyle ({\mathcal {K}}_{\phi }v_{t})(x)={\mathcal {F}}^{-1}(R_{\phi }\cdot ({\mathcal {F}}v_{t}))(x),} where F {\displaystyle {\mathcal {F}}} represents the Fourier transform and R ϕ {\displaystyle R_{\phi }} represents the Fourier transform of some periodic function κ ϕ {\displaystyle \kappa _{\phi }} . That is, FNO parameterizes the kernel integration directly in Fourier space, using a prescribed number of Fourier modes. When the grid at which the input function is presented is uniform, the Fourier transform can be approximated using the discrete Fourier transform (DFT) with frequencies below some specified threshold. The discrete Fourier transform can be computed using a fast Fourier transform (FFT) implementation. == Training == Training neural operators is similar to the training process for a traditional neural network. Neural operators are typically trained in some Lp norm or Sobolev norm. In particular, for a dataset { ( a i , u i ) } i = 1 N {\displaystyle \{(a_{i},u_{i})\}_{i=1}^{N}} of size N {\displaystyle N} , neural operators minimize (a discretization of) L U ( { ( a i , u i ) } i = 1 N ) := ∑ i = 1 N ‖ u i − G θ ( a i ) ‖ U 2 {\displaystyle {\mathcal {L}}_{\mathca
Infone
Infone was a service launched by Metro One Telecommunications in 2003. The service was discontinued effective December 14, 2005. == How it worked == Infone included directory assistance and other services via a toll-free phone number. A user could call 888-411-1111 to request directory assistance, directions, traffic information, movie times, call completion, dinner reservation assistance and other services. Infone provided a number of innovative 411 'concierge'-like services, including movie listings from a live operator, and offered a feature where they could provide information from a linked Microsoft Outlook calendar when set up in advance. For a period of time they advertised heavily on U.S. television, featuring ads with then Governor of Minnesota Jesse Ventura, emphasizing their use of all U.S. based operators. The price offered was $0.89 per call up to 15 minutes (for use when the operator connects you to the requested number, as well as for additional information requests afterwards), with $0.05 for each additional minute, making Infone also a competitively priced long-distance service. New users received 5–10 free calls. Infone identified a registered user (along with billing information; the service was only payable by credit card) by caller ID (numbers were registered on signing up) and by an advanced voiceprint recognition system (VPRS) from SpeechWorks that identified the user when the user called from an unregistered telephone number (or no caller ID) through the use of a personal phrase spoken by the user (e.g., "Hello Infone!") after the welcome tone.