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  Virtual assistant

  A virtual assistant (VA) is a software agent that can perform a range of tasks or services for a user based on user input, such as commands or questions, including verbal ones. Such technologies often incorporate chatbot capabilities to streamline task execution. The interaction may be via text, graphical interface, or voice, as some virtual assistants are able to interpret human speech and respond via synthesized voices. In many cases, users can ask their virtual assistants questions, control home automation devices and media playback, and manage other basic tasks such as email, to-do lists, and calendars – all with verbal commands. In recent years, prominent virtual assistants for direct consumer use have included Apple Siri, Amazon Alexa, Google Assistant (Gemini), Microsoft Copilot and Samsung Bixby. Also, companies in various industries often incorporate some kind of virtual assistant technology into their customer service or support. Into the 2020s, the emergence of artificial intelligence based chatbots, such as ChatGPT, has brought increased capability and interest to the field of virtual assistant products and services. == History == === Experimental decades: 1910s–1980s === Radio Rex was the first voice-activated toy, patented in 1916 and released in 1922. It was a wooden toy in the shape of a dog that would come out of its house when its name is called. In 1952, Bell Labs presented "Audrey", the Automatic Digit Recognition machine. It occupied a six-foot-high relay rack, consumed substantial power, had streams of cables and exhibited the myriad maintenance problems associated with complex vacuum-tube circuitry. It could recognize the fundamental units of speech, phonemes. It was limited to the accurate recognition of digits spoken by designated talkers. It could therefore be used for voice dialing, but in most cases, push-button dialing was cheaper and faster, rather than speaking the consecutive digits. Another early tool which was enabled to perform digital speech recognition was the IBM Shoebox voice-activated calculator, presented to the general public during the 1962 Seattle World's Fair after its initial market launch in 1961. This early computer, developed almost 20 years before the introduction of the first IBM Personal Computer in 1981, was able to recognize 16 spoken words and the digits 0 to 9. The first natural language processing computer program or the chatbot ELIZA was developed by MIT professor Joseph Weizenbaum in the 1960s. It was created to "demonstrate that the communication between man and machine was superficial". ELIZA used pattern matching and substitution methodology into scripted responses to simulate conversation, which gave an illusion of understanding on the part of the program. Weizenbaum's own secretary reportedly asked Weizenbaum to leave the room so that she and ELIZA could have a real conversation. Weizenbaum was surprised by this, later writing: "I had not realized ... that extremely short exposures to a relatively simple computer program could induce powerful delusional thinking in quite normal people. This gave name to the ELIZA effect, the tendency to unconsciously assume computer behaviors are analogous to human behaviors; that is, anthropomorphisation, a phenomenon present in human interactions with virtual assistants. The next milestone in the development of voice recognition technology was achieved in the 1970s at the Carnegie Mellon University in Pittsburgh, Pennsylvania with substantial support of the United States Department of Defense and its DARPA agency, funded five years of a Speech Understanding Research program, aiming to reach a minimum vocabulary of 1,000 words. Companies and academia including IBM, Carnegie Mellon University (CMU) and Stanford Research Institute took part in the program. The result was "Harpy", it mastered about 1000 words, the vocabulary of a three-year-old and it could understand sentences. It could process speech that followed pre-programmed vocabulary, pronunciation, and grammar structures to determine which sequences of words made sense together, and thus reducing speech recognition errors. In 1986, Tangora was an upgrade of the Shoebox, it was a voice recognizing typewriter. Named after the world's fastest typist at the time, it had a vocabulary of 20,000 words and used prediction to decide the most likely result based on what was said in the past. IBM's approach was based on a hidden Markov model, which adds statistics to digital signal processing techniques. The method makes it possible to predict the most likely phonemes to follow a given phoneme. Still each speaker had to individually train the typewriter to recognize their voice, and pause between each word. In 1983, Gus Searcy invented the "Butler in a Box", an electronic voice home controller system. === Birth of smart virtual assistants: 1990s–2010s === In the 1990s, digital speech recognition technology became a feature of the personal computer with IBM, Philips and Lernout & Hauspie fighting for customers. Much later the market launch of the first smartphone IBM Simon in 1994 laid the foundation for smart virtual assistants as we know them today. In 1997, Dragon's NaturallySpeaking software could recognize and transcribe natural human speech without pauses between each word into a document at a rate of 100 words per minute. A version of Naturally Speaking is still available for download and it is still used today, for instance, by many doctors in the US and the UK to document their medical records. In 2001 Colloquis publicly launched SmarterChild, on platforms like AIM and MSN Messenger. While entirely text-based SmarterChild was able to play games, check the weather, look up facts, and converse with users to an extent. The first modern digital virtual assistant installed on a smartphone was Siri, which was introduced as a feature of the iPhone 4S on 4 October 2011. Apple Inc. developed Siri following the 2010 acquisition of Siri Inc., a spin-off of SRI International, which is a research institute financed by DARPA and the United States Department of Defense. Its aim was to aid in tasks such as sending a text message, making phone calls, checking the weather or setting up an alarm. Over time, it has developed to provide restaurant recommendations, search the internet, and provide driving directions. In November 2014, Amazon announced Alexa alongside the Echo. In 2016, Salesforce debuted Einstein, developed from a set of technologies underlying the Salesforce platform. Einstein was replaced by Agentforce, an agentic AI, in September 2024. In April 2017 Amazon released a service for building conversational interfaces for any type of virtual assistant or interface. === Large Language Models: 2020s-present === In the 2020s, artificial intelligence (AI) systems like ChatGPT have gained popularity for their ability to generate human-like responses to text-based conversations. In February 2020, Microsoft introduced its Turing Natural Language Generation (T-NLG), which was then the "largest language model ever published at 17 billion parameters." On November 30, 2022, ChatGPT was launched as a prototype and quickly garnered attention for its detailed responses and articulate answers across many domains of knowledge. The advent of ChatGPT and its introduction to the wider public increased interest and competition in the space. In February 2023, Google began introducing an experimental service called "Bard" which is based on its LaMDA program to generate text responses to questions asked based on information gathered from the web. While ChatGPT and other generalized chatbots based on the latest generative AI are capable of performing various tasks associated with virtual assistants, there are also more specialized forms of such technology that are designed to target more specific situations or needs. == Method of interaction == Virtual assistants work via: Text, including: online chat (especially in an instant messaging application or other application ), SMS text, e-mail or other text-based communication channel, for example Conversica's intelligent virtual assistants for business. Voice: for example with Amazon Alexa on Amazon Echo devices, Siri on an iPhone, Google Assistant on Google-enabled Android devices, or Bixby on Samsung devices. Images: some assistants, such as Google Assistant (which includes Google Lens) and Bixby on the Samsung Galaxy series, have the added capability of performing image processing to recognize objects in images. Many virtual assistants are accessible via multiple methods, offering versatility in how users can interact with them, whether through chat, voice commands, or other integrated technologies. Virtual assistants use natural language processing (NLP) to match user text or voice input to executable commands. Some continually learn using artificial intelligence techniques including machine learning and ambient intelligence. To activate a virtual assistant u

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  Chandy–Misra–Haas algorithm resource model

  The Chandy–Misra–Haas algorithm resource model checks for deadlock in a distributed system. It was developed by K. Mani Chandy, Jayadev Misra and Laura M. Haas. == Locally dependent == Consider the n processes P1, P2, P3, P4, P5,, ... ,Pn which are performed in a single system (controller). P1 is locally dependent on Pn, if P1 depends on P2, P2 on P3, so on and Pn−1 on Pn. That is, if P 1 → P 2 → P 3 → … → P n {\displaystyle P_{1}\rightarrow P_{2}\rightarrow P_{3}\rightarrow \ldots \rightarrow P_{n}} , then P 1 {\displaystyle P_{1}} is locally dependent on P n {\displaystyle P_{n}} . If P1 is said to be locally dependent to itself if it is locally dependent on Pn and Pn depends on P1: i.e. if P 1 → P 2 → P 3 → … → P n → P 1 {\displaystyle P_{1}\rightarrow P_{2}\rightarrow P_{3}\rightarrow \ldots \rightarrow P_{n}\rightarrow P_{1}} , then P 1 {\displaystyle P_{1}} is locally dependent on itself. == Description == The algorithm uses a message called probe(i,j,k) to transfer a message from controller of process Pj to controller of process Pk. It specifies a message started by process Pi to find whether a deadlock has occurred or not. Every process Pj maintains a boolean array dependent which contains the information about the processes that depend on it. Initially the values of each array are all "false". === Controller sending a probe === Before sending, the probe checks whether Pj is locally dependent on itself. If so, a deadlock occurs. Otherwise it checks whether Pj, and Pk are in different controllers, are locally dependent and Pj is waiting for the resource that is locked by Pk. Once all the conditions are satisfied it sends the probe. === Controller receiving a probe === On the receiving side, the controller checks whether Pk is performing a task. If so, it neglects the probe. Otherwise, it checks the responses given Pk to Pj and dependentk(i) is false. Once it is verified, it assigns true to dependentk(i). Then it checks whether k is equal to i. If both are equal, a deadlock occurs, otherwise it sends the probe to next dependent process. == Algorithm == In pseudocode, the algorithm works as follows: === Controller sending a probe === if Pj is locally dependent on itself then declare deadlock else for all Pj,Pk such that (i) Pi is locally dependent on Pj, (ii) Pj is waiting for 'Pk and (iii) Pj, Pk are on different controllers. send probe(i, j, k). to home site of Pk === Controller receiving a probe === if (i)Pk is idle / blocked (ii) dependentk(i) = false, and (iii) Pk has not replied to all requests of to Pj then begin "dependents""k"(i) = true; if k == i then declare that Pi is deadlocked else for all Pa,Pb such that (i) Pk is locally dependent on Pa, (ii) Pa is waiting for 'Pb and (iii) Pa, Pb are on different controllers. send probe(i, a, b). to home site of Pb end == Example == P1 initiates deadlock detection. C1 sends the probe saying P2 depends on P3. Once the message is received by C2, it checks whether P3 is idle. P3 is idle because it is locally dependent on P4 and updates dependent3(2) to True. As above, C2 sends probe to C3 and C3 sends probe to C1. At C1, P1 is idle so it update dependent1(1) to True. Therefore, deadlock can be declared. == Complexity == Suppose there are n {\displaystyle n} controllers and m {\displaystyle m} processes, at most m ( n − 1 ) / 2 {\displaystyle m(n-1)/2} messages need to be exchanged to detect a deadlock, with a delay of O ( n ) {\displaystyle O(n)} messages.

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  Adaptive algorithm

  An adaptive algorithm is an algorithm that changes its behavior at the time it is run, based on information available and on a priori defined reward mechanism (or criterion). Such information could be the story of recently received data, information on the available computational resources, or other run-time acquired (or a priori known) information related to the environment in which it operates. Among the most used adaptive algorithms is the Widrow-Hoff’s least mean squares (LMS), which represents a class of stochastic gradient-descent algorithms used in adaptive filtering and machine learning. In adaptive filtering the LMS is used to mimic a desired filter by finding the filter coefficients that relate to producing the least mean square of the error signal (difference between the desired and the actual signal). For example, stable partition, using no additional memory is O(n lg n) but given O(n) memory, it can be O(n) in time. As implemented by the C++ Standard Library, stable_partition is adaptive and so it acquires as much memory as it can get (up to what it would need at most) and applies the algorithm using that available memory. Another example is adaptive sort, whose behavior changes upon the presortedness of its input. An example of an adaptive algorithm in radar systems is the constant false alarm rate (CFAR) detector. In machine learning and optimization, many algorithms are adaptive or have adaptive variants, which usually means that the algorithm parameters such as learning rate are automatically adjusted according to statistics about the optimisation thus far (e.g. the rate of convergence). Examples include adaptive simulated annealing, adaptive coordinate descent, adaptive quadrature, AdaBoost, Adagrad, Adadelta, RMSprop, and Adam. In data compression, adaptive coding algorithms such as Adaptive Huffman coding or Prediction by partial matching can take a stream of data as input, and adapt their compression technique based on the symbols that they have already encountered. In signal processing, the Adaptive Transform Acoustic Coding (ATRAC) codec used in MiniDisc recorders is called "adaptive" because the window length (the size of an audio "chunk") can change according to the nature of the sound being compressed, to try to achieve the best-sounding compression strategy.

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  Web data integration

  Web data integration (WDI) is the process of aggregating and managing data from different websites into a single, homogeneous workflow. This process includes data access, transformation, mapping, quality assurance and fusion of data. Data that is sourced and structured from websites is referred to as "web data". WDI is an extension and specialization of data integration that views the web as a collection of heterogeneous databases. Data integration techniques in the context of the web, forms the foundation for businesses taking advantage of data available on the ever-increasing number of publicly-accessible websites. Corporate spending on this area amounted to about USD 2.5bn in 2017, and it is expected that by 2020 the market will reach almost USD 7bn.

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  Availability zone

  In cloud computing, an availability region is a group of data centres that are located in the same geographical region. Availability regions comprise multiple availability zones, which are groups of data centres that are located far enough from each other to prevent large-scale outages in the event of failure of a single zone, whilst still being close enough to each other to enable low-latency connections. Distributed systems spanning multiple availability zones allow for high availability, even in the event of catastrophic failure, such as natural disasters. Services offering distinct availability zones include Amazon Web Services, Microsoft Azure and Google Cloud.

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  Upper ontology

  In information science, an upper ontology (also known as a top-level ontology, upper model, or foundation ontology) is an ontology (in the sense used in information science) that consists of very general terms (such as "object", "property", "relation") that are common across all domains. An important function of an upper ontology is to support broad semantic interoperability among a large number of domain-specific ontologies by providing a common starting point for the formulation of definitions. Terms in the domain ontology are ranked under the terms in the upper ontology, e.g., the upper ontology classes are superclasses or supersets of all the classes in the domain ontologies. A number of upper ontologies have been proposed, each with its own proponents. Library classification systems predate upper ontology systems. Though library classifications organize and categorize knowledge using general concepts that are the same across all knowledge domains, neither system is a replacement for the other. == Development == Any standard foundational ontology is likely to be contested among different groups, each with its own idea of "what exists". One factor exacerbating the failure to arrive at a common approach has been the lack of open-source applications that would permit the testing of different ontologies in the same computational environment. The differences have thus been debated largely on theoretical grounds, or are merely the result of personal preferences. Foundational ontologies can however be compared on the basis of adoption for the purposes of supporting interoperability across domain ontologies. No particular upper ontology has yet gained widespread acceptance as a de facto standard. Different organizations have attempted to define standards for specific domains. The 'Process Specification Language' (PSL) created by the National Institute of Standards and Technology (NIST) is one example. Another important factor leading to the absence of wide adoption of any existing upper ontology is the complexity. Some upper ontologies—Cyc is often cited as an example in this regard—are very large, ranging up to thousands of elements (classes, relations), with complex interactions among them and with a complexity similar to that of a human natural language, and the learning process can be even longer than for a natural language because of the unfamiliar format and logical rules. The motivation to overcome this learning barrier is largely absent because of the paucity of publicly accessible examples of use. As a result, those building domain ontologies for local applications tend to create the simplest possible domain-specific ontology, not related to any upper ontology. Such domain ontologies may function adequately for the local purpose, but they are very time-consuming to relate accurately to other domain ontologies. To solve this problem, some genuinely top level ontologies have been developed, which are deliberately designed to have minimal overlap with any domain ontologies. Examples are Basic Formal Ontology and the DOLCE (see below). === Arguments for the infeasibility of an upper ontology === Historically, many attempts in many societies have been made to impose or define a single set of concepts as more primal, basic, foundational, authoritative, true or rational than all others. A common objection to such attempts points out that humans lack the sort of transcendent perspective — or God's eye view — that would be required to achieve this goal. Humans are bound by language or culture, and so lack the sort of objective perspective from which to observe the whole terrain of concepts and derive any one standard. Thomasson, under the headline "1.5 Skepticism about Category Systems", wrote: "category systems, at least as traditionally presented, seem to presuppose that there is a unique true answer to the question of what categories of entity there are – indeed the discovery of this answer is the goal of most such inquiries into ontological categories. [...] But actual category systems offered vary so much that even a short survey of past category systems like that above can undermine the belief that such a unique, true and complete system of categories may be found. Given such a diversity of answers to the question of what the ontological categories are, by what criteria could we possibly choose among them to determine which is uniquely correct?" Another objection is the problem of formulating definitions. Top level ontologies are designed to maximize support for interoperability across a large number of terms. Such ontologies must therefore consist of terms expressing very general concepts, but such concepts are so basic to our understanding that there is no way in which they can be defined, since the very process of definition implies that a less basic (and less well understood) concept is defined in terms of concepts that are more basic and so (ideally) more well understood. Very general concepts can often only be elucidated, for example by means of examples, or paraphrase. There is no self-evident way of dividing the world up into concepts, and certainly no non-controversial one There is no neutral ground that can serve as a means of translating between specialized (or "lower" or "application-specific") ontologies Human language itself is already an arbitrary approximation of just one among many possible conceptual maps. To draw any necessary correlation between English words and any number of intellectual concepts, that we might like to represent in our ontologies, is just asking for trouble. (WordNet, for instance, is successful and useful, precisely because it does not pretend to be a general-purpose upper ontology; rather, it is a tool for semantic / syntactic / linguistic disambiguation, which is richly embedded in the particulars and peculiarities of the English language.) Any hierarchical or topological representation of concepts must begin from some ontological, epistemological, linguistic, cultural, and ultimately pragmatic perspective. Such pragmatism does not allow for the exclusion of politics between persons or groups, indeed it requires they be considered as perhaps more basic primitives than any that are represented. Those who doubt the feasibility of general purpose ontologies are more inclined to ask "what specific purpose do we have in mind for this conceptual map of entities and what practical difference will this ontology make?" This pragmatic philosophical position surrenders all hope of devising the encoded ontology version of "The world is everything that is the case." (Wittgenstein, Tractatus Logico-Philosophicus). Finally, there are objections similar to those against artificial intelligence. Technically, the complex concept acquisition and the social / linguistic interactions of human beings suggest any axiomatic foundation of "most basic" concepts must be cognitive biological or otherwise difficult to characterize since we don't have axioms for such systems. Ethically, any general-purpose ontology could quickly become an actual tyranny by recruiting adherents into a political program designed to propagate it and its funding means, and possibly defend it by violence. Historically, inconsistent and irrational belief systems have proven capable of commanding obedience to the detriment or harm of persons both inside and outside a society that accepts them. How much more harmful would a consistent rational one be, were it to contain even one or two basic assumptions incompatible with human life? === Arguments for the feasibility of an upper ontology === Many of those who doubt the possibility of developing wide agreement on a common upper ontology fall into one of two traps: they assert that there is no possibility of universal agreement on any conceptual scheme; but they argue that a practical common ontology does not need to have universal agreement, it only needs a large enough user community (as is the case for human languages) to make it profitable for developers to use it as a means to general interoperability, and for third-party developer to develop utilities to make it easier to use; and they point out that developers of data schemes find different representations congenial for their local purposes; but they do not demonstrate that these different representations are in fact logically inconsistent. In fact, different representations of assertions about the real world (though not philosophical models), if they accurately reflect the world, must be logically consistent, even if they focus on different aspects of the same physical object or phenomenon. If any two assertions about the real world are logically inconsistent, one or both must be wrong, and that is a topic for experimental investigation, not for ontological representation. In practice, representations of the real world are created as and known to be approximations to the basic reality, and their use is circumscribed by the limits of e

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  AI: When a Robot Writes a Play

  AI: When a Robot Writes a Play (in Czech: AI: Když robot píše hru) is a 2021 experimental theatre play, where 90% of its script was automatically generated by artificial intelligence (the GPT-2 language model). The play is in Czech language, but an English version of the script also exists. == Creation == The play is the first result of the THEaiTRE research project, aiming to commemorate the centenary of the R.U.R. play by Karel Čapek by investigating to what extent artificial intelligence could be used to create theatre play scripts. The script of the play was created using the THEaiTRobot tool, based on the GPT-2 language model. First, the play dramaturge, David Košťák, described the initial setting of each scene in a few sentences, and wrote the first line for each character. Next, THEaiTRobot suggested a continuation of the script, which the dramaturge could use, reject, or use part of it and let the tool generate a new continuation. Another option was to manually insert another line or a scenic remark. The script was generated in English and was automatically translated to Czech by the state-of-the-art CUBBITT machine translation tool. The resulting script was then further post-edited by the dramaturge. The resulting script was made freely available for non-commercial use both in English and in Czech, with marked manually inserted texts and manual edits. The analysis shows that 90% of the English script is automatically generated, with 10% manually written or manually post-edited. In the Czech script, a larger amount of edits were made, but the analysis claims that these additional edits are corrections of errors of the automated translation and stylistic corrections which do not change the meaning of the lines as represented by the English script, but rather bring the Czech script closer to the English one. == Characters == The play contains 9 characters. The Robot appears in all the scenes, while each of the other characters appears in only one scene. Robot – The lead character, a male humanoid robot. Master – An old man, the creator of the Robot. Boy – A schoolboy. Masseuse – A sex worker in a brothel. Stranger – An engineer. Man. Psychologist. Administrator – A female clerk at an employment agency. Actress – A film actress and a model in a robot-like costume. == Plot == The play is composed of 8 scenes. It tells the story of a humanoid robot, who encounters 8 other characters and engages into various typically human situations and activities, related to death, love, sex, violence, etc. The individual scenes are not tightly linked, but there are some linking points, such as the central character of the robot or some repeated and developing themes, such as the robot's search for love. The scenes often contain some absurd turns and it is often hard to find sense in them. It is therefore a very complicated piece interpretationally, requiring the director and the actors to invest a lot of effort and creativity in finding a meaningful interpretation which would not deviate from the script. In the interpretation by Švanda theatre, who premiered the play and who also participated on the creation of the script, the scenes typically contain non-verbally expressed content which can add a lot to the meaning of the scene compared to what is contained in the actual script (as the script only contains the lines said by the characters). === Scene 1: Death === The play opens by the Robot parting with his dying Master. The Master gives the Robot several last lessons and talks with him about death, soul, and love. === Scene 2: Sense of Humour === In the second scene, the Robot meets a sad and angry Boy, who complains that he wants to go to school, that his girlfriend is crazy, that he wants to buy a car, etc. The Robot tries to help the Boy by giving him advice, but the Boy's reactions are quite negative and irritated. The Boy then repeatedly asks the Robot to tell him a joke; the Robot keeps refusing, but ultimately tells the following joke: When you are dead. When your children are dead. When your grandchildren are dead, I will be still alive. === Scene 3: Nightclub === The Robot wants to feel pleasure, so he goes to a "night club" (a brothel), where he meets a "Masseuse" (a prostitute). The Robot is initially "a bit cold", but eventually manages to enjoy the experience and falls in love with the Masseuse. In the Švanda theatre performance, the Robot and the Masseuse seem to have a sort of virtual sex without touching each other, reminiscent of the sex scene in Demolition Man. === Scene 4: Fear of the Dark === It is the night. The Robot is standing under a lamp, unable to move away from the light as he finds that he is afraid of the dark. He meets a Stranger, an engineer who tells him that robots don't have feelings and that people cannot be trusted, and keeps hurting him. In the Švanda theatre performance, the Man repeatedly zaps the Robot with some kind of electric pulse. === Scene 5: Killer Robot === A Man approaches the Robot and repeatedly asks him to kill him. Instead, the Robot sticks a finger into the Man's anus, which leads to an argument between the Man and the Robot. === Scene 6: Burn Out === The Robot meets a Psychologist, who keeps asking him lots of questions regarding his life, burnout feeling, love, relationships, and emotions. They also talk about the Robot using a device called emotion machine which helps him to get rid of stress. === Scene 7: Search for Job === The Robot comes to an employment agency. He meets an Administrator and asks her to help him find a job. He expresses the wish to become an actor, and talks about his experience as a clown. He reveals his name to be Troy McClure, which is a character from The Simpsons who is an actor. In the Švanda theatre performance, the Administrator starts to seduce the Robot once his name is revealed, which he keeps ignoring; the Administrator then becomes irritated. === Scene 8: Love at First Sight === The Robot meets a human Actress in a robotic costume and falls in love with her immediately. The Actress is first reluctant, but the Robot manages to seduce her and she also falls in love with him. The Robot tells her about a binary world, in which he lives and where he will also take her. Ultimately, the Actress agrees, and the whole play concludes by the Robot and the Actress promising each to other to always be together. In the Švanda theatre performance, the Robot does not have a physical body in this scene, we can only hear his voice and see a pulsating light (based on the line in the script where the Robot says: "I have no body. So I don't need to wear clothes. You can't see me, you only hear me."), and the Actress eventually also agrees to lose her physical body so that she can be with the Robot forever. == Theatrical performances == The play premiered on 26 February 2021 in Švanda Theatre in Prague, Czech Republic, directed by Daniel Hrbek. Due to the COVID-19 pandemic, the play was not played in front of a live audience, but it was broadcast online, in Czech language with English subtitles. The play was followed by a panel discussion by the project members and experts on artificial intelligence. The premiere was viewed by 13,498 spectators worldwide. A short trailer of the premiere is available on YouTube. In 2021, after the opening of the theatres in the Czech Republic to spectators, the play can be viewed at Švanda Theatre. The performance takes approximately 60 minutes, and is followed by a discussion of the creators with the audience. The derniere is planned for 4 February 2023. == Reception == The play received a number of reviews, both in its country of origin as well as internationally. It is praised as first of its kind, although some reviewers note the similarity to previous works, such as the musical Beyond the Fence, the play Lifestyle of the Richard and Family, or the short movie Sunspring; however, these works used less advanced technology, and either were very short (Sunspring) or necessitated a larger amount of human interventions. The reviewers note that the script is far from perfect, with many inconsistencies and nonsensical parts, and conclude that the technology is definitely not yet ready to replace human authors; however, some find some parts of the script frighteningly human-like. The amount of human intervention is a somewhat controversial topic, with some reviewers finding the human influence too large (especially in interpreting the script and putting the play on scene), while others feel that a greater amount of human intervention would have been favorable as this could greatly improve the quality of the play. The reviews also frequently comment on the amount of sex, violence and strong language in the play; this can be attributed to the method used for creating the script, where the GPT-2 language model reflects topics and language common in the human-written articles on the internet that were used to train the model. Furthermore, some r

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  Operational system

  An operational system is a term used in data warehousing to refer to a system that is used to process the day-to-day transactions of an organization. These systems are designed in a manner that processing of day-to-day transactions is performed efficiently and the integrity of the transactional data is preserved. == Synonyms == Sometimes operational systems are referred to as operational databases, transaction processing systems, or online transaction processing systems (OLTP). However, the use of the last two terms as synonyms may be confusing, because operational systems can be batch processing systems as well. Any enterprise must necessarily maintain a lot of data about its operation.

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  Color picker

  A color picker (also color chooser or color tool) is a graphical user interface widget, usually found within graphics software or online, used to select colors and, in some cases, to create color schemes (the color picker might be more sophisticated than the palette included with the program). Operating systems such as Microsoft Windows or macOS have a system color picker, which can be used by third-party programs (e.g., Adobe Photoshop). == History == The concept of color pickers dates back to the early days of computer graphics and digital design. Early versions were rudimentary, often featuring basic color palettes and limited functionality. One of the first drawing programs to include a color picker was SketchPad (also referred to as LisaSketch), designed by Bill Atkinson in 1983 to showcase LisaGraf's capabilities. It used a black and white pattern system, using dithering to create the illusion of color depth. With the increased popularity of personal computers with color graphics, there soon came software similar to SketchPad that supported more than two colors, like Broderbund's Dazzle Draw for the Apple II or Electronic Arts' Deluxe Paint. However, the color pickers present in those programs relied on indexed colors. Color pickers, resembling ones used in modern software with support for direct, 24-bit color, appeared soon after the release of the Macintosh II, with the release of programs like Adobe Photoshop and Corel Painter. As the increase of color depth allowed the choice of significantly more colors, the shape and form of color pickers started to diverge. For example, Adobe Photoshop used a hue-saturation color wheel with a slider for brightness in version 0.63, later on switching to a rectangular design accompanied by a hue slider. Corel Painter pioneered the triangular saturation and brightness picker with a hue ring around it, aiming to better represent the continuity of the hue spectrum and the relationship between saturation and brightness. == Purpose == A color picker is used to select and adjust color values. In graphic design and image editing, users typically choose colors via an interface with a visual representation of a color—organized with quasi-perceptually-relevant hue, saturation and lightness dimensions (HSL) – instead of keying in alphanumeric text values. Because color appearance depends on comparison of neighboring colors (see color vision), many interfaces attempt to clarify the relationships between colors. == Interface == Color tools can vary in their interface. Some may use sliders, buttons, text boxes for color values, or direct manipulation. Often a two-dimensional square is used to create a range of color values (such as lightness and saturation) that can be clicked on or selected in some other manner. Drag and drop, color droppers, and various other forms of interfaces are commonly used as well. Usually, color values are also displayed numerically, so they can be precisely remembered and keyed-in later, such as three values of 0-255 representing red, green, and blue, respectively. === Eyedropper === The eyedropper is a tool present in most color pickers and graphics software that allows a user to read a color at a specific point in an image, or position on a display. This enables the color to be transferred to other applications particularly quickly. Modern implementations of eyedropper tools are also available as browser extensions, allowing users to pick colors directly from web pages, such as in Google Chrome and Microsoft Edge. == Working == A color picker has two main parts, first a color slider and second a color canvas. The color slider has a linear or radial gradient of the seven rainbow colors i.e. Violet, Indigo, Blue, Green, Yellow, Orange and Red. It allows one to choose any of the seven primary colors. The color value chosen from the color slider instantly reflects in the color canvas. The color canvas is a mixture of two linear color gradients. First a linear gradient of the current chosen color and second a linear gradient of the black color. This mixture of color gradients lets one choose a lighter and darker version of the current chosen color from the color slider.

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  Semantic translation

  Semantic translation is the process of using semantic information to aid in the translation of data in one representation or data model to another representation or data model. Semantic translation takes advantage of semantics that associate meaning with individual data elements in one dictionary to create an equivalent meaning in a second system. An example of semantic translation is the conversion of XML data from one data model to a second data model using formal ontologies for each system such as the Web Ontology Language (OWL). This is frequently required by intelligent agents that wish to perform searches on remote computer systems that use different data models to store their data elements. The process of allowing a single user to search multiple systems with a single search request is also known as federated search. Semantic translation should be differentiated from data mapping tools that do simple one-to-one translation of data from one system to another without actually associating meaning with each data element. Semantic translation requires that data elements in the source and destination systems have "semantic mappings" to a central registry or registries of data elements. The simplest mapping is of course where there is equivalence. There are three types of Semantic equivalence: Class Equivalence - indicating that class or "concepts" are equivalent. For example: "Person" is the same as "Individual" Property Equivalence - indicating that two properties are equivalent. For example: "PersonGivenName" is the same as "FirstName" Instance Equivalence - indicating that two individual instances of objects are equivalent. For example: "Dan Smith" is the same person as "Daniel Smith" Semantic translation is very difficult if the terms in a particular data model do not have direct one-to-one mappings to data elements in a foreign data model. In that situation, an alternative approach must be used to find mappings from the original data to the foreign data elements. This problem can be alleviated by centralized metadata registries that use the ISO-11179 standards such as the National Information Exchange Model (NIEM).

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  Information audit

  The information audit (IA) extends the concept of auditing from a traditional scope of accounting and finance to the organisational information management system. Information is representative of a resource which requires effective management and this led to the development of interest in the use of an IA. Prior the 1990s and the methodologies of Orna, Henczel, Wood, Buchanan and Gibb, IA approaches and methodologies focused mainly upon an identification of formal information resources (IR). Later approaches included an organisational analysis and the mapping of the information flow. This gave context to analysis within an organisation's information systems and a holistic view of their IR and as such could contribute to the development of the information systems architecture (ISA). In recent years the IA has been overlooked in favour of the systems development process which can be less expensive than the IA, yet more heavily technically focused, project specific (not holistic) and does not favour the top-down analysis of the IA. == Definition == A definition for the Information Audit cannot be universally agreed-upon amongst scholars, however the definition offered by ASLIB received positive support from a few notable scholars including Henczel, Orna and Wood; “(the IA is a) systematic examination of information use, resources and flows, with a verification by reference to both people and existing documents, in order to establish the extent to which they are contributing to an organisation’s objectives” In summary, the term audit itself implies a counting, the IA being much the same yet it counts IR and analyses how they are used and how critical they are to the success of a given task. == Role and scope of an IA == In much the same way as the IA is difficult to define, it can be utilised in a range of contexts by the information professional, from complying with freedom of information legislation to identifying any existing gaps, duplications, bottlenecks or other inefficiencies in information flows and to understand how existing channels can be used for knowledge transfer In 2007 Buchanan and Gibb developed upon their 1998 examination of the IA process by outlining a summary of its main objectives: To identify an organisation’s information resource To identify an organisation’s information needs Furthermore, Buchanan and Gibb went on to state that the IA also had to meet the following additional objectives: To identify the cost/benefits of information resources To identify the opportunities to use the information resources for strategic competitive advantage To integrate IT investment with strategic business initiatives To identify information flow and processes To develop an integrated information strategy and/or policy To create an awareness of the importance of Information Resource Management (IRM) To monitor/evaluate conformance to information related standards, legislations, policy and guidelines. == Methodology evolution == === Overview === In 1976 Riley first published a definition of IA as a way of analysing IR based on a cost-benefit model. Since Riley, scholars have outlined further developed methodologies. Henderson took a cost-benefit approach hoping to draw focus from manpower-costing to information storage and acquisition which he felt was being overlooked. In 1985 Gillman focused upon identifying the relationships which existed between various components in order to map them to one another. Neither Henderson nor Gillman’s methods offered alternative approaches beyond the existing organisational frameworks. Quinn took a hybrid-approach combining Gillman and Henderson’s methods to identify the purpose of existing IR and to position them within the organisation, as did Worlock. The differentiator between Quinn and Worlock lay in Worlock’s consideration of solutions outside of the current organisational structure. These approaches had thus far had paid little attention to the needs of the user or in making structured recommendations for the development of a corporate information strategy. Therefore, here follows a brief outline and overall comparison of four published strategic approaches in order that one might understand the development of the IA methodology. === Burk and Horton === In 1988 Burk and Horton developed InfoMap, the first IA methodology developed for widespread use. It aimed to discover, map and evaluate the IR within an organisation using a 4-stage process: Survey staff using questionnaires/interviews Measure the IR against cost/value Analyse resources Synthesise the findings and map the strengths and weaknesses of the IR against the objectives of the organisation. Although the method inventoried all IR (and therefore met standard ISO 1779) this bottom-up approach revealed limited analysis of the organisation holistically and the steps were not explicit enough. === Orna === Orna produced a top-down methodology in contrast to Burk and Horton, placing emphasis upon the importance of organisational analysis and aimed to assist in the production of a corporate information policy. Initially the method had just 4-stages, this later revised to a 10-stage process which included pre and post-audit stages as below: Conduct a preliminary review to confirm operational/strategic direction Gain support/resource from management Gain commitment from the other stakeholders (staff) Planning including the project, team, tools and techniques Identify the IR, information flow and produce a cost/value assessment Interpret findings based upon current versus desired state Produce a report to present findings Implement recommendations Monitor effects of change Repeat the IA Orna’s method introduced the need for a cyclical IA to be put in place in order for the IR to be continually tracked and improvements made regularly. Again this method was criticised for lacking some practical application and in 2004 Orna revised the methodology once more to try to rectify this problem === Buchanan and Gibb === In 1998, similarly to Orna's earlier publication, Buchanan and Gibb took a top-down approach, drawing techniques from established management disciplines to provide a framework and a level of familiarity for information professionals. This set of techniques was a notable contribution to IA methodologies and understood the need to be flexible for each organisation. Theirs was a 5-stage process: Promote benefits of the IA through seminars/surveys/CEO letter for cooperation Identify the mission objectives of the organisation, define environment (PEST), map information flow and examine organisation culture. Analyse and formulate action plan for problem areas, flow diagrams and a report of findings and recommendations Account for cost of IR and related services using Activity Based Costing (ABC) and Output Based Specification (OBS). Synthesise the whole process in final audit report and provide an information strategy (strategic direction) in relation to the organisation’s mission statement. This was the introduction of a new approach to costing the IR and had an integrated strategic direction, yet the scholars admitted that this method may be impractical for smaller organisations. === Henczel === Henczel’s methodology drew upon the strengths of Orna and Buchanan and Gibb to produce a 7-stage process: Planning and submission of business case for approval to proceed Data collection and development of an IR database and population through survey techniques Structured data analysis Data evaluation, interpretation and formulation of recommendations Communication of recommendations through a report Implementing recommendations through a devised programme The IA as a continuum-establishment of a cyclical process Focus was made once more on the strategic direction of the organisation conducting the IA. Furthermore, Henczel made examination into the use of the IA as a first-step in the development of a knowledge audit or knowledge management strategy as discussed in the later section. == Case studies == Scholars and information professionals have since tested the above methodologies with varied results. An early case study produced by Soy and Bustelo in a Spanish financial institution in 1999 aimed to identify the use of information resources for qualitative and quantitative data analysis due to the rapid expansion of the organisation within a six-year period. Although the methodology was not explicitly credited to any of the above-mentioned scholars, it did follow a strategic (post 1990's) IA process including gaining support from management, the use of questionnaires for data collection, analysis and evaluation of the data, identification and mapping of the IR, cost-analysis and outlining recommendations to assist with the establishment of an Information policy. In addition the IA report suggested that the process would need to be continual (cyclical as Orna, Henczel and Buchanan and Gibb suggest). Conclusions of this case-study stated that th

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  Archival bond

  The archival bond is a concept in archival theory referring to the relationship that each archival record has with the other records produced as part of the same transaction or activity and located within the same grouping. These bonds are a core component of each individual record and are necessary for transforming a document into a record, as a document will only acquire meaning (and become a record) through its interrelationships with other records. == Description == The concept of the archival bond is primarily associated with the work of Luciana Duranti along with Heather MacNeil, as part of research into the integrity of electronic records. Duranti resumed and extended the concept of vincolo archivistico (archival bond), first expressed in 1937 by archivist Giorgio Cencetti of the Italian archival school. This bond emerges from the fact that electronic records are not physically arranged like traditional records. For traditional, analog records, their bond is implicit in their arrangement. But for electronic records, this bond must be made explicit due to the lack of a single sequential order of records in a digital environment. The archival bond was one of the core concepts of the subsequent International Research on Permanent Authentic Records in Electronic Systems (InterPARES) project and can be found in the InterPARES glossary. As Duranti notes, the archival bond is not to be confused with the broader term "context" as context exists independently of a record, while "the archival bond is an essential part of the record, which would not exist without it."

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  Kernel (image processing)

  In image processing, a kernel, convolution matrix, or mask is a small matrix used for blurring, sharpening, embossing, edge detection, and more. This is accomplished by doing a convolution between the kernel and an image. Or more simply, when each pixel in the output image is a function of the nearby pixels (including itself) in the input image, the kernel is that function. == Details == The general expression of a convolution is g x , y = ω ∗ f x , y = ∑ i = − a a ∑ j = − b b ω i , j f x − i , y − j , {\displaystyle g_{x,y}=\omega f_{x,y}=\sum _{i=-a}^{a}{\sum _{j=-b}^{b}{\omega _{i,j}f_{x-i,y-j}}},} where g ( x , y ) {\displaystyle g(x,y)} is the filtered image, f ( x , y ) {\displaystyle f(x,y)} is the original image, ω {\displaystyle \omega } is the filter kernel. Every element of the filter kernel is considered by − a ≤ i ≤ a {\displaystyle -a\leq i\leq a} and − b ≤ j ≤ b {\displaystyle -b\leq j\leq b} . Depending on the element values, a kernel can cause a wide range of effects: The above are just a few examples of effects achievable by convolving kernels and images. === Origin === The origin is the position of the kernel which is above (conceptually) the current output pixel. This could be outside of the actual kernel, though usually it corresponds to one of the kernel elements. For a symmetric kernel, the origin is usually the center element. == Convolution == Convolution is the process of adding each element of the image to its local neighbors, weighted by the kernel. This is related to a form of mathematical convolution. The matrix operation being performed—convolution—is not traditional matrix multiplication, despite being similarly denoted by . For example, if we have two three-by-three matrices, the first a kernel, and the second an image piece, convolution is the process of flipping both the rows and columns of the kernel and multiplying locally similar entries and summing. The element at coordinates [2, 2] (that is, the central element) of the resulting image would be a weighted combination of all the entries of the image matrix, with weights given by the kernel: ( [ a b c d e f g h i ] ∗ [ 1 2 3 4 5 6 7 8 9 ] ) [ 2 , 2 ] = {\displaystyle \left({\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}}{\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}}\right)[2,2]=} ( i ⋅ 1 ) + ( h ⋅ 2 ) + ( g ⋅ 3 ) + ( f ⋅ 4 ) + ( e ⋅ 5 ) + ( d ⋅ 6 ) + ( c ⋅ 7 ) + ( b ⋅ 8 ) + ( a ⋅ 9 ) . {\displaystyle (i\cdot 1)+(h\cdot 2)+(g\cdot 3)+(f\cdot 4)+(e\cdot 5)+(d\cdot 6)+(c\cdot 7)+(b\cdot 8)+(a\cdot 9).} The other entries would be similarly weighted, where we position the center of the kernel on each of the boundary points of the image, and compute a weighted sum. The values of a given pixel in the output image are calculated by multiplying each kernel value by the corresponding input image pixel values. This can be described algorithmically with the following pseudo-code: for each image row in input image: for each pixel in image row: set accumulator to zero for each kernel row in kernel: for each element in kernel row: if element position corresponding to pixel position then multiply element value corresponding to pixel value add result to accumulator endif set output image pixel to accumulator corresponding input image pixels are found relative to the kernel's origin. If the kernel is symmetric then place the center (origin) of the kernel on the current pixel. The kernel will overlap the neighboring pixels around the origin. Each kernel element should be multiplied with the pixel value it overlaps with and all of the obtained values should be summed. This resultant sum will be the new value for the current pixel currently overlapped with the center of the kernel. If the kernel is not symmetric, it has to be flipped both around its horizontal and vertical axis before calculating the convolution as above. The general form for matrix convolution is [ x 11 x 12 ⋯ x 1 n x 21 x 22 ⋯ x 2 n ⋮ ⋮ ⋱ ⋮ x m 1 x m 2 ⋯ x m n ] ∗ [ y 11 y 12 ⋯ y 1 n y 21 y 22 ⋯ y 2 n ⋮ ⋮ ⋱ ⋮ y m 1 y m 2 ⋯ y m n ] = ∑ i = 0 m − 1 ∑ j = 0 n − 1 x ( m − i ) ( n − j ) y ( 1 + i ) ( 1 + j ) {\displaystyle {\begin{bmatrix}x_{11}&x_{12}&\cdots &x_{1n}\\x_{21}&x_{22}&\cdots &x_{2n}\\\vdots &\vdots &\ddots &\vdots \\x_{m1}&x_{m2}&\cdots &x_{mn}\\\end{bmatrix}}{\begin{bmatrix}y_{11}&y_{12}&\cdots &y_{1n}\\y_{21}&y_{22}&\cdots &y_{2n}\\\vdots &\vdots &\ddots &\vdots \\y_{m1}&y_{m2}&\cdots &y_{mn}\\\end{bmatrix}}=\sum _{i=0}^{m-1}\sum _{j=0}^{n-1}x_{(m-i)(n-j)}y_{(1+i)(1+j)}} === Edge handling === Kernel convolution usually requires values from pixels outside of the image boundaries. There are a variety of methods for handling image edges. Extend The nearest border pixels are conceptually extended as far as necessary to provide values for the convolution. Corner pixels are extended in 90° wedges. Other edge pixels are extended in lines. Wrap The image is conceptually wrapped (or tiled) and values are taken from the opposite edge or corner. Mirror The image is conceptually mirrored at the edges. For example, attempting to read a pixel 3 units outside an edge reads one 3 units inside the edge instead. Crop / Avoid overlap Any pixel in the output image which would require values from beyond the edge is skipped. This method can result in the output image being slightly smaller, with the edges having been cropped. Move kernel so that values from outside of image is never required. Machine learning mainly uses this approach. Example: Kernel size 10x10, image size 32x32, result image is 23x23. Kernel Crop Any pixel in the kernel that extends past the input image isn't used and the normalizing is adjusted to compensate. Constant Use constant value for pixels outside of image. Usually black or sometimes gray is used. Generally this depends on application. === Normalization === Normalization is defined as the division of each element in the kernel by the sum of all kernel elements, so that the sum of the elements of a normalized kernel is unity. This will ensure the average pixel in the modified image is as bright as the average pixel in the original image. === Optimization === Fast convolution algorithms include: separable convolution ==== Separable convolution ==== 2D convolution with an M × N kernel requires M × N multiplications for each sample (pixel). If the kernel is separable, then the computation can be reduced to M + N multiplications. Using separable convolutions can significantly decrease the computation by doing 1D convolution twice instead of one 2D convolution. === Implementation === Here a concrete convolution implementation done with the GLSL shading language :

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  Unrestricted algorithm

  An unrestricted algorithm is an algorithm for the computation of a mathematical function that puts no restrictions on the range of the argument or on the precision that may be demanded in the result. The idea of such an algorithm was put forward by C. W. Clenshaw and F. W. J. Olver in a paper published in 1980. In the problem of developing algorithms for computing, as regards the values of a real-valued function of a real variable (e.g., g[x] in "restricted" algorithms), the error that can be tolerated in the result is specified in advance. An interval on the real line would also be specified for values when the values of a function are to be evaluated. Different algorithms may have to be applied for evaluating functions outside the interval. An unrestricted algorithm envisages a situation in which a user may stipulate the value of x and also the precision required in g(x) quite arbitrarily. The algorithm should then produce an acceptable result without failure.

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  Zassenhaus algorithm

  In mathematics, the Zassenhaus algorithm is a method to calculate a basis for the intersection and sum of two subspaces of a vector space. It is named after Hans Zassenhaus, but no publication of this algorithm by him is known. It is used in computer algebra systems. == Algorithm == === Input === Let V be a vector space and U, W two finite-dimensional subspaces of V with the following spanning sets: U = ⟨ u 1 , … , u n ⟩ {\displaystyle U=\langle u_{1},\ldots ,u_{n}\rangle } and W = ⟨ w 1 , … , w k ⟩ . {\displaystyle W=\langle w_{1},\ldots ,w_{k}\rangle .} Finally, let B 1 , … , B m {\displaystyle B_{1},\ldots ,B_{m}} be linearly independent vectors so that u i {\displaystyle u_{i}} and w i {\displaystyle w_{i}} can be written as u i = ∑ j = 1 m a i , j B j {\displaystyle u_{i}=\sum _{j=1}^{m}a_{i,j}B_{j}} and w i = ∑ j = 1 m b i , j B j . {\displaystyle w_{i}=\sum _{j=1}^{m}b_{i,j}B_{j}.} === Output === The algorithm computes the base of the sum U + W {\displaystyle U+W} and a base of the intersection U ∩ W {\displaystyle U\cap W} . === Algorithm === The algorithm creates the following block matrix of size ( ( n + k ) × ( 2 m ) ) {\displaystyle ((n+k)\times (2m))} : ( a 1 , 1 a 1 , 2 ⋯ a 1 , m a 1 , 1 a 1 , 2 ⋯ a 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a n , 1 a n , 2 ⋯ a n , m a n , 1 a n , 2 ⋯ a n , m b 1 , 1 b 1 , 2 ⋯ b 1 , m 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ b k , 1 b k , 2 ⋯ b k , m 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,m}&a_{1,1}&a_{1,2}&\cdots &a_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,m}&a_{n,1}&a_{n,2}&\cdots &a_{n,m}\\b_{1,1}&b_{1,2}&\cdots &b_{1,m}&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\b_{k,1}&b_{k,2}&\cdots &b_{k,m}&0&0&\cdots &0\end{pmatrix}}} Using elementary row operations, this matrix is transformed to the row echelon form. Then, it has the following shape: ( c 1 , 1 c 1 , 2 ⋯ c 1 , m ∙ ∙ ⋯ ∙ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ c q , 1 c q , 2 ⋯ c q , m ∙ ∙ ⋯ ∙ 0 0 ⋯ 0 d 1 , 1 d 1 , 2 ⋯ d 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 d ℓ , 1 d ℓ , 2 ⋯ d ℓ , m 0 0 ⋯ 0 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}c_{1,1}&c_{1,2}&\cdots &c_{1,m}&\bullet &\bullet &\cdots &\bullet \\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\c_{q,1}&c_{q,2}&\cdots &c_{q,m}&\bullet &\bullet &\cdots &\bullet \\0&0&\cdots &0&d_{1,1}&d_{1,2}&\cdots &d_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&d_{\ell ,1}&d_{\ell ,2}&\cdots &d_{\ell ,m}\\0&0&\cdots &0&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&0&0&\cdots &0\end{pmatrix}}} Here, ∙ {\displaystyle \bullet } stands for arbitrary numbers, and the vectors ( c p , 1 , c p , 2 , … , c p , m ) {\displaystyle (c_{p,1},c_{p,2},\ldots ,c_{p,m})} for every p ∈ { 1 , … , q } {\displaystyle p\in \{1,\ldots ,q\}} and ( d p , 1 , … , d p , m ) {\displaystyle (d_{p,1},\ldots ,d_{p,m})} for every p ∈ { 1 , … , ℓ } {\displaystyle p\in \{1,\ldots ,\ell \}} are nonzero. Then ( y 1 , … , y q ) {\displaystyle (y_{1},\ldots ,y_{q})} with y i := ∑ j = 1 m c i , j B j {\displaystyle y_{i}:=\sum _{j=1}^{m}c_{i,j}B_{j}} is a basis of U + W {\displaystyle U+W} and ( z 1 , … , z ℓ ) {\displaystyle (z_{1},\ldots ,z_{\ell })} with z i := ∑ j = 1 m d i , j B j {\displaystyle z_{i}:=\sum _{j=1}^{m}d_{i,j}B_{j}} is a basis of U ∩ W {\displaystyle U\cap W} . === Proof of correctness === First, we define π 1 : V × V → V , ( a , b ) ↦ a {\displaystyle \pi _{1}:V\times V\to V,(a,b)\mapsto a} to be the projection to the first component. Let H := { ( u , u ) ∣ u ∈ U } + { ( w , 0 ) ∣ w ∈ W } ⊆ V × V . {\displaystyle H:=\{(u,u)\mid u\in U\}+\{(w,0)\mid w\in W\}\subseteq V\times V.} Then π 1 ( H ) = U + W {\displaystyle \pi _{1}(H)=U+W} and H ∩ ( 0 × V ) = 0 × ( U ∩ W ) {\displaystyle H\cap (0\times V)=0\times (U\cap W)} . Also, H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} is the kernel of π 1 | H {\displaystyle {\pi _{1}|}_{H}} , the projection restricted to H. Therefore, dim ⁡ ( H ) = dim ⁡ ( U + W ) + dim ⁡ ( U ∩ W ) {\displaystyle \dim(H)=\dim(U+W)+\dim(U\cap W)} . The Zassenhaus algorithm calculates a basis of H. In the first m columns of this matrix, there is a basis y i {\displaystyle y_{i}} of U + W {\displaystyle U+W} . The rows of the form ( 0 , z i ) {\displaystyle (0,z_{i})} (with z i ≠ 0 {\displaystyle z_{i}\neq 0} ) are obviously in H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} . Because the matrix is in row echelon form, they are also linearly independent. All rows which are different from zero ( ( y i , ∙ ) {\displaystyle (y_{i},\bullet )} and ( 0 , z i ) {\displaystyle (0,z_{i})} ) are a basis of H, so there are dim ⁡ ( U ∩ W ) {\displaystyle \dim(U\cap W)} such z i {\displaystyle z_{i}} s. Therefore, the z i {\displaystyle z_{i}} s form a basis of U ∩ W {\displaystyle U\cap W} . == Example == Consider the two subspaces U = ⟨ ( 1 − 1 0 1 ) , ( 0 0 1 − 1 ) ⟩ {\displaystyle U=\left\langle \left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right\rangle } and W = ⟨ ( 5 0 − 3 3 ) , ( 0 5 − 3 − 2 ) ⟩ {\displaystyle W=\left\langle \left({\begin{array}{r}5\\0\\-3\\3\end{array}}\right),\left({\begin{array}{r}0\\5\\-3\\-2\end{array}}\right)\right\rangle } of the vector space R 4 {\displaystyle \mathbb {R} ^{4}} . Using the standard basis, we create the following matrix of dimension ( 2 + 2 ) × ( 2 ⋅ 4 ) {\displaystyle (2+2)\times (2\cdot 4)} : ( 1 − 1 0 1 1 − 1 0 1 0 0 1 − 1 0 0 1 − 1 5 0 − 3 3 0 0 0 0 0 5 − 3 − 2 0 0 0 0 ) . {\displaystyle \left({\begin{array}{rrrrrrrr}1&-1&0&1&&1&-1&0&1\\0&0&1&-1&&0&0&1&-1\\\\5&0&-3&3&&0&0&0&0\\0&5&-3&-2&&0&0&0&0\end{array}}\right).} Using elementary row operations, we transform this matrix into the following matrix: ( 1 0 0 0 ∙ ∙ ∙ ∙ 0 1 0 − 1 ∙ ∙ ∙ ∙ 0 0 1 − 1 ∙ ∙ ∙ ∙ 0 0 0 0 1 − 1 0 1 ) {\displaystyle \left({\begin{array}{rrrrrrrrr}1&0&0&0&&\bullet &\bullet &\bullet &\bullet \\0&1&0&-1&&\bullet &\bullet &\bullet &\bullet \\0&0&1&-1&&\bullet &\bullet &\bullet &\bullet \\\\0&0&0&0&&1&-1&0&1\end{array}}\right)} (Some entries have been replaced by " ∙ {\displaystyle \bullet } " because they are irrelevant to the result.) Therefore ( ( 1 0 0 0 ) , ( 0 1 0 − 1 ) , ( 0 0 1 − 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\0\\0\\0\end{array}}\right),\left({\begin{array}{r}0\\1\\0\\-1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right)} is a basis of U + W {\displaystyle U+W} , and ( ( 1 − 1 0 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right)\right)} is a basis of U ∩ W {\displaystyle U\cap W} .

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