AI Face Verification Generator

AI Face Verification Generator — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Database virtualization

  Database virtualization is the decoupling of the database layer, which lies between the storage and application layers within the application stack. Virtualization of the database layer enables a shift away from the physical, toward the logical or virtual. Virtualization enables compute and storage resources to be pooled and allocated on demand. This enables both the sharing of single server resources for multi-tenancy, as well as the pooling of server resources into a single logical database or cluster. In both cases, database virtualization provides increased flexibility, more granular and efficient allocation of pooled resources, and more scalable computing. == Virtual data partitioning == The act of partitioning data stores as a database grows has been in use for several decades. There are two primary ways that data has been partitioned inside legacy data management systems: Shared-data databases: an architecture that assumes all database cluster nodes share a single partition. Inter-node communications are used to synchronize update activities performed by different nodes on the cluster. Shared-data data management systems are limited to single-digit node clusters. Shared-nothing databases: an architecture in which all data is segregated to internally managed partitions with clear, well-defined data location boundaries. Shared-nothing databases require manual partition management. In virtual partitioning, logical data is abstracted from physical data by autonomously creating and managing large numbers of data partitions (100s to 1000s). Because they are autonomously maintained, the resources required to manage the partitions are minimal. This kind of massive partitioning results in: Partitions that are small, efficiently managed, and load-balanced. Systems that do not require re-partitioning events to define additional partitions, even when the hardware is changed. “Shared-data” and “shared-nothing” architectures allow scalability through multiple data partitions and cross-partition querying and transaction processing without full partition scanning. == Horizontal data partitioning == Partitioning database sources from consumers is a fundamental concept. With greater numbers of database sources, inserting a horizontal data virtualization layer between the sources and consumers helps address this complexity. Rick van der Lans, the author of multiple books on SQL and relational databases, has defined data virtualization as "the process of offering data consumers a data access interface that hides the technical aspects of stored data, such as location, storage structure, API, access language, and storage technology." == Advantages == Added flexibility and agility for existing computing infrastructure. Enhanced database performance. Pooling and sharing computing resources, either splitting them (multi-tenancy) or combining them (clustering). Simplification of administration and management. Increased fault tolerance.

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  ArcObjects

  ArcObjects is a development environment of the ArcGIS family of applications. Using Visual Basic for Applications, C# or Java SDK for ArcGIS, it allows developers to extend these applications.ArcObjects is a library of Component Object Model (COM) components that build up the foundation of Esri's ArcGIS platform. ArcObjects is written primarily in the C++ programming language. Since ArcGIS is completely built on top of ArcObjects, the ArcGIS platform can be fully customized and extended by making use of its COM services and capabilities. This allows for easy extension of the ArcObjects data model with any programming language that is compatible with COM, such as Visual Basic, C#, Visual Basic.NET, Java and Python. COM enables components to be reused at a binary level, meaning developers do not require access to the source code of ArcObjects in order to extend the ArcGIS platform. For this reason, an ArcObjects programmer can make use of any type inside the ArcObjects system without knowing the implementation details of the type, only needing to know what the type is able to do. The ArcObjects data model is based on the COM standard, which makes it compatible with other COM objects and applications. This allows for easy integration and collaboration with other systems that are also based on the COM standard. The ArcGIS platform was built using ArcObjects types, such as classes, interfaces, and enumerations. ArcObjects use COM interfaces to organize and communicate properties and methods of its classes, ensuring compatibility with other COM-based objects and systems. When working with an ArcObjects COM class, its properties and methods are accessed solely through one of its implemented interfaces via the process of Query Interface (QI). Multiple interfaces are commonly available for classes in ArcObjects. For example, it is possible to query for additional interfaces implemented by an object after instantiation via the process of QI. Although only one interface can be used when instantiating an object, multiple interfaces are often available for classes in ArcObjects, allowing for greater flexibility and compatibility with other systems based on the COM standard.

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  SmartQVT

  SmartQVT is a unmaintained (since 2013) full Java open-source implementation of the QTV-Operational language which is dedicated to express model-to-model transformations. This tool compiles QVT transformations into Java programs to be able to run QVT transformations. The compiled Java programs are EMF-based applications. It is provided as Eclipse plug-ins running on top of the EMF metamodeling framework and is licensed under EPL. == Components == SmartQVT contains 3 main components: a code editor: this component helps the user to write QVT code by highlighting key words. a parser: this component converts QVT code files into model representations of the QVT programs (abstract syntax). a compiler: this component converts model representations of the QVT program into executable Java programs.

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  ObjectVision

  ObjectVision was a forms-based programming language and environment for Windows 3.x developed by Borland. The latest version, 2.1, was released in 1992. An ObjectVision application is composed by forms designed in a graphic way that contains objects and events to provide interactivity. Forms are connected together with logic in the form of decision trees. ObjectVision applications also can interact with databases using multiple engines, like Paradox and dBase. A finished project is saved as an OVD file, that is executed by an interpreted runtime that can be freely distributed. ObjectVision was not used broadly except in some niche segments, but the visual programming ideas were the basis for Borland Delphi.

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  Computer-aided lean management

  Computer-aided lean management, in business management, is a methodology of developing and using software-controlled, lean systems integration. Its goal is to drive innovation towards cost and cycle-time savings. It attempts to create an efficient use of capital and resources through the development and use of one integrated system model to run a business's planning, engineering, design, maintenance, and operations. == Overview == Computer-Aided Lean Management (CALM) is a management philosophy that uses software to reduce risk and inefficiencies. CALM acts on uncertainties and business inefficiencies to increase profitability through the use of computational decision-making tools that enable opportunities for additional value creation. It is based on the application of software to enable continuous improvement through an Integrated System Model (ISM) of the business’s physical assets, business processes, and machine learning. This integration of software applications using lean principles was developed in the aerospace industry and has migrated to the energy industry. The creation of an ISM removes the barriers posed by the silos or stovepipes inherent in the departmentalization of most companies. Integration enables lean uses of information for the creation of actionable knowledge. CALM strives to create such a lean management approach to running the company through the rigors of software enforcement. From this software enforcement comes clear policy and procedures that are adhered to, activity-based costing, measurement of effectiveness, and the capability of using advanced algorithms for dramatic improvements in optimization of resources. CALM creates business capabilities through software to enable technology application, streamlining of processes, and a lean organizational structure. The methodology is based on a common sense approach for running a business, by measuring actions taken and using those measurements to design more efficient processes. == History == CALM was inspired by lean processes and techniques that were already dominant management technologies with a wide diversity of applications and successes. Motorola and General Electric had been known for the concepts of Six Sigma; Boeing had been managing mass (using modular and flexible assembly options), and Toyota combined elements of these methodologies to create the Toyota Production System. Boeing then took the Toyota model and added computer-aided enforcement of lean methodologies throughout the manufacturing process. One of the major sources for CALM's outgrowth was integrated definition (IDEF) modeling in aerospace manufacturing that was pioneered by the U.S. Air Force in the 1970s. IDEF is a methodology designed to model the end-to-end decisions, actions, and activities of an organization or system so that costs, performance, and cycle times can be optimized. IDEF methods have been adapted for wider use in automotive, aerospace, pharmaceuticals, and software development industries. IDEF methods serve as a starting point to understand lean management through semantic data modeling. The IDEF process begins by mapping the existing functions of an enterprise, creating a graphical model, or road map, that shows what controls each important function, who performs it, what resources are required for carrying it out, what it produces, how much it costs, and what relationships it has to other functions of the organization. IDEF simulations have been found to be efficient at streamlining and modernizing both companies and governmental agencies. Perhaps the best-developed evolution of the IDEF model beyond Toyota was at Boeing. Their project life-cycle process has grown into a rigorous software system that links people, tasks, tools, materials, and the environmental impact of any newly planned project, before any building is allowed to begin. Routinely, more than half of the time for any given project is spent building the precedence diagrams, or three-dimensional process maps, integrating with outside suppliers, and designing the implementation plan–all on the computer. Once real activity is initiated, an action tracker is used to monitor inputs and outputs versus the schedule and delivery metrics in real time throughout the organization. When the execution of a new airplane design begins, it is so well organized that it consistently cuts both costs and build time in half for each successive generation of airframe. Boeing created a complex lean management process called 'define and control airplane configuration/manufacturing resource management' (DCAC/MRM). The process was built with the help of the operations research and computer sciences departments of the University of Pittsburgh. The manufacture of the Boeing 777 was ultimately a success, and it became the precursor to succeeding generations of CALM at Boeing. The methodology of CALM has recently been applied to field orientated infrastructure based businesses with highly interdependent systems, such as electric utilities where a smart grid concept is being researched and developed. The management of infrastructure-based industries like oil, gas, electricity, water, transportation, and renewables requires massive investments in interdependent, physical infrastructure, as well as simultaneous attention to disparate market forces. In infrastructure businesses that manage field assets, uncertainty is the biggest impediment to profitability, rather than the maintenance of efficient supply chains or the management of factory assembly lines. These businesses are dominated by risk from uncertainties such as weather, market variations, transportation disruptions, government actions, logistic difficulties, geology, and asset reliability. CALM has been applied to deal with these types of infrastructure based challenges.

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  ShowScoop

  ShowScoop is a website and mobile app platform on which users can rate and review artists, concerts, and music festivals that they have seen/attended. The reviews and ratings are designed to be informative of how well such performances are live. This helps concert-goers decide which live music events they want to attend. == History == ShowScoop was founded in August 2012 by Micah Smurthwaite and is based out of San Diego, CA. In February 2013, ShowScoop launched its mobile app at the SF Music Tech Summit. The application is currently available on the iPhone, with plans to expand into the Android market in the future. == Services == ShowScoop uses crowdsourcing to provide accurate ratings of live concert experiences. In addition to viewing ratings, users are encouraged to rate and review concerts they have attended. The ShowScoop database includes nearly one million artists and over 2.5 million live music events. ShowScoop users can rate artists on four aspects of the performance: stage presence, crowd interaction, sound quality, and visual effects. The rating system uses an ascending scale from one to five in each of the aspects, with five being the highest score. In addition to the quantitative ratings, ShowScoop users are also free to write qualitative reviews in a provided comment section. This allows users to explain their ratings and add further insight or opinion. ShowScoop incorporates several facets of social media into its services. Users can create a user profile to share limited personal information and store their ratings and reviews. Users are also given the option of sharing their evaluations with their social networks on Facebook and Twitter. Users can "like" reviews, follow artists, and follow other ShowScoop users. The mobile app allows users to take photos, apply filters, and share the final image in conjunction with reviews and through Instagram. == Road Crew == ShowScoop's "Road Crew" is a group made up of top contributors within the ShowScoop community. The Road Crew assists in curating artist pages, assuring information quality and accuracy. In return, members of the Road Crew are given incentives, including free tickets to concerts and personal invitations to exclusive shows. Applicants to the Road Crew are judged on the number and quality of their reviews, the photos and videos they have posted, and their general engagement with the ShowScoop community in following and liking users and reviews.

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  Azure Data Lake

  Azure Data Lake is a scalable data storage and analytics service. The service is hosted in Azure, Microsoft's public cloud. == History == Azure Data Lake service was released on November 16, 2016. It is based on COSMOS, which is used to store and process data for applications such as Azure, AdCenter, Bing, MSN, Skype and Windows Live. COSMOS features a SQL-like query engine called SCOPE upon which U-SQL was built. == Storage == Data Lake Storage is a cloud service to store structured, semi-structured or unstructured data produced from applications including social networks, relational data, sensors, videos, web apps, mobile or desktop devices. A single account can store trillions of files where a single file can be greater than a petabyte in size. == Analytics == Data Lake Analytics is a parallel on-demand job service. The parallel processing system is based on Microsoft Dryad. Dryad can represent arbitrary Directed Acyclic Graphs (DAGs) of computation. Data Lake Analytics provides a distributed infrastructure that can dynamically allocate resources so that customers pay for only the services they use. The system uses Apache YARN, the part of Apache Hadoop which governs resource management across clusters. Data Lake Store supports any application that uses the Hadoop Distributed File System (HDFS) interface. == U-SQL == U-SQL is a query language for Data Lake Analytics parallel data transformation and processing programs. It combines SQL and C#: it is and an evolution of the declarative SQL language with native extensibility through user code written in C#. U-SQL uses C# data types and the C# expression language. == Retirement == In 2021, Microsoft announced the 2024 retirement of the original Azure Data Lake Storage, now called "Gen1". The related Azure Data Lake Analytics / U-SQL technologies are also being retired. Azure Data Lake Storage Gen2, an extension of Azure Storage, will continue. The suggested replacement technologies are Azure Synapse Analytics and Apache Spark.

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  WiPay

  WiPay is a Caribbean-based payment technology company that specializes in electronic payments for businesses. WiPay was founded in 2016 by Aldwyn Wayne Jr., a Trinidadian businessman and graduate of Georgia Tech Institute. In September 2019, WiPay partnered with MasterCard. As a result, WiPay became the only licensed Payment Facilitator (PAYFAC) on both the MasterCard and Visa networks in the region.

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  TalkBack

  TalkBack is an accessibility service for the Android operating system that helps blind and visually impaired users to interact with their devices. It uses spoken words, vibration and other audible feedback to allow the user to know what is happening on the screen allowing the user to better interact with their device. The service is pre-installed on many Android devices, and it became part of the Android Accessibility Suite in 2017. According to the Google Play Store, the Android Accessibility Suite has been downloaded over five billion times, including devices that have the suite preinstalled. == Open-source == Google releases the source code of TalkBack with some releases of the accessibility service to GitHub, with the latest of these changes being from May 6, 2021. The source for these versions of Google TalkBack have been released under the Apache License version 2.0. == Release history ==

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  Fuse Mediation Router

  Fuse Mediation Router is an open source tool for integrating services using Enterprise Integration Patterns based on Apache Camel for use in enterprise IT organizations. It is certified, productized and fully supported by the people who wrote the code. Fuse Mediation Router uses a standard method of notation to go from diagram to implementation without coding. Fuse Mediation Router is a rule-based routing and process mediation engine that combines the ease of basic POJO development with the clarity of the standard Enterprise Integration Patterns. It can be deployed inside any container or be used stand-alone, and works directly with any kind of transport or messaging model to rapidly integrate existing services and applications. Fuse Mediation Router is now a part of Red Hat JBoss Fuse. == Tooling == FuseSource offers graphical, Eclipse-based tooling for Apache Camel for download.

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  AlternativeTo

  AlternativeTo is a website which lists alternatives to web-based software, desktop computer software, and mobile apps, and sorts the alternatives by various criteria, including the number of registered users who have "Liked" each of them on AlternativeTo. Users can search the site to find better alternatives to an application they are using or previously have used, including free alternatives such as free web applications (cloud computing) which don't require any installation and can be accessed from any browser. == Differences == Unlike a number of other software directory websites, the software is not arranged into categories, but each individual piece of software has its own list of alternatives. However, users can also search by tag to find software, which offers an alternative way of finding related software. Users can also narrow their search by focusing on particular platforms and license types (such as "free for non-commercial use"). AlternativeTo lists basic information such as platform and license type at the top of each entry, but does not have comparison tables listing software features side by side. AlternativeTo does not host software for download but it provides links to official websites to where you can download or buy them. AlternativeTo allows anyone to register and suggest new alternatives, or to update the information held about existing entries. Suggestions and alterations are reviewed before being made publicly visible. Users can register using either email and password or OpenID. Login with Facebook has been discontinued. As AlternativeTo is itself a web application, it even has a page for alternatives to itself. == Features == Tweets from anyone mentioning particular pieces of software are also pulled in dynamically from Twitter. Each application has an RSS feed for notifying users of newly listed alternatives to that application. After a user has clicked the Like button next to an application, they are offered the opportunity to tell their friends on Facebook or their followers on Twitter that they liked it. The site also has a forum. For software developers, a JSON API used to be available, but has been taken down indefinitely.

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  Dropbox Paper

  Dropbox Paper, or simply Paper, is a collaborative document-editing service developed by Dropbox. Originating from the company's acquisition of document collaboration company Hackpad in April 2014, Dropbox Paper was officially announced in October 2015, and launched in January 2017. It offers a web application, as well as mobile apps for Android and iOS. Dropbox Paper was described in the official announcement post as "a flexible workspace that brings people and ideas together. With Paper, teams can create, review, revise, manage, and organize — all in shared documents". Reception of Dropbox Paper has been mixed. Critics praised collaboration functionality, including content available immediately, the ability to mention specific collaborators, assign tasks, write comments, as well as editing attribution, and revision history. It received particular praise for its support for rich media from a variety of sources, with one reviewer noting that the Paper's support for rich media exceeds the capabilities of most of its competitors. However, it was criticized for a lack of formatting options and editing features. While the user interface was liked for being minimal, reviewers cited the lack of a fixed formatting bar and missing features present in competitors' products as making Dropbox Paper seem like a "light" tool. == History == Dropbox acquired document collaboration company Hackpad in April 2014. A year later, Dropbox launched a Dropbox Notes note-taking product in beta testing phase. Dropbox Paper was officially announced on October 15, 2015, followed by an open beta and release of mobile Android and iOS apps in August 2016. Dropbox Paper was officially released on January 30, 2017. == Reception == In a comparison between Dropbox Paper and Evernote, PC World's Michael Ansaldo wrote that "With its emphasis on document creation, you might expect formatting to be front and center in Dropbox Paper. That's not the case." Ansaldo noted the lack of a "fixed formatting toolbar as you'd find in Evernote or a word processor like Google Docs or Microsoft Word. Instead, the text editor appears as a floating ribbon only when you highlight selected text." The only formatting options available for emphasis were bolding, strikethrough, bulleted and numbered lists, and H1 and H2 tags. Users can also add links, convert text to checklists, and add comments. Ansaldo wrote that "Both Evernote and Dropbox Paper make it easy to add images to a document", but also noted that "Dropbox Paper doesn't support any image editing". Paper supports rich media, and users can "add rich content to your document just by pasting a link to the file. In addition to Dropbox, Paper supports media from a variety of popular services including YouTube, Spotify, Vimeo, SoundCloud, Facebook, and Google's productivity suite. Once the file appears, you can delete the link for a cleaner display." To start working with other people, Paper "allows you to invite people via email from within a document", with sharing options for who can view the link (anyone with the link or just the invited person), and action permissions (edit or only comment). Regarding collaboration, Ansaldo wrote that "Creative collaboration is Paper’s marquee feature, and it provides a variety of ways to work effectively with others in real time". Users can "make any content immediately visible and accessible to a specific collaborator with "@mentions"", and "You can also use @mentions to create and assign task lists within a document." Paper also "boasts essential collaboration tools including comments, editing attribution, and revision history." Writing for TechRadar, John Brandon wrote that Dropbox Paper "might be a 'light' tool for now without the extensive templates of Microsoft Office or the integration with other apps in the Zoho suite, but it does work well with the Dropbox storage service that's so popular with office workers these days." Kyle Wiggers of Digital Trends wrote that Paper is "all about minimizing distractions. Its interface is quite literally a big, blank canvas on which you tap out your agenda. You can organize notes by title and create to-do lists, but even basic formatting tools are obscured from view", noting Paper's "floating box above words and phrases highlighted by your cursor". Wiggers stated that "Paper is not a to-do organizer", but that it's "well suited to the purpose thanks to a bevy of labor-saving conveniences", highlighting that Paper "supports more media than most of its to-do and note-taking counterparts". He praised the collaboration tools, writing that they "are as extensive as you'd hope, and then some", citing its invitation system with permission controls, lists of changes and revision history, comment and chat support, and "perhaps best of all", the ability to assign tasks with a "@" mention. Business Insider's Alex Heath praised that "Paper's interface is spotless and friendly to write in. You don't feel overwhelmed with formatting options", but criticized the available features, writing that "Google Docs is much more full-featured in the formatting department, so Paper has some catching up to do if it wants to be on par with the competition". Writing for The Verge, Casey Newton praised Paper's handling of rich media, complimenting it for being "great", and added that "I imagine that creative types who work on teams will appreciate having rich media embedded in the documents they're working on rather than in a series of infinite tabs".

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  Semantic analytics

  Semantic analytics, also termed semantic relatedness, is the use of ontologies to analyze content in web resources. This field of research combines text analytics and Semantic Web technologies like RDF. Semantic analytics measures the relatedness of different ontological concepts. Some academic research groups that have active project in this area include Kno.e.sis Center at Wright State University among others. == History == An important milestone in the beginning of semantic analytics occurred in 1996, although the historical progression of these algorithms is largely subjective. In his seminal study publication, Philip Resnik established that computers have the capacity to emulate human judgement. Spanning the publications of multiple journals, improvements to the accuracy of general semantic analytic computations all claimed to revolutionize the field. However, the lack of a standard terminology throughout the late 1990s was the cause of much miscommunication. This prompted Budanitsky & Hirst to standardize the subject in 2006 with a summary that also set a framework for modern spelling and grammar analysis. In the early days of semantic analytics, obtaining a large enough reliable knowledge bases was difficult. In 2006, Strube & Ponzetto demonstrated that Wikipedia could be used in semantic analytic calculations. The usage of a large knowledge base like Wikipedia allows for an increase in both the accuracy and applicability of semantic analytics. == Methods == Given the subjective nature of the field, different methods used in semantic analytics depend on the domain of application. No singular methods is considered correct, however one of the most generally effective and applicable method is explicit semantic analysis (ESA). ESA was developed by Evgeniy Gabrilovich and Shaul Markovitch in the late 2000s. It uses machine learning techniques to create a semantic interpreter, which extracts text fragments from articles into a sorted list. The fragments are sorted by how related they are to the surrounding text. Latent semantic analysis (LSA) is another common method that does not use ontologies, only considering the text in the input space. == Applications == Entity linking Ontology building / knowledge base population Search and query tasks Natural language processing Spoken dialog systems (e.g., Amazon Alexa, Google Assistant, Microsoft's Cortana) Artificial intelligence Knowledge management The application of semantic analysis methods generally streamlines organizational processes of any knowledge management system. Academic libraries often use a domain-specific application to create a more efficient organizational system. By classifying scientific publications using semantics and Wikipedia, researchers are helping people find resources faster. Search engines like Semantic Scholar provide organized access to millions of articles.

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  Zamzar

  Zamzar is an online file converter and compressor, created by brothers Mike and Chris Whyley in England in 2006. It allows users to convert files online, without downloading a software tool, and supports over 1,200 different conversion types. Since its formation, the service has converted over 510 million files for users from 245 different countries. The service supports the conversion of documents, images, audio, video, e-Books, CAD files and compressed file formats. Users can type in a URL or upload one or more files (if they are all of the same format) from their computer; Zamzar will then convert the file(s) to another user-specified format, such as an Adobe PDF file to a Microsoft Word document. Once conversion is complete, users can immediately download the file from their web browser. Users can also choose to receive an email with a link to download the converted file. In February 2021 Zamzar expanded their tool and announced a new file compression service. The compressor is visually similar to the conversion tool with a drag and drop download feature. As with the converter, users have the option to subscribe for a paid plan if they wish to compress multiple or larger files than the free service permits == File conversion API == in 2015 Zamzar launched a file conversion API, allowing users to integrate file conversion capabilities into their own websites and applications. Sample code is provided to allow users to integrate file conversion capabilities in C#, Java, Node.js, PHP, Python and cURL. Zamzar also maintains a project on GitHub which allows users to perform file conversion from the command line on Linux, MacOS or Windows systems. == Email file conversion == It is also possible to send files for conversion by emailing them to Zamzar. Zamzar launched this capability in 2012, allowing users to email files to dedicated email addresses for the file to be automatically converted to a different format. A link is then emailed back to the end user to allow them to download their converted file. == User privilege levels == Zamzar is currently free to use, but there is a limit of two conversions per hour for files up to 100MB. Users can pay a monthly subscription in order to access preferential features, such as unlimited file conversions, online file management, shorter response and queuing times and other benefits. == Name == Its name comes from Franz Kafka's The Metamorphosis. Its main character is called Gregor Samsa and it is from his surname that Zamzar is derived. The founders of the service considered three other names – Konvertieren, Khamailen and Obrogo – before settling on Zamzar.

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  Circular convolution

  Circular convolution, also known as cyclic convolution, is a special case of periodic convolution, which is the convolution of two periodic functions that have the same period. Periodic convolution arises, for example, in the context of the discrete-time Fourier transform (DTFT). In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences. And each DTFT is a periodic summation of a continuous Fourier transform function (see Discrete-time Fourier transform § Relation to Fourier Transform). Although DTFTs are usually continuous functions of frequency, the concepts of periodic and circular convolution are also directly applicable to discrete sequences of data. In that context, circular convolution plays an important role in maximizing the efficiency of a certain kind of common filtering operation. == Definitions == The periodic convolution of two T-periodic functions, h T ( t ) {\displaystyle h_{_{T}}(t)} and x T ( t ) {\displaystyle x_{_{T}}(t)} can be defined as: ∫ t o t o + T h T ( τ ) ⋅ x T ( t − τ ) d τ , {\displaystyle \int _{t_{o}}^{t_{o}+T}h_{_{T}}(\tau )\cdot x_{_{T}}(t-\tau )\,d\tau ,} where t o {\displaystyle t_{o}} is an arbitrary parameter. An alternative definition, in terms of the notation of normal linear or aperiodic convolution, follows from expressing h T ( t ) {\displaystyle h_{_{T}}(t)} and x T ( t ) {\displaystyle x_{_{T}}(t)} as periodic summations of aperiodic components h {\displaystyle h} and x {\displaystyle x} , i.e.: h T ( t ) ≜ ∑ k = − ∞ ∞ h ( t − k T ) = ∑ k = − ∞ ∞ h ( t + k T ) . {\displaystyle h_{_{T}}(t)\ \triangleq \ \sum _{k=-\infty }^{\infty }h(t-kT)=\sum _{k=-\infty }^{\infty }h(t+kT).} Then: Both forms can be called periodic convolution. The term circular convolution arises from the important special case of constraining the non-zero portions of both h {\displaystyle h} and x {\displaystyle x} to the interval [ 0 , T ] . {\displaystyle [0,T].} Then the periodic summation becomes a periodic extension, which can also be expressed as a circular function: x T ( t ) = x ( t m o d T ) , t ∈ R {\displaystyle x_{_{T}}(t)=x(t_{\mathrm {mod} \ T}),\quad t\in \mathbb {R} \,} (any real number) And the limits of integration reduce to the length of function h {\displaystyle h} : ( h ∗ x T ) ( t ) = ∫ 0 T h ( τ ) ⋅ x ( ( t − τ ) m o d T ) d τ . {\displaystyle (hx_{_{T}})(t)=\int _{0}^{T}h(\tau )\cdot x((t-\tau )_{\mathrm {mod} \ T})\ d\tau .} == Discrete sequences == Similarly, for discrete sequences, and a parameter N, we can write a circular convolution of aperiodic functions h {\displaystyle h} and x {\displaystyle x} as: ( h ∗ x N ) [ n ] ≜ ∑ m = − ∞ ∞ h [ m ] ⋅ x N [ n − m ] ⏟ ∑ k = − ∞ ∞ x [ n − m − k N ] {\displaystyle (hx_{_{N}})[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }h[m]\cdot \underbrace {x_{_{N}}[n-m]} _{\sum _{k=-\infty }^{\infty }x[n-m-kN]}} This function is N-periodic. It has at most N unique values. For the special case that the non-zero extent of both x and h are ≤ N, it is reducible to matrix multiplication where the kernel of the integral transform is a circulant matrix. == Example == A case of great practical interest is illustrated in the figure. The duration of the x sequence is N (or less), and the duration of the h sequence is significantly less. Then many of the values of the circular convolution are identical to values of x∗h, which is actually the desired result when the h sequence is a finite impulse response (FIR) filter. Furthermore, the circular convolution is very efficient to compute, using a fast Fourier transform (FFT) algorithm and the circular convolution theorem. There are also methods for dealing with an x sequence that is longer than a practical value for N. The sequence is divided into segments (blocks) and processed piecewise. Then the filtered segments are carefully pieced back together. Edge effects are eliminated by overlapping either the input blocks or the output blocks. To help explain and compare the methods, we discuss them both in the context of an h sequence of length 201 and an FFT size of N = 1024. === Overlapping input blocks === This method uses a block size equal to the FFT size (1024). We describe it first in terms of normal or linear convolution. When a normal convolution is performed on each block, there are start-up and decay transients at the block edges, due to the filter latency (200-samples). Only 824 of the convolution outputs are unaffected by edge effects. The others are discarded, or simply not computed. That would cause gaps in the output if the input blocks are contiguous. The gaps are avoided by overlapping the input blocks by 200 samples. In a sense, 200 elements from each input block are "saved" and carried over to the next block. This method is referred to as overlap-save, although the method we describe next requires a similar "save" with the output samples. When an FFT is used to compute the 824 unaffected DFT samples, we don't have the option of not computing the affected samples, but the leading and trailing edge-effects are overlapped and added because of circular convolution. Consequently, the 1024-point inverse FFT (IFFT) output contains only 200 samples of edge effects (which are discarded) and the 824 unaffected samples (which are kept). To illustrate this, the fourth frame of the figure at right depicts a block that has been periodically (or "circularly") extended, and the fifth frame depicts the individual components of a linear convolution performed on the entire sequence. The edge effects are where the contributions from the extended blocks overlap the contributions from the original block. The last frame is the composite output, and the section colored green represents the unaffected portion. === Overlapping output blocks === This method is known as overlap-add. In our example, it uses contiguous input blocks of size 824 and pads each one with 200 zero-valued samples. Then it overlaps and adds the 1024-element output blocks. Nothing is discarded, but 200 values of each output block must be "saved" for the addition with the next block. Both methods advance only 824 samples per 1024-point IFFT, but overlap-save avoids the initial zero-padding and final addition.

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