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Jais (language model)

Jais is an open-source large language model launched in August 2023. Developed as a collaboration between Emirati AI company G42, the Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), and US-based Cerebras Systems, Jais was designed to produce high-quality Arabic text and was also trained on English data. The model's creation was motivated by the underrepresentation of the Arabic language in the field of generative artificial intelligence. It aims to provide a more culturally and linguistically accurate model for the world's 400 million Arabic speakers. Its name is a reference to Jebel Jais, the highest mountain in the UAE. == Background and development == Jais was developed in response to the limited availability of advanced generative artificial intelligence models for the Arabic language, despite it being spoken by over 400 million people. Existing models were often trained on limited or low-quality Arabic web content, resulting in poor performance. The project represents a significant investment by the United Arab Emirates in the field of AI as part of its national strategy. The model was created through a partnership between Inception (now Core42), a subsidiary of the Abu Dhabi-based AI company G42; the Mohamed bin Zayed University of Artificial Intelligence (MBZUAI); and Cerebras Systems, a US company specializing in AI hardware. The model is named after Jebel Jais, the highest peak in the UAE. == Training == The initial version of Jais released in August 2023 had 13 billion parameters. In November 2023, Core42 released Jais 30B, an improved version with 30 billion parameters. Both models were trained on a subset of the Cerebras Condor Galaxy 1 supercomputer. The training dataset consisted of a mix of Arabic, English, and computer code. According to Timothy Baldwin, a professor of natural language processing at MBZUAI, training the model on a diverse Arabic dataset allows it to switch between dialects. == Features == Jais is designed to generate text in both English and Arabic. The project has also released instruction-tuned "Chat" variants for both the 13B and 30B models, which are specifically optimized for conversational applications. Additional functionality for working with images, graphs, and tabular data is planned for future releases.
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Web application

A web application (or web app) is application software that is created with web technologies and runs via a web browser. Web applications emerged during the late 1990s and allowed for the server to dynamically build a response to the request, in contrast to static web pages. Web applications are commonly distributed via a web server. There are several different tier systems that web applications use to communicate between the web browsers, the client interface, and server data. Each system has its own uses as they function in different ways. However, there are many security risks that developers must be aware of during development; proper measures to protect user data are vital. Web applications are often constructed with the use of a web application framework. Single-page applications (SPAs) and progressive web apps (PWAs) are two architectural approaches to creating web applications that provide a user experience similar to native apps, including features such as smooth navigation, offline support, and faster interactions. Web applications are often fully hosted on remote cloud services, can require a constant connection to them, and can replace conventional desktop applications for operating systems such as Microsoft Windows, thus facilitating the operation of software as a service as it grants the developer the power to tightly control billing based on use of the remote services as well as vendor lock-in by hosting data remotely. Modern browsers such as Chrome offer sandboxing for every browser tab which improves security and restricts access to local resources. No software installation is required as the app runs within the browser which reduces the need for managing software installations. With the use of remote cloud services, customers do not need to manage servers as that can be left to the developer and the cloud service and can use the software with a relatively low power, low-resource PC such as a thin client. The source code of the application can stay the same across operating systems and devices of users with the use of responsive web design, since it only needs to be compatible with web browsers which adhere to web standards, making the code highly portable and saving on development time. Numerous JavaScript frameworks and CSS frameworks facilitate development. == History == The concept of a "web application" was first introduced in the Java language in the Servlet Specification version 2.2, which was released in 1999. At that time, both JavaScript and XML had already been developed, but the XMLHttpRequest object had only been recently introduced on Internet Explorer 5 as an ActiveX object. Beginning around the early 2000s, applications such as "Myspace (2003), Gmail (2004), Digg (2004), [and] Google Maps (2005)," started to make their client sides more and more interactive. A web page script is able to contact the server for storing/retrieving data without downloading an entire web page. The practice became known as Ajax in 2005. Eventually this was replaced by web APIs using JSON, accessed via JavaScript asynchronously on the client side. In earlier computing models like client-server, the processing load for the application was shared between code on the server and code installed on each client locally. In other words, an application had its own pre-compiled client program which served as its user interface and had to be separately installed on each user's personal computer. An upgrade to the server-side code of the application would typically also require an upgrade to the client-side code installed on each user workstation, adding to the support cost and decreasing productivity. Additionally, both the client and server components of the application were bound tightly to a particular computer architecture and operating system, which made porting them to other systems prohibitively expensive for all but the largest applications. Later, in 1995, Netscape introduced the client-side scripting language called JavaScript, which allowed programmers to add dynamic elements to the user interface that ran on the client side. Essentially, instead of sending data to the server in order to generate an entire web page, the embedded scripts of the downloaded page can perform various tasks such as input validation or showing/hiding parts of the page. "Progressive web apps", the term coined by designer Frances Berriman and Google Chrome engineer Alex Russell in 2015, refers to apps taking advantage of new features supported by modern browsers, which initially run inside a web browser tab but later can run completely offline and can be launched without entering the app URL in the browser. == Structure == Traditional PC applications are typically single-tiered, residing solely on the client machine. In contrast, web applications inherently facilitate a multi-tiered architecture. Though many variations are possible, the most common structure is the three-tiered application. In its most common form, the three tiers are called presentation, application and storage. The first tier, presentation, refers to a web browser itself. The second tier refers to any engine using dynamic web content technology (such as ASP, CGI, ColdFusion, Dart, JSP/Java, Node.js, PHP, Python or Ruby on Rails). The third tier refers to a database that stores data and determines the structure of a user interface. Essentially, when using the three-tiered system, the web browser sends requests to the engine, which then services them by making queries and updates against the database and generates a user interface. The 3-tier solution may fall short when dealing with more complex applications, and may need to be replaced with the n-tiered approach; the greatest benefit of which is how business logic (which resides on the application tier) is broken down into a more fine-grained model. Another benefit would be to add an integration tier, which separates the data tier and provides an easy-to-use interface to access the data. For example, the client data would be accessed by calling a "list_clients()" function instead of making an SQL query directly against the client table on the database. This allows the underlying database to be replaced without making any change to the other tiers. There are some who view a web application as a two-tier architecture. This can be a "smart" client that performs all the work and queries a "dumb" server, or a "dumb" client that relies on a "smart" server. The client would handle the presentation tier, the server would have the database (storage tier), and the business logic (application tier) would be on one of them or on both. While this increases the scalability of the applications and separates the display and the database, it still does not allow for true specialization of layers, so most applications will outgrow this model. == Security == Security breaches on these kinds of applications are a major concern because it can involve both enterprise information and private customer data. Protecting these assets is an important part of any web application, and there are some key operational areas that must be included in the development process. This includes processes for authentication, authorization, asset handling, input, and logging and auditing. Building security into the applications from the beginning is sometimes more effective and less disruptive in the long run. == Development == Writing web applications is simplified with the use of web application frameworks. These frameworks facilitate rapid application development by allowing a development team to focus on the parts of their application which are unique to their goals without having to resolve common development issues such as user management. In addition, there is potential for the development of applications on Internet operating systems, although currently there are not many viable platforms that fit this model.
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Apache Hama

Apache Hama is a distributed computing framework based on bulk synchronous parallel computing techniques for massive scientific computations e.g., matrix, graph and network algorithms. Originally a sub-project of Hadoop, it became an Apache Software Foundation top level project in 2012. It was created by Edward J. Yoon, who named it (short for "Hadoop Matrix Algebra"), and Hama also means hippopotamus in Yoon's native Korean language (하마), following the trend of naming Apache projects after animals and zoology (such as Apache Pig). Hama was inspired by Google's Pregel large-scale graph computing framework described in 2010. When executing graph algorithms, Hama showed a fifty-fold performance increase relative to Hadoop. Retired in April 2020, project resources are made available as part of the Apache Attic. Yoon cited issues of installation, scalability, and a difficult programming model for its lack of adoption. == Architecture == Hama consists of three major components: BSPMaster, GroomServers and Zookeeper. === BSPMaster === BSPMaster is responsible for: Maintaining groom server status Controlling super steps in a cluster Maintaining job progress information Scheduling jobs and assigning tasks to groom servers Disseminating execution class across groom servers Controlling fault Providing users with the cluster control interface. A BSP Master and multiple grooms are started by the script. Then, the bsp master starts up with a RPC server for groom servers. Groom servers starts up with a BSPPeer instance and a RPC proxy to contact the bsp master. After started, each groom periodically sends a heartbeat message that encloses its groom server status, including maximum task capacity, unused memory, and so on. Each time the BSP master receives a heartbeat message, it brings the groom server status up-to-date. The bsp master makes use of groom servers' status in order to assign tasks to idle groom servers - and returns a heartbeat response containing assigned tasks and others actions for a groom server to do. Currently BSP master has a FIFO job scheduler and simple task assignment algorithms. === GroomServer === A groom server (shortly referred to as groom) is a process that performs BSP tasks assigned by BSPMaster. Each groom contacts the BSPMaster, and it takes assigned tasks and reports its status by means of periodical piggybacks with BSPMaster. Each groom is designed to run with HDFS or other distributed storages. Basically, a groom server and a data node should be run on one physical node. === Zookeeper === A Zookeeper is used to manage the efficient barrier synchronisation of the BSPPeers.
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BioBIKE

BioBike(nee. BioLingua ) is a cloud-based, through-the-web programmable (Paas) symbolic biocomputing and bioinformatics platform that aims to make computational biology, and especially intelligent biocomputing (that is, the application of Artificial Intelligence to computational biology) accessible to research scientists who are not expert programmers. == Unique capabilities == BioBIKE is an integrated symbolic biocomputing and bioinformatics platform, built from the start as an entirely (what is now called) cloud-based architecture where all computing is done in remote servers, and all user access is accomplished through web browsers. BioBIKE has a built-in frame system in which all objects, data, and knowledge are represented. This enables code written either in the native Lisp, in the visual programming language, or systems of rules expressed in the SNARK theorem prover to access the whole of biological knowledge in an integrated manner. For its time (released in 2002) it was unique in permitting users to create fully functional biocomputing programs that run on the back-end servers entirely through the web browser UI. (In modern terms it was one of the first PaaS (Platform as a Service) systems, predating even Salesforce in this capability.) Initially this programming was carried out in raw Lisp, but Jeff Elhai's team at VCU, with NSF funding, created an entirely graphical programming environment on top of BioBIKE based upon the Boxer-style programming environments. Being a multi-headed, multi-threaded, multi-user, multi-tenancy cloud-based system, BioBIKE users were able to directly work together through their web browsers, remotely sharing the same listener and memory space. This permitted a unique sort of collaboration, discussed in Shrager (2007). A specialized offshoot of BioBIKE called "BioDeducta" includes SRI's SNARK theorem prover, offering unique "deductive biocomputing" capabilities. == Implementation == BioBIKE is open-source software implemented using the Lisp programming language. Continuing development takes place by the BioBIKE team centered at Virginia Commonwealth University . == History == BioBIKE was originally called "BioLingua", and was developed by Jeff Shrager at The Carnegie Inst. of Washington Dept. of Plant Biology, and JP Massar with funding from NASA's Astrobiology Division. Shrager and Massar wanted to create a web-based, multi-user Lisp Machine, specialized for bioinformatics. Other early contributors to the project included Mike Travers, and Jeff Elhai of VCU. Elhai obtained continuing funding from the National Science Foundation for the project, which was renamed BioBIKE. Elhai and colleagues added BioBIKE's unique visual programming language. Shrager, meanwhile, collaborated with Richard Waldinger at SRI to build SRI's (SNARK) theorem prover into BioBIKE, creating a deductive biocomputing system, called BioDeducta. == Instances == There used to be a number of BioBIKE verticals in different biological domains, including viral pathogens, cyanobacteria and other bacteria, Arabidopsis thaliana, and several others described in the references.
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RFinder

RFinder ("repeater finder") is a subscription-based website and mobile app. RFinder's main service is the World Wide Repeater Directory (WWRD), which is a directory of amateur radio repeaters. RFinder is the official repeater directory of several amateur radio associations. RFinder has listings for several amateur radio modes, including FM, D-STAR, DMR, and ATV. == World Wide Repeater Directory == Repeaters are listed in the directory along with its call sign, Maidenhead Locator System and GPS coordinates, transmit/receive offset ("split"), CTCSS and DCS squelch settings, and VoIP settings (IRLP and Echolink nodes). The directory has over 50,000 repeater listings in over 170 countries. === Website === The RFinder website has several search options including for routes. === Forums === RFinder user forums is for help and support for the app and hardware. === Mobile app === RFinder has mobile apps for Android and iOS. When using the mobile app, RFinder can display the distance to repeaters, based on the mobile device's current location. === ARRL Repeater Directory === The ARRL publishes the ARRL Repeater Directory which contains over 31,000 repeater listings for the US and Canada with listings provided by RFinder. == Subscription == RFinder requires a subscription. A one-year subscription is US$12.99. == Radio programming software == Some radio programming software applications can query RFinder and download repeater listing to program radios. Compatible software includes: CHIRP RT Systems == Radio associations == RFinder is the official repeater directory of the following associations: Amateur Radio Society Italy American Radio Relay League Cayman Amateur Radio Society Deutscher Amateur Radio Club Federacion Mexicana de Radio Experimentadores L’association Réseau des Émetteurs Français Lietuvos Radijo Mėgėjų Draugija Liga de Amadores Brasilieros de Radio Emissão Radio Amateurs of Canada Radio Society of Great Britain Rede dos Emissores Portugueses Unión de Radioaficionados Españoles
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Caspio

Caspio, Inc. is an American software company providing a low-code platform for building cloud-based business applications. Founded in 2000 by Frank Zamani, the company is headquartered in Sunnyvale, California, with operations in Poland, the Philippines, and Spain. Caspio’s platform allows organizations to create online database applications and workflow tools without extensive coding. == History == Caspio was founded by Frank Zamani in 2000. The company initially focused on simplifying custom cloud applications and reducing development time and cost as compared to traditional software development. Caspio released the first version of its platform, Caspio Bridge, in 2001. In 2014, Caspio released a HIPAA-Compliant Edition of its low-code application development platform. Caspio also released an EU General Data Protection Regulation (GDPR) Compliance Edition of its low-code application development platform in 2016. Caspio's second European Software Development Center opened in Kraków, Poland in 2017. In 2019, Forrester Research listed Caspio and three other platforms in its highest of four ranked tiers of twelve low-code platforms for business developers based on rankings of offerings and strategy at that time. Caspio also opened data centers in Montreal, Canada and India in 2020.
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Personal cloud

A personal cloud is a collection of digital content and services that are accessible from any device through the Internet. It is not a tangible entity, but a place that gives users the ability to store, synchronize, stream and share content on a relative core, moving from one platform, screen and location to another. Created on connected services and applications, it reflects and sets consumer expectations for how next-generation computing services will work. The four primary types of personal cloud in use today are: Online cloud, NAS device cloud, server device cloud, and home-made clouds. == Online cloud == The online cloud is sometimes referred to as the public cloud. It is the cloud computing model where online resources like software and data storage are made available over the Internet. Typically, an individual or organization has little control over the ecosystem in which the online cloud is hosted, and the core infrastructure is shared between many individuals and organizations. The data and applications provided by the service provider are logically segregated so that only those authorized are allowed access. == NAS device cloud == A network-attached storage (NAS) device is a computer connected to a network that provides only file-based data storage services to other devices on the network. Although it may technically be possible to run other software on a NAS device, it is not designed to be a general purpose server. Cloud NAS is remote storage that is accessed over the Internet as if it were local. A cloud NAS is often used for backups and archiving. One of the benefits of NAS Cloud is that data in the cloud can be accessed at any time from anywhere. The main drawback, however, is that the speed of the transfer rate is only as fast as the network connection the data is accessed over and can therefore be fairly slow. == Server device cloud == In many ways cloud servers work in the same way as physical servers but the functions they perform can be very different. Typically, the cloud server is an on-premises device that is connected to the Internet and gives users the functions available on the online cloud but with the added benefit and security of the files being in their control on their premises. The server cloud has been historically enterprise-based deployed by businesses needing an in-house cloud. However, there are also in-house options available for individual users. == Home-made clouds == For the more technologically proficient user a common solution for using a personal cloud is to create a home-made cloud system by connecting an external USB hard drive to a Wi-Fi router. This enables both wired and wireless computers to access the USB hard drive and use it for storage or for retrieving files a user needs to share on the network thereby acting like a cloud. Setting up a personal cloud requires a user to have particular skills in technology and network setup. One of the risks associated with improper setup is security, and leaving the files accessible to anyone with technical knowledge. Not every router supports this type of access and modification.
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Toggl Track

Toggl Track (formerly Toggl) is a time tracking software developed by Toggl OÜ which is headquartered in Tallinn, Estonia. The company offers online time tracking and reporting services through their website along with mobile and desktop applications. Time can be tracked through a start/stop button, manual entry, or dragging and resizing time blocks in a calendar view. == History == According to Alari Aho, Toggl's CEO and founder, the application has been fully self-funded from the start. The name was created using a random name generator.
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Umoove

Umoove is a high tech startup company that has developed and patented a software-only face and eye tracking technology. The idea was first conceived as an attempt to aid people with disabilities but has since evolved. The only compatibility qualification for tablet computers and smartphones to run Umoove software is a front-facing camera. Umoove headquarters are in Israel on Jerusalem’s Har Hotzvim. Umoove has 15 employees and received two million dollars in financing in 2012. The company's original founders invested around $800,000 to start the business in 2010. In 2013 Umoove was named one of the top three most promising Israeli start ups by Newsgeeks magazine. The company also participated in the 2013 LeWeb conference in Paris, France, where innovative technology startups are showcased. == Technology == The technology uses information extracted from previous frames, such as the angle of the user's head to predict where to look for facial targets in the next frame. This anticipation minimizes the amount of computation needed to scan each image. Umoove accounts for variances in environment, lighting conditions and user hand shake/movement. The technology is designed to provide a consistent experience, whether you're in a brightly lit area or a darkened basement, and to work fluidly between them by adapting its processing when it detects color and brightness shifts. It uses an active stabilization technique to filter out natural body movements from an unstable camera in order to minimize false-positive motion detection. Running the Umoove software on a Samsung Galaxy S3 is said to take up only 2% CPU. Umoove works exclusively with software and there is no hardware add-on necessary. It can be run on any smartphone or tablet computer that has a front-facing camera. Umoove claims that even a low-quality camera on an old device will run their software flawlessly. == Umoove Experience == In January 2014 Umoove released its first game onto the app store. The Umoove Experience game lets users control where they are 'flying' in the game through simple gestures and motions with their head. The avatar will basically go toward wherever the user looks. The game was created to showcase the technology for game developers but that did not stop some from criticizing its simplicity. Umoove also announced that they raised another one million dollars and that they are opening offices in Silicon Valley, California. In February 2014, Umoove announced that their face-tracking software development kit is available for Android developers as well as iOS. == Reviews == The Umoove Experience garnered mostly positive reviews from bloggers and mainstream media with some predicting that it could be the future of mobile gaming. Mashable wrote that Umoove's technology could be the emergence of gesture recognition technology in the mobile space, similar to Kinect with console gaming and what Leap Motion has done with desktop computers. Some, however, remain skeptical. CNET, for example, did not give the game a positive review and called the eye tracking technology 'freaky but cool'. They also noted that pioneering technologies have been known to fall short of expectations, citing Apple Inc’s Siri as an example. The technology blog GigaOM said that the Umoove Experience is ’awesome’ and technology evangelist Robert Scoble has called Umoove "brilliant". == uHealth == In January 2015, Umoove released uHealth, a mobile application that uses eye tracking game-like exercise to challenge the user's ability to be attentive, continuously focus, follow commands and avoid distractions. The app is designed in the form of two games, one to improve attention and another that hones focus. uHealth is a training tool, not a diagnostic. Umoove has stated that they want to use their technology for diagnosing neurological disorders but this will depend on clinical tests and FDA approval. The company cites the direct relationship between eye movements and brain activity as well as various vision-based therapies have been backed by many scientific studies conducted over the past decades. uHealth is the first time this type of therapy is delivered right to the end user through a simple download. == Collaboration rumors == In March 2013 there were rumors on the internet that Umoove would be the functioning software embedded into the Samsung Galaxy S4, which was due to launch that month. This rumor was perpetrated by, among others, New York Times, Techcrunch and Yahoo. Once Samsung launched without the Umoove technology rumors about a potential collaboration with Apple Inc hit the web. It has been said that due to the fact that Apple Inc is losing market share and stock value to Samsung they will be more aggressive and eye tracking is a logical place to make that move.
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DocuWare

DocuWare is cloud-based Software as a Service (SaaS) provider. DocuWare software provides document management, repository, and workflow automation functions (also referred to as enterprise content management (ECM) or content services). The company is headquartered in Germany and the United States. DocuWare is also the name of the flagship product offered by the company. == Company history == On October 27, 1988, DOCUNET GmbH was founded in Germering, Germany (near Munich) by President Jürgen Biffar. Since 1990, Biffar has been managing the company with his colleague, Thomas Schneck. DOCUNET AG has since been renamed and is now known as DocuWare. Since 1999, DocuWare has outsourced parts of its development to Sofia, Bulgaria. As of 2016, Nemetschek OOD had 42 employees working on the DocuWare product. DocuWare GmbH holds a 20 percent stake in Nemetschek OOD. In April 2012, an investment agreement was signed between the company and Morgan Stanley Expansion Capital LP, a Morgan Stanley Investment Management private equity fund. Its aim was promoting and accelerating the global growth of DocuWare. The legal form, AG (Public Holding Company) changed to GmbH (limited liability corporation). The company acquired U.S.-based Westbrook Technologies Inc., developer of Fortis ECM software in August 2013. In 2014, Westbrook Technologies Inc. was merged into DocuWare Corporation. At the beginning of 2016, DocuWare appointed Dr. Michael Berger as its Chief Technology Officer (CTO). Dr. Berger joined the company in 2008 as Vice President Research & Development. On January 1, 2019, Jürgen Biffar and Thomas Schneck stepped back from their operational roles after 30 years, and Dr. Michael Berger and Max Ertl started their new roles as co-presidents. On August 6, 2019, DocuWare was acquired by Ricoh. DocuWare continues to operate as a standalone subsidiary of Ricoh. In 2020, the company received approval to move its U.S. headquarters from New Windsor to Beacon, New York. === Subsidiaries === DocuWare Corporation (Beacon, NY), founded January 1, 2001 DocuWare Ltd (Nottinghamshire), founded April 1, 2005 DocuWare SARL (Paris), founded September 1, 2008 DocuWare S.L. (Barcelona), founded July 1, 2009
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Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle fg} , as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f ∗ g {\displaystyle fg} differs from cross-correlation f ⋆ g {\displaystyle f\star g} only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} . For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, computer vision and human vision, geophysics, engineering, physics, and differential equations. The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. == Definition == The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle fg} , denoting the operator with the symbol ∗ {\displaystyle } . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform: ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} An equivalent definition is (see commutativity): ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( t − τ ) g ( τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .} While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of the input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along the τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ for f , g : [ 0 , ∞ ) → R . {\displaystyle (fg)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .} For the multi-dimensional formulation of convolution, see domain of definition (below). === Notation === A common engineering notational convention is: f ( t ) ∗ g ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ ⏟ ( f ∗ g ) ( t ) , {\displaystyle f(t)g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(fg)(t)},} which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)g(t-t_{0})} is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (fg)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (fg)(t-2t_{0})} . === Relations with other transforms === Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u} and G ( s ) = ∫ − ∞ ∞ e − s v g ( v ) d v {\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v} respectively, the convolution operation ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u ⋅ ∫ − ∞ ∞ e − s v g ( v ) d v = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s ( u + v ) f ( u ) g ( v ) d u d v {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}} Let t = u + v {\displaystyle t=u+v} , then F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s t f ( u ) g ( t − u ) d u d t = ∫ − ∞ ∞ e − s t ∫ − ∞ ∞ f ( u ) g ( t − u ) d u ⏟ ( f ∗ g ) ( t ) d t = ∫ − ∞ ∞ e − s t ( f ∗ g ) ( t ) d t . {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(fg)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(fg)(t)\ {\text{d}}t.\end{aligned}}} Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. == Visual explanation == == Historical developments == One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: ∫ f ( u ) ⋅ g ( x − u ) d u {\displaystyle \int f(u)\cdot g(x-u)\,du} is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: ∫ 0 t φ ( s ) ψ ( t − s ) d s , 0 ≤ t < ∞ , {\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,} is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. == Circular c
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List of Ruby software and tools

This is a list of software and programming tools for the Ruby programming language, which includes libraries, web frameworks, implementations, tools, and related projects. == Web tools == Capistrano (software) – remote server automation tool Mongrel – Ruby web server Rack – interface between web servers and web applications Ruby on Rails – full-stack web application framework Sinatra – lightweight Ruby web application framework Spree Commerce – e-commerce platform WEBrick – Ruby HTTP server toolkit == Libraries == BioRuby – bioinformatics and computational biology library for Ruby Bogus – Ruby library for creating reliable test doubles with contract verification ERuby – embedded Ruby templating EventMachine – event-driven I/O library Factory Bot – test fixtures library Fat comma – Ruby library for JSON-like hash syntax Geocoder – Ruby library for geocoding and reverse geocoding addresses Haml – HTML templating engine Markaby – HTML generation via Ruby Nokogiri – XML/HTML parsing library RSpec – behavior-driven testing framework for Ruby RubyGems – package manager for Ruby libraries and applications Sass – CSS preprocessor Sidekiq – background job framework for Ruby, used to handle asynchronous tasks. Uconv – Unicode text conversion library Watir – web application testing framework == Ruby implementations == HotRuby – Ruby interpreter implemented in JavaScript, enabling Ruby code to run in web browsers. IronRuby – Ruby for .NET platform JRuby – Ruby on the Java Virtual Machine MacRuby – Ruby implementation for macOS Mod ruby – Apache module that embeds the Ruby interpreter to improve performance of Ruby web applications Mruby – lightweight Ruby interpreter Rubinius – alternative Ruby implementation, based loosely on the Smalltalk-80 Blue Book design. Ruby MRI – the standard Ruby interpreter YARV – "Yet Another Ruby VM," the bytecode interpreter used in modern Ruby implementations == Tools == Homebrew – package manager for macOS and Linux written in Ruby Pry – interactive Ruby shell Rake – build and task management Ruby Version Manager – environment manager RubyCocoa – bridge between Ruby and Cocoa RubyForge – project hosting site RubyMotion – for iOS/macOS development RubySpec – language specification tests == Integrated Development Environments == Aptana Studio — integrated RadRails plugin for Ruby on Rails development Eclipse DLTK Ruby Plugin — Ruby development plugin for Eclipse Eric — open-source Python-based IDE with Ruby support Komodo IDE — commercial cross-platform IDE with Ruby support RubyMine — commercial IDE for Ruby and Rails by JetBrains SlickEdit — commercial cross-platform IDE with Ruby support == List of websites using Ruby on Rails == Airbnb Basecamp Diaspora – decentralized social network application built with Ruby on Rails Discourse – open-source discussion platform built with Ruby on Rails Fiverr GitHub Hulu Shopify SoundCloud Twitch Zendesk
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Blackmagic Design

Blackmagic Design Pty Ltd is an Australian company that develops digital cinema technology and manufactures professional video production hardware and software. Headquartered in South Melbourne, it is known for producing high-end digital movie cameras and a range of broadcast and post-production equipment. The company also develops software applications, including the DaVinci Resolve application for non-linear video editing, color correction, color grading, visual effects, and audio post-production. == History == Blackmagic Design Pty Ltd was founded on 7 September 2001 by Grant Petty. Its first product, DeckLink, introduced in 2002, was a video capture card for macOS that supported uncompressed 10-bit video, marking a shift toward professional-grade yet affordable video workflows. Subsequent versions—including the DeckLink 2, Pro SDI, HD Plus, and Multibridge—added capabilities such as color correction, Windows support, and compatibility with major editing software like Adobe Premiere Pro, to broaden the product's appeal. At the 2012 NAB Show, Blackmagic announced its first Cinema Camera, a digital movie camera. Blackmagic made several acquisitions over the next decade. In 2009, it acquired da Vinci Systems, known for its color-grading tools. In 2010, it acquired Echolab's ATEM switcher line, in 2014, it added eyeon Software (developer of the Blackmagic Fusion compositing software) and London's Cintel (film scanning and restoration), and in 2016, it acquired Fairlight, an audio technology company known for its CMI synthesizers as well as mixing consoles. == Products == List of all products developed by the company. Editing, Color Correction and Audio Post Production DaVinci Resolve (free version) and DaVinci Resolve Studio (paid version), computer software for non-linear video editing, color correction, color grading, visual effects, and audio post-production. Audio/Video Controller Consoles: Editor Keyboard, Speed Editor, DaVinci Resolve Replay Editor, Micro Panel, Mini Panel, DaVinci Resolve Micro Color Panel, Advanced Panel, Fairlight Console Channel Fader, Fairlight Console Channel Control, Fairlight Console LCD Monitor, Fairlight Console Audio Editor, Fairlight Desktop Audio Editor, Fairlight Desktop Console, Fairlight Audio Interface Cintel Film Scanner (Generations 1-3) Live Production Home Streaming: ATEM Mini, ATEM Mini Pro/ISO, ATEM Mini Extreme, ATEM Mini Extreme ISO (The ATEM Mini series has both HDMI and SDI variants) Production Switchers: ATEM 1,2 & 4 M/E Constellation HD, ATEM 1,2 & 4 M/E Constellation 4K, ATEM Constellation 8K, ATEM 1,2 & 4 M/E Production Studio 4K, ATEM Television Studio HD8 & HD8 ISO Switcher & Camera Controllers: ATEM Camera Control Panel, ATEM 1 M/E Advanced Panel, ATEM 2 M/E Advanced Panel, ATEM 4 M/E Advanced Panel Chroma Keyers: Ultimatte 12 HD Mini, Ultimatte 12 HD, Ultimatte 12 4K, Ultimatte 12 8K Recording and Storage: HyperDeck Studio HD Mini, HyperDeck Studio HD Plus, HyperDeck Studio HD Plus, HyperDeck Studio 4K Pro, HyperDeck Extreme 8K HDR, HyperDeck Extreme 4K HDR, HyperDeck Extreme Control, HyperDeck Shuttle HD, Duplicator 4K, MultiDock 10G, Video Assist 7" 12G HDR, Video Assist 5" 12G HDR Capture and Playback UltraStudio: 3G, HD Mini, 4K Mini, 4K Extreme 3 DeckLink (PCIe cards): Mini Recorder, Mini Monitor, Mini Monitor 4K, Mini Recorder 4K, Duo 2 Mini, Duo 2, Quad 2, SDI 4K, Studio 4K, 4K Extreme 12G, 8K Pro, Quad HDMI Recorder Network Storage Cloud Store Cloud Pod Broadcast Converters Micro Converter: BiDirectional SDI/HDMI 3G wPSU, HDMI to SDI 3G wPSU, SDI to HDMI 3G wPSU, BiDirectional SDI/HDMI 3G, HDMI to SDI 3G, SDI to HDMI 3G Mini Converters: Audio to SDI, Optical Fiber 12G, SDI Multiplex 4K, Quad SDI to HDMI 4K, SDI Distribution 4K, SDI to Analog 4K, Audio to SDI 4K, SDI to Audio 4K, HDMI to SDI 6G, SDI to HDMI 6G Teranex Mini: SDI Distribution 12G, SDI to HDMI 12G, Audio to SDI 12G, SDI to Analog 12G, SDI to HDMI 8K HDR, SDI to DisplayPort 8K HDR 2110 IP Converters Routing and Distribution Videohub
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Princh

Princh is a Danish software company, which is headquartered in Aarhus, Denmark. Founded in 2015, Princh develops cloud printing and electronic payment products. The company is headquartered in the city of Aarhus. While utilizing a smartphone or web app, users can locate a nearby printer to their current location, get directions to access said printer, and/or authorize a print and pay for the print job in question. The product is available as a native mobile apps for Android and iOS, as well as on web and desktop products for businesses and libraries. The app connects a network of printer owners and users around the world. Princh supports an array of printable files. == History == The company was founded in 2015. The company is currently based in the southern part of Aarhus. The Princh printing service was officially launched on June 23, 2015. Currently, Princh is available as a service in a multitude of locations such as print shops, libraries, hotels, or universities. Princh is a popular printing and payment product among libraries and can among other places be found in Denmark, Sweden, Norway, Germany, United Kingdom, United States, and Canada. == How it works == With the Princh app, users will be able to locate their nearest printer. Once the user is at the printer, the user chooses the document to be printed out and shares it with the Princh app. The user then selects the desired nearby printer entering the printer ID number or scanning the QR-code located on top of the printer, pays electronically and the print job is processed by the printer. Printer owners get access to a personal control panel where they can set printing prices and monitor all Princh activity for their business. == Notes and references ==
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Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle fg} , as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f ∗ g {\displaystyle fg} differs from cross-correlation f ⋆ g {\displaystyle f\star g} only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} . For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, computer vision and human vision, geophysics, engineering, physics, and differential equations. The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. == Definition == The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle fg} , denoting the operator with the symbol ∗ {\displaystyle } . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform: ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} An equivalent definition is (see commutativity): ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( t − τ ) g ( τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .} While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of the input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along the τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ for f , g : [ 0 , ∞ ) → R . {\displaystyle (fg)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .} For the multi-dimensional formulation of convolution, see domain of definition (below). === Notation === A common engineering notational convention is: f ( t ) ∗ g ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ ⏟ ( f ∗ g ) ( t ) , {\displaystyle f(t)g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(fg)(t)},} which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)g(t-t_{0})} is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (fg)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (fg)(t-2t_{0})} . === Relations with other transforms === Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u} and G ( s ) = ∫ − ∞ ∞ e − s v g ( v ) d v {\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v} respectively, the convolution operation ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u ⋅ ∫ − ∞ ∞ e − s v g ( v ) d v = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s ( u + v ) f ( u ) g ( v ) d u d v {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}} Let t = u + v {\displaystyle t=u+v} , then F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s t f ( u ) g ( t − u ) d u d t = ∫ − ∞ ∞ e − s t ∫ − ∞ ∞ f ( u ) g ( t − u ) d u ⏟ ( f ∗ g ) ( t ) d t = ∫ − ∞ ∞ e − s t ( f ∗ g ) ( t ) d t . {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(fg)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(fg)(t)\ {\text{d}}t.\end{aligned}}} Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. == Visual explanation == == Historical developments == One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: ∫ f ( u ) ⋅ g ( x − u ) d u {\displaystyle \int f(u)\cdot g(x-u)\,du} is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: ∫ 0 t φ ( s ) ψ ( t − s ) d s , 0 ≤ t < ∞ , {\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,} is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. == Circular c
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