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  • Automatic taxonomy construction

    Automatic taxonomy construction

    Automatic taxonomy construction (ATC) is the use of software programs to generate taxonomical classifications from a body of texts called a corpus. ATC is a branch of natural language processing, which in turn is a branch of artificial intelligence. A taxonomy (or taxonomical classification) is a scheme of classification, especially, a hierarchical classification, in which things are organized into groups or types. Among other things, a taxonomy can be used to organize and index knowledge (stored as documents, articles, videos, etc.), such as in the form of a library classification system, or a search engine taxonomy, so that users can more easily find the information they are searching for. Many taxonomies are hierarchies (and thus, have an intrinsic tree structure), but not all are. Manually developing and maintaining a taxonomy is a labor-intensive task requiring significant time and resources, including familiarity of or expertise in the taxonomy's domain (scope, subject, or field), which drives the costs and limits the scope of such projects. Also, domain modelers have their own points of view which inevitably, even if unintentionally, work their way into the taxonomy. ATC uses artificial intelligence techniques to quickly automatically generate a taxonomy for a domain in order to avoid these problems and remove limitations. == Approaches == There are several approaches to ATC. One approach is to use rules to detect patterns in the corpus and use those patterns to infer relations such as hyponymy. Other approaches use machine learning techniques such as Bayesian inferencing and Artificial Neural Networks. === Keyword extraction === One approach to building a taxonomy is to automatically gather the keywords from a domain using keyword extraction, then analyze the relationships between them (see Hyponymy, below), and then arrange them as a taxonomy based on those relationships. === Hyponymy and "is-a" relations === In ATC programs, one of the most important tasks is the discovery of hypernym and hyponym relations among words. One way to do that from a body of text is to search for certain phrases like "is a" and "such as". In linguistics, is-a relations are called hyponymy. Words that describe categories are called hypernyms and words that are examples of categories are hyponyms. For example, dog is a hypernym and Fido is one of its hyponyms. A word can be both a hyponym and a hypernym. So, dog is a hyponym of mammal and also a hypernym of Fido. Taxonomies are often represented as is-a hierarchies where each level is more specific than (in mathematical language "a subset of") the level above it. For example, a basic biology taxonomy would have concepts such as mammal, which is a subset of animal, and dogs and cats, which are subsets of mammal. This kind of taxonomy is called an is-a model because the specific objects are considered instances of a concept. For example, Fido is-a instance of the concept dog and Fluffy is-a cat. == Applications == ATC can be used to build taxonomies for search engines, to improve search results. ATC systems are a key component of ontology learning (also known as automatic ontology construction), and have been used to automatically generate large ontologies for domains such as insurance and finance. They have also been used to enhance existing large networks such as Wordnet to make them more complete and consistent. == ATC software == == Other names == Other names for automatic taxonomy construction include: Automated outline building Automated outline construction Automated outline creation Automated outline extraction Automated outline generation Automated outline induction Automated outline learning Automated outlining Automated taxonomy building Automated taxonomy construction Automated taxonomy creation Automated taxonomy extraction Automated taxonomy generation Automated taxonomy induction Automated taxonomy learning Automatic outline building Automatic outline construction Automatic outline creation Automatic outline extraction Automatic outline generation Automatic outline induction Automatic outline learning Automatic taxonomy building Automatic taxonomy creation Automatic taxonomy extraction Automatic taxonomy generation Automatic taxonomy induction Automatic taxonomy learning Outline automation Outline building Outline construction Outline creation Outline extraction Outline generation Outline induction Outline learning Semantic taxonomy building Semantic taxonomy construction Semantic taxonomy creation Semantic taxonomy extraction Semantic taxonomy generation Semantic taxonomy induction Semantic taxonomy learning Taxonomy automation Taxonomy building Taxonomy construction Taxonomy creation Taxonomy extraction Taxonomy generation Taxonomy induction Taxonomy learning

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  • Small data

    Small data

    Small data is data that is 'small' enough for human comprehension. It is data in a volume and format that makes it accessible, informative and actionable. The term "big data" is about machines and "small data" is about people. This is to say that eyewitness observations or five pieces of related data could be small data. Small data is what we used to think of as data. The only way to comprehend Big data is to reduce the data into small, visually-appealing objects representing various aspects of large data sets (such as histogram, charts, and scatter plots). Big Data is all about finding correlations, but Small Data is all about finding the causation, the reason why. A formal definition of small data has been proposed by Allen Bonde, former vice-president of Innovation at Actuate - now part of OpenText: "Small data connects people with timely, meaningful insights (derived from big data and/or “local” sources), organized and packaged – often visually – to be accessible, understandable, and actionable for everyday tasks." Another definition of small data is: The small set of specific attributes produced by the Internet of Things. These are typically a small set of sensor data such as temperature, wind speed, vibration and status. It was estimated (2016) that “If one takes the top 100 biggest innovations of our time, perhaps around 60% to 65% percent are really based on Small Data.” as Martin Lindstrom puts it. Small data includes everything from Snapchat to simple objects such as the post-it note. Lindstrom believes we become so focused on Big-Data that we tend to forget about more basic concepts and creativity. Lindstrom defines Small Data "as seemingly insignificant observations you identify in consumers’ homes, is everything from how you place your shoes on how you hang your paintings". He thus considers that one should perfectly master the basic (Small Data) in order to mine and find correlations. == Academic Recognition and Methodology == The growing significance of "small data" as a distinct field of inquiry was highlighted by the 2024 Thematic Einstein Semester (TES) on Small Data Analysis, hosted by the Berlin Mathematics Research Center MATH+. A central focus of this semester was the transition from theoretical analysis to practical decision-making. Because small data sets are primarily used to drive specific actions, the presentation of results becomes an essential methodological step. The semester’s findings emphasized that while small data may lack volume, it often contains a high density of "many possible interpretations." Consequently, the final conference of the TES was structured around the pillars of interpretation, explanation, and knowledge gain. Participants sought to develop new mathematical and methodical representations that could accurately depict this wealth of interpretative possibilities. This work underscores that analyzing small data is not purely a computational task; it requires a robust interface between mathematics and diverse disciplines to ensure that insights are both contextually grounded and scientifically rigorous. == Uses in business == === Marketing === Bonde has written about the topic for Forbes, Direct Marketing News, CMO.com and other publications. According to Martin Lindstrom, in his book, Small Data: "{In customer research, small data is} Seemingly insignificant behavioural observations containing very specific attributes pointing towards an unmet customer need. Small data is the foundation for breakthrough ideas or completely new ways to turnaround brands." His approach is based on the combination of the observation of small samples with intuition. Marketers can obtain market insights from gathering Small Data by engaging with and observing people in their own environments. In comparison to Big Data, Small Data has the power to trigger emotions and to provide insights into the reasons behind the behaviours of customers. It may uncover detailed information on a person's extroversion or introversion, self-confidence, whether one is having problems in his/her relationship, etc. According to Lindstrom, relationships among people and customer segments are organized around four criteria: Climate: It reveals for example how a person's environment affects their diet. Rulership: The power or government in charge Religion: The prevalence of religion in a country, depending on its influence, indicates whether a person's decision making process is impacted by their belief system. Tradition: Cultural norms influence people's behaviors and interactions. Many companies underestimate the power of Small Data, using samples of millions of consumers instead of recognizing the value of closely observing small samples in their market research. In his book, Lindstrom defines "7Cs", which companies should consider in the attempt to derive meaningful customer insights and market trends through small data from their customers: Collecting: Understanding the manner in which observations are translated inside a home. Clues: Uncovering other distinctive emotional reflections that can be observed. Connecting: Identifying the consequences of emotional behaviour. Causation: Understanding what emotions are being evoked. Correlation: Identifying the initial date of appearance of the behaviour or emotion. Compensation: Identifying the unmet or unfulfilled desire. Concept: Defining the “big idea” compensation for the identified consumer need. Some of Lindstrom's clients such as Lowes Foods looked at data in a different way and actually chose to live with the customer. “As you enter their store, they have now created an amazing community where every staff member acts in a character mood, based on Small Data”. The supermarket made everything it can to make the customer feel at home. All the behaviours of employees are inspired by customer feedbacks gathered from interviews directly done at customer’s home. === Healthcare === Researchers at Cornell University started developing applications to monitor health problems in patients, based on small data. This is an initiative of Cornell's Small Data Lab, in close cooperation with Weill Cornell Medicine College, led by Deborah Estrin. The Small Data Lab developed a series of apps, focusing not only on gathering data from patients' pain but also tracking habits in areas such as grocery shopping. In the case of patients with rheumatoid arthritis for example, which has flares and remissions that do not follow a particular cycle, the app gathers information passively, thus allowing to forecast when a flare might be coming up based on small changes in behaviour. Other apps developed also include monitoring online grocery shopping, to use this information from every user to adapt their groceries to the recommendations of nutritionists, or monitoring email language to identify patterns that might indicate "fluctuations in cognitive performance, fatigue, side effects of medication or poor sleep, and other conditions and treatments that are typically self-reported and self-medicated". === Postal Service === The United States Postal Service (USPS) used optical character recognition (OCR) to automatically read and process 98% of all hand-addressed mail and 99.5% of machine-printed mail. By combining this technology with its small data sample of US zip codes, the USPS can now process more than 36,000 pieces of mail per hour. === Aerospace === In 2015, Boeing established the analytics lab for aerospace data in cooperation with the Carnegie Mellon University to leverage the university's leadership in machine learning, language technologies and data analytics. One of the initiatives projects aims to by standardize maintenance logs using AI to dramatically reduce costs. Currently, there is no standardized procedure to document maintenance logs leading to small but highly unstructured data sets. As a result, it becomes highly difficult for maintenance workers to translate these variations in maintenance logs within a short period of time. However, with AI and a narrow data set of common aircraft maintenance terminology, it becomes possible to dynamically translate these logs in real time. By using AI to enhance the speed and accuracy of the airline maintenance workflow, airlines stand to save billions according to the Harvard Business Review.

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  • Small data

    Small data

    Small data is data that is 'small' enough for human comprehension. It is data in a volume and format that makes it accessible, informative and actionable. The term "big data" is about machines and "small data" is about people. This is to say that eyewitness observations or five pieces of related data could be small data. Small data is what we used to think of as data. The only way to comprehend Big data is to reduce the data into small, visually-appealing objects representing various aspects of large data sets (such as histogram, charts, and scatter plots). Big Data is all about finding correlations, but Small Data is all about finding the causation, the reason why. A formal definition of small data has been proposed by Allen Bonde, former vice-president of Innovation at Actuate - now part of OpenText: "Small data connects people with timely, meaningful insights (derived from big data and/or “local” sources), organized and packaged – often visually – to be accessible, understandable, and actionable for everyday tasks." Another definition of small data is: The small set of specific attributes produced by the Internet of Things. These are typically a small set of sensor data such as temperature, wind speed, vibration and status. It was estimated (2016) that “If one takes the top 100 biggest innovations of our time, perhaps around 60% to 65% percent are really based on Small Data.” as Martin Lindstrom puts it. Small data includes everything from Snapchat to simple objects such as the post-it note. Lindstrom believes we become so focused on Big-Data that we tend to forget about more basic concepts and creativity. Lindstrom defines Small Data "as seemingly insignificant observations you identify in consumers’ homes, is everything from how you place your shoes on how you hang your paintings". He thus considers that one should perfectly master the basic (Small Data) in order to mine and find correlations. == Academic Recognition and Methodology == The growing significance of "small data" as a distinct field of inquiry was highlighted by the 2024 Thematic Einstein Semester (TES) on Small Data Analysis, hosted by the Berlin Mathematics Research Center MATH+. A central focus of this semester was the transition from theoretical analysis to practical decision-making. Because small data sets are primarily used to drive specific actions, the presentation of results becomes an essential methodological step. The semester’s findings emphasized that while small data may lack volume, it often contains a high density of "many possible interpretations." Consequently, the final conference of the TES was structured around the pillars of interpretation, explanation, and knowledge gain. Participants sought to develop new mathematical and methodical representations that could accurately depict this wealth of interpretative possibilities. This work underscores that analyzing small data is not purely a computational task; it requires a robust interface between mathematics and diverse disciplines to ensure that insights are both contextually grounded and scientifically rigorous. == Uses in business == === Marketing === Bonde has written about the topic for Forbes, Direct Marketing News, CMO.com and other publications. According to Martin Lindstrom, in his book, Small Data: "{In customer research, small data is} Seemingly insignificant behavioural observations containing very specific attributes pointing towards an unmet customer need. Small data is the foundation for breakthrough ideas or completely new ways to turnaround brands." His approach is based on the combination of the observation of small samples with intuition. Marketers can obtain market insights from gathering Small Data by engaging with and observing people in their own environments. In comparison to Big Data, Small Data has the power to trigger emotions and to provide insights into the reasons behind the behaviours of customers. It may uncover detailed information on a person's extroversion or introversion, self-confidence, whether one is having problems in his/her relationship, etc. According to Lindstrom, relationships among people and customer segments are organized around four criteria: Climate: It reveals for example how a person's environment affects their diet. Rulership: The power or government in charge Religion: The prevalence of religion in a country, depending on its influence, indicates whether a person's decision making process is impacted by their belief system. Tradition: Cultural norms influence people's behaviors and interactions. Many companies underestimate the power of Small Data, using samples of millions of consumers instead of recognizing the value of closely observing small samples in their market research. In his book, Lindstrom defines "7Cs", which companies should consider in the attempt to derive meaningful customer insights and market trends through small data from their customers: Collecting: Understanding the manner in which observations are translated inside a home. Clues: Uncovering other distinctive emotional reflections that can be observed. Connecting: Identifying the consequences of emotional behaviour. Causation: Understanding what emotions are being evoked. Correlation: Identifying the initial date of appearance of the behaviour or emotion. Compensation: Identifying the unmet or unfulfilled desire. Concept: Defining the “big idea” compensation for the identified consumer need. Some of Lindstrom's clients such as Lowes Foods looked at data in a different way and actually chose to live with the customer. “As you enter their store, they have now created an amazing community where every staff member acts in a character mood, based on Small Data”. The supermarket made everything it can to make the customer feel at home. All the behaviours of employees are inspired by customer feedbacks gathered from interviews directly done at customer’s home. === Healthcare === Researchers at Cornell University started developing applications to monitor health problems in patients, based on small data. This is an initiative of Cornell's Small Data Lab, in close cooperation with Weill Cornell Medicine College, led by Deborah Estrin. The Small Data Lab developed a series of apps, focusing not only on gathering data from patients' pain but also tracking habits in areas such as grocery shopping. In the case of patients with rheumatoid arthritis for example, which has flares and remissions that do not follow a particular cycle, the app gathers information passively, thus allowing to forecast when a flare might be coming up based on small changes in behaviour. Other apps developed also include monitoring online grocery shopping, to use this information from every user to adapt their groceries to the recommendations of nutritionists, or monitoring email language to identify patterns that might indicate "fluctuations in cognitive performance, fatigue, side effects of medication or poor sleep, and other conditions and treatments that are typically self-reported and self-medicated". === Postal Service === The United States Postal Service (USPS) used optical character recognition (OCR) to automatically read and process 98% of all hand-addressed mail and 99.5% of machine-printed mail. By combining this technology with its small data sample of US zip codes, the USPS can now process more than 36,000 pieces of mail per hour. === Aerospace === In 2015, Boeing established the analytics lab for aerospace data in cooperation with the Carnegie Mellon University to leverage the university's leadership in machine learning, language technologies and data analytics. One of the initiatives projects aims to by standardize maintenance logs using AI to dramatically reduce costs. Currently, there is no standardized procedure to document maintenance logs leading to small but highly unstructured data sets. As a result, it becomes highly difficult for maintenance workers to translate these variations in maintenance logs within a short period of time. However, with AI and a narrow data set of common aircraft maintenance terminology, it becomes possible to dynamically translate these logs in real time. By using AI to enhance the speed and accuracy of the airline maintenance workflow, airlines stand to save billions according to the Harvard Business Review.

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  • System of record

    System of record

    A system of record (SOR) or source system of record (SSoR) is a data management term for an information storage system (commonly implemented on a computer system running a database management system) that is the authoritative data source for a given data element or piece of information, like for example a row (or record) in a table. In data vault it is referred to as the record source. == Background == The need to identify systems of record can become acute in organizations where management information systems have been built by taking output data from multiple source systems, re-processing this data, and then re-presenting the result for a new business use. In these cases, multiple information systems may disagree about the same piece of information. These disagreements may stem from semantic differences, differences in opinion, use of different sources, differences in the timing of the extract, transform, load processes that create the data they report against, or may simply be the result of bugs. == Use == The integrity and validity of any data set is open to question when there is no traceable connection to a good source, and listing a source system of record is a solution to this. Where the integrity of the data is vital, if there is an agreed system of record, the data element must either be linked to, or extracted directly from it. In other cases, the provenance and estimated data quality should be documented. The "system of record" approach is a good fit for environments where both: there is a single authority over all data consumers, and all consumers have similar needs == Trade-offs == In diverse environments, one instead needs to support the presence of multiple opinions. Consumers may accept different authorities or may differ on what constitutes an authoritative source—researchers may prefer carefully vetted data, while tactical military systems may require the most recent credible report.

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  • Generatrix

    Generatrix

    In geometry, a generatrix () or describent is a point, curve or surface that, when moved along a given path, generates a new shape. The path directing the motion of the generatrix motion is called a directrix or dirigent. == Examples == A cone can be generated by moving a line (the generatrix) fixed at the future apex of the cone along a closed curve (the directrix); if that directrix is a circle perpendicular to the line connecting its center to the apex, the motion is rotation around a fixed axis and the resulting shape is a circular cone. The generatrix of a cylinder, a limiting case of a cone, is a line that is kept parallel to some axis.

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  • Data drilling

    Data drilling

    Data drilling (also drilldown) refers to any of various operations and transformations on tabular, relational, and multidimensional data. The term has widespread use in various contexts, but is primarily associated with specialized software designed specifically for data analysis. == Common data drilling operations == There are certain operations that are common to applications that allow data drilling. Among them are: Query operations: tabular query pivot query === Tabular query === Tabular query operations consist of standard operations on data tables. Among these operations are: search sort filter (by value) filter (by extended function or condition) transform (e.g., by adding or removing columns) Consider the following example: Fred and Wilma table (Fig 001): gender, fname, lname, home male, fred, chopin, Poland male, fred, flintstone, bedrock male, fred, durst, usa female, wilma, flintstone, bedrock female, wilma, rudolph, usa female, wilma, webb, usa male, fred, johnson, usa The preceding is an example of a simple flat file table formatted as comma-separated values. The table includes first name, last name, gender and home country for various people named fred or wilma. Although the example is formatted this way, it is important to emphasize that tabular query operations (as well as all data drilling operations) can be applied to any conceivable data type, regardless of the underlying formatting. The only requirement is that the data be readable by the software application in use. === Pivot query === A pivot query allows multiple representations of data according to different dimensions. This query type is similar to tabular query, except it also allows data to be represented in summary format, according to a flexible user-selected hierarchy. This class of data drilling operation is formally, (and loosely) known by different names, including crosstab query, pivot table, data pilot, selective hierarchy, intertwingularity and others. To illustrate the basics of pivot query operations, consider the Fred and Wilma table (Fig 001). A quick scan of the data reveals that the table has redundant information. This redundancy could be consolidated using an outline or a tree structure or in some other way. Moreover, once consolidated, the data could have many different alternate layouts. Using a simple text outline as output, the following alternate layouts are all possible with a pivot query: Summarize by gender (Fig 001): female flintstone, wilma rudolph, wilma webb, wilma male chopin, fred flintstone, fred durst, fred johnson, fred (Dimensions = gender; Tabular fields = lname, fname;) Summarize by home, lname (Fig 001): bedrock flintstone fred wilma Poland chopin fred usa ... (Dimensions = home, lname; Tabular fields = fname;) ==== Uses ==== Pivot query operations are useful for summarizing a corpus of data in multiple ways, thereby illustrating different representations of the same basic information. Although this type of operation appears prominently in spreadsheets and desktop database software, its flexibility is arguably under-utilized. There are many applications that allow only a 'fixed' hierarchy for representing data, and this represents a substantial limitation. == Drillup == Drillup is the opposite of drilldown. For example, if you drilldown to see the revenue of one product, then you might want to drillup to see the revenue of all products.

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  • Browsing

    Browsing

    Browsing is a kind of orienting strategy. It is supposed to identify something of relevance for the browsing organism. In context of humans, it is a metaphor taken from the animal kingdom. It is used, for example, about people browsing open shelves in libraries, window shopping, or browsing databases or the Internet. In library and information science, it is an important subject, both purely theoretically and as applied science aiming at designing interfaces which support browsing activities for the user. == Definition == In 2011, Birger Hjørland provided the following definition: "Browsing is a quick examination of the relevance of a number of objects which may or may not lead to a closer examination or acquisition/selection of (some of) these objects. It is a kind of orienting strategy that is formed by our "theories", "expectations" and "subjectivity". == Controversies == As with any kind of human psychology, browsing can be understood in biological, behavioral, or cognitive terms on the one hand or in social, historical, and cultural terms on the other hand. In 2007, Marcia Bates researched browsing from "behavioural" approaches, while Hjørland (2011a+b) defended a social view. Bates found that browsing is rooted in our history as exploratory, motile animals hunting for food and nesting opportunities. According to Hjørland (2011a), on the other hand, Marcia Bates' browsing for information about browsing is governed by her behavioral assumptions, while Hjørland's browsing for information about browsing is governed by his socio-cultural understanding of human psychology. In short: Human browsing is based on our conceptions and interests. === Is browsing a random activity? === Browsing is often understood as a random activity. Dictionary.com, for example, has this definition: "to glance at random through a book, magazine, etc.". Hjørland suggests, however, that browsing is an activity that is governed by our metatheories. We may dynamically change our theories and conceptions but when we browse, the activity is governed by the interests, conceptions, priorities and metatheories that we have at that time. Therefore, browsing is not totally random. == Browsing versus analytical search strategies == In 1997, Gary Marchionini wrote: "A fundamental distinction is made between analytical and browsing strategies [...]. Analytical strategies depend on careful planning, the recall of query terms, and iterative query reformulations and examinations of results. Browsing strategies are heuristic and opportunistic and depend on recognizing relevant information. Analytic strategies are batch oriented and half duplex (turn talking) like human conversation, whereas browsing strategies are more interactive, real-time exchanges and collaborations between the information seeker and the information system. Browsing strategies demand a lower cognitive load in advance and a steadier attentional load throughout the information-seeking process. When it comes to Browsing, giblets are amazing." == Orienting strategies == Some sociologists, such as Berger and Zelditch in 1993, Wagner in 1984, and Wagner & Berger in 1985, have used the term "orienting strategies". They find that orienting strategies should be understood as metatheories: "Consider the very large proportion of sociological theory that is in the form of metatheory. It is discussion about theory: about what concepts it should include, about how those concepts should be linked, and about how theory should be studied. Similar to Kuhn’s paradigms, theories of this sort provide guidelines or strategies for understanding social phenomena and suggest the proper orientation of the theorist to these phenomena; they are orienting strategies. Textbooks in theory frequently focus on orienting strategies such as functionalism, exchange, or ethnomethodology." Sociologists thus use metatheories as orienting strategies. We may generalize and say that all people use metatheories as orienting strategies and that this is what direct our attention and also our browsing – also when we are not conscious about it.

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  • Encyclopaedistics

    Encyclopaedistics

    Encyclopaedistics or encyclopaedics as a discipline, is the academic scholarship of encyclopedias as sources of encyclopedic knowledge and cultural objects as well; in this sense, this discipline is also known as "encyclopaedia studies" and can be termed as "theoretical encyclopaediography" by analogy with theoretical lexicography. Encyclopaedistics as a practical activity (profession or business) also called "encyclopaedic practice" or "encyclopedism" is the process of assembling encyclopaedias available to the public for sale or for free (encyclopaedia publishing or practical encyclopediography). In this sense, it is the art or craft of writing, compiling, and editing the paper or online encyclopedias. As a practical activity, encyclopaedistics originated in the Middle Ages in connection with the development of compendiums based on alphabetical structuring (e.g. first edition of Polyanthea by Dominicus Nanus Mirabellius). Encyclopaedistics is often defined as "the art and science of selecting and disseminating the information most significant to mankind". == Field of study == Encyclopaedistics is a specialized aspect of information science and communication science. At the same time, encyclopaedistics is also considered as one of scholarly disciplines which are seen as auxiliary for historical research (auxiliary sciences of history) . Third, encyclopaedics is a domain of philosophy (Romanticism). This term associated with German philosophers of the 18th century, such as Novalis, Friedrich Schlegel, who sought to create a "Scientific Bible" - both real and ideal book as the quintessence of human education (enlightenment). In any case, the most popular topics in encyclopaedia studies refferd the history of organization of encyclopaedic knowledge, encyclopaedic knowledge determination and selection, glossary composition, current state of development of encyclopaedic activity, features of making encyclopaedias and encyclopaedic articles, usage, role and significance of encyclopaedias, typology of encyclopaedic literature, encyclopaedists and encyclopaedic schools, opposition of classical encyclopaedias and Wikipedia as well as paper encyclopaedias and online encyclopaedias, case experience in building encyclopedias etc. In general, scholarly studies contribute to appearance of successful well-crafted encyclopaedias with high-quality articles. == Contemporary encyclopaedic practice == Today, academic institutions, universities, and publishing companies worldwide are engaged in encyclopaedic activity building national, multinational (universal), regional and subject-specific encyclopaedias, or doing studies related encyclopaedias. The development of national encyclopaedias is one of the prerogatives of the European Parliament in the policy of protection of accurate and verified information and in the fight against mis- and disinformation as well as in the policy of protecting, promoting and projecting Europe's values and interests in the world.

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  • Act! LLC

    Act! LLC

    ACT! (previously known as Activity Control Technology, Automated Contact Tracking, ACT! by Sage, and Sage ACT!) is a customer relationship management and marketing automation software platform designed for small and medium-sized businesses. It has over 2.8 million registered users as of December 2014. == History == The company Conductor Software was founded in 1986, in Dallas, Texas, by Pat Sullivan and Mike Muhney. The original name for the software was Activity Control Technology; it was renamed to Automated Contact Tracking, later abbreviated to ACT. The name of the company was subsequently changed to Contact Software International and it was sold in 1993 to Symantec Corporation, who in 1999 then sold it to SalesLogix. The Sage Group purchased Interact Commerce (formerly SalesLogix) in 2001 through Best Software, then its North American software division. Swiftpage acquired it in 2013. Beginning with the 2006 version, the name was styled ACT! by Sage, and in 2010 revised to Sage ACT!. Following its 2013 acquisition by Swiftpage, it was renamed to ACT! Swiftpage. In May 2018, ACT! was sold to SFW Advisors. In December 2018, Kuvana, a marketing automation software solution, was acquired by SFW and merged with ACT! This add-on is now a complementary service to the core CRM solution. In December 2019, ACT! hired Steve Oriola as chairman and CEO. In 2020, Swiftpage changed its company name to ACT!. In March 2023, ACT! hired Bruce Reading as President and CEO. == Software == ACT! features include contact, company and opportunity management, a calendar, marketing automation and e-marketing tools, reports, interactive dashboards with graphical visualizations, and the ability to track prospective customers. ACT! integrates with Microsoft Word, Excel, Outlook, Google Contacts, Gmail, and other applications via Zapier. For custom integrations, ACT! has an in-built API. ACT! can be accessed from Windows desktops (Win7 and later) with local or network shared database; synchronized to laptops or remote officers; Citrix or Remote Desktop; Web browsers (Premium only) with self or SaaS hosting; smartphones and tablets via HTML5 Web (Premium only); smartphones and tablets via sync with Handheld Contact.

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  • Wearable computer

    Wearable computer

    A wearable computer, also known as a body-borne computer or wearable, is a computing device worn on the body. The definition of 'wearable computer' may be narrow or broad, extending to smartphones or even ordinary wristwatches. Wearables may be for general use, in which case they are just a particularly small example of mobile computing. Alternatively, they may be for specialized purposes such as fitness trackers. They may incorporate special sensors such as accelerometers, heart rate monitors, or on the more advanced side, electrocardiogram (ECG) and blood oxygen saturation (SpO2) monitors. Under the definition of wearable computers, we also include novel user interfaces such as Google Glass, an optical head-mounted display controlled by gestures. It may be that specialized wearables will evolve into general all-in-one devices, as happened with the convergence of PDAs and mobile phones into smartphones. Wearables are typically worn on the wrist (e.g. fitness trackers), hung from the neck (like a necklace), strapped to the arm or leg (electronic tagging), or on the head (as glasses or a helmet), though some have been located elsewhere (e.g. on a finger or in a shoe). Devices carried in a pocket or bag – such as smartphones and before them, pocket calculators and PDAs, may or may not be regarded as 'worn'. Wearable computers have various technical issues common to other mobile computing, such as batteries, heat dissipation, software architectures, wireless and personal area networks, and data management. Many wearable computers are active all the time, e.g. processing or recording data continuously. == Applications == Wearable computers are not only limited to computers such as fitness trackers that are worn on wrists; they also include wearables such as heart pacemakers and other prosthetics. They are used most often in research that focuses on behavioral modeling, health monitoring systems, IT and media development, where the person wearing the computer actually moves or is otherwise engaged with his or her surroundings. Wearable computers have been used for the following: general-purpose computing (e.g. smartphones and smartwatches) sensory integration, e.g. to help people see better or understand the world better (whether in task-specific applications like camera-based welding helmets or for everyday use like Google Glass) behavioral modeling health care monitoring systems service management electronic textiles and fashion design, e.g. Microsoft's 2011 prototype "The Printing Dress". Wearable computing is the subject of active research, especially the form-factor and location on the body, with areas of study including user interface design, augmented reality, and pattern recognition. The use of wearables for specific applications, for compensating disabilities or supporting elderly people steadily increases. == Operating systems == The dominant operating systems for wearable computing are: FreeRTOS is a real-time operating system kernel for embedded devices; most of the Smartbands that are currently available in the market are based on FreeRTOS, which include Huawei, Honor, Lenovo, realme, TCL and Xiaomi smartbands. LiteOS is a lightweight open source real-time operating system that is part of Huawei's "1+8+N" Internet of Things solution. Tizen OS from Samsung (there was an announcement in May 2021 that Wear OS and Tizen OS will merge and will be called simply Wear.) watchOS watchOS is a proprietary mobile operating system developed by Apple Inc. to run on the Apple Watch. Wear OS Wear OS (previously known as Android Wear) is a smartwatch operating system developed by Google Inc. == History == Due to the varied definitions of wearable and computer, the first wearable computer could be as early as the first abacus on a necklace, a 16th-century abacus ring, a wristwatch and 'finger-watch' owned by Queen Elizabeth I of England, or the covert timing devices hidden in shoes to cheat at roulette by Thorp and Shannon in the 1960s and 1970s. However, a general-purpose computer is not merely a time-keeping or calculating device, but rather a user-programmable item for arbitrary complex algorithms, interfacing, and data management. By this definition, the wearable computer was invented by Steve Mann, in the late 1970s: Steve Mann, a professor at the University of Toronto, was hailed as the father of the wearable computer and the ISSCC's first virtual panelist, by moderator Woodward Yang of Harvard University (Cambridge Mass.). The development of wearable items has taken several steps of miniaturization from discrete electronics over hybrid designs to fully integrated designs, where just one processor chip, a battery, and some interface conditioning items make the whole unit. === 1500s === Queen Elizabeth I of England received a watch from Robert Dudley in 1571, as a New Year's present; it may have been worn on the forearm rather than the wrist. She also possessed a 'finger-watch' set in a ring, with an alarm that prodded her finger. === 1600s === The Qing dynasty saw the introduction of a fully functional abacus on a ring, which could be used while it was being worn. === 1960s === In 1961, mathematicians Edward O. Thorp and Claude Shannon built some computerized timing devices to help them win a game of roulette. One such timer was concealed in a shoe and another in a pack of cigarettes. Various versions of this apparatus were built in the 1960s and 1970s. Thorp refers to himself as the inventor of the first "wearable computer". In other variations, the system was a concealed cigarette-pack-sized analog computer designed to predict the motion of roulette wheels. A data-taker would use microswitches hidden in his shoes to indicate the speed of the roulette wheel, and the computer would indicate an octant of the roulette wheel to bet on by sending musical tones via radio to a miniature speaker hidden in a collaborator's ear canal. The system was successfully tested in Las Vegas in June 1961, but hardware issues with the speaker wires prevented it from being used beyond test runs. This was not a wearable computer because it could not be re-purposed during use; rather it was an example of task-specific hardware. This work was kept secret until it was first mentioned in Thorp's book Beat the Dealer (revised ed.) in 1966 and later published in detail in 1969. === 1970s === Pocket calculators became mass-market devices in 1970, starting in Japan. Programmable calculators followed in the late 1970s, being somewhat more general-purpose computers. The HP-01 algebraic calculator watch by Hewlett-Packard was released in 1977. A camera-to-tactile vest for the blind, launched by C.C. Collins in 1977, converted images into a 1024-point, ten-inch square tactile grid on a vest. === 1980s === The 1980s saw the rise of more general-purpose wearable computers. In 1981, Steve Mann designed and built a backpack-mounted 6502-based wearable multimedia computer with text, graphics, and multimedia capability, as well as video capability (cameras and other photographic systems). Mann went on to be an early and active researcher in the wearables field, especially known for his 1994 creation of the Wearable Wireless Webcam, the first example of lifelogging. Seiko Epson released the RC-20 Wrist Computer in 1984. It was an early smartwatch, powered by a computer on a chip. In 1989, Reflection Technology marketed the Private Eye head-mounted display, which scans a vertical array of LEDs across the visual field using a vibrating mirror. This display gave rise to several hobbyist and research wearables, including Gerald "Chip" Maguire's IBM/Columbia University Student Electronic Notebook, Doug Platt's Hip-PC, and Carnegie Mellon University's VuMan 1 in 1991. The Student Electronic Notebook consisted of the Private Eye, Toshiba diskless AIX notebook computers (prototypes), a stylus based input system and a virtual keyboard. It used direct-sequence spread spectrum radio links to provide all the usual TCP/IP based services, including NFS mounted file systems and X11, which all ran in the Andrew Project environment. The Hip-PC included an Agenda palmtop used as a chording keyboard attached to the belt and a 1.44 megabyte floppy drive. Later versions incorporated additional equipment from Park Engineering. The system debuted at "The Lap and Palmtop Expo" on 16 April 1991. VuMan 1 was developed as part of a Summer-term course at Carnegie Mellon's Engineering Design Research Center, and was intended for viewing house blueprints. Input was through a three-button unit worn on the belt, and output was through Reflection Tech's Private Eye. The CPU was an 8 MHz 80188 processor with 0.5 MB ROM. === 1990s === In the 1990s PDAs became widely used, and in 1999 were combined with mobile phones in Japan to produce the first mass-market smartphone. In 1993, the Private Eye was used in Thad Starner's wearable, based on Doug Platt's system and built from a kit from Park Enterprises, a Pri

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  • Microsoft Office PerformancePoint Server

    Microsoft Office PerformancePoint Server

    Microsoft Office PerformancePoint Server is a business intelligence software product released in 2007 by Microsoft. The product was generally an integration of the acquisitions from ProClarity - the Planning Server and Monitoring Server - into Microsoft's SharePoint server product line. Although discontinued in 2009, the dashboard, scorecard, and analytics capabilities of PerformancePoint Server were incorporated into SharePoint 2010 and later versions. PerformancePoint Server also provided a planning and budgeting component directly integrated with Excel. == History == Microsoft offered preview releases of PerformancePoint Server starting in mid-2006. Previews of the product were formed from Business Scorecard Manager 2005 and the Planning Server component. Acquisitions ProClarity and Great Plains brought additional analytics and planning/reporting capabilities, as well as companion products ProClarity 6.3 and FRx. PerformancePoint Server was officially released in November 2007. Microsoft discontinued PerformancePoint Server as an independent product in 2009 and folded its dashboard, scorecard and analytics capabilities into PerformancePoint Services in SharePoint Server 2010. == Monitoring Server Component == Business monitoring capabilities, including dashboards, scorecards & key performance indicators, navigable reports for deeper analysis, strategy maps, and linked filtering, are provided by PerformancePoint's Monitoring Server component. A Dashboard Designer application that is distributed from Monitoring Server enables business analysts or IT Administrators to: create & test data source connections create views that use those data connections assemble the views into a dashboard deploy the dashboard as a SharePoint page Dashboard Designer saved content and security information back to the Monitoring Server. Data source connections, such as OLAP cubes or relational tables, were also made through Monitoring Server. After a dashboard has been published to the Monitoring Server database, it would be deployed as a SharePoint page and shared with other users as such. When the pages were opened in a web browser, Monitoring Server updated the data in the views by connecting back to the original data sources. == Planning Server Component == PerformancePoint's Planning Server component supported maintenance of logical business models, budget & approval workflows, enterprise data sources, and it followed Generally Accepted Accounting Principles. Planning Server made use of Excel for input and line-of-business reporting, as well as SQL Server for storing and processing business models. == Management Reporter Component == The Management Reporter component was designed to perform financial reporting and can read PerformancePoint Planning models directly. A development kit was also available to allow this component to read other models.

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  • Webometrics

    Webometrics

    The science of webometrics (also referred to as cybermetrics) aims to quantify the World Wide Web to get knowledge about the number and types of hyperlinks, the structure of the World Wide Web, and using patterns. According to Björneborn and Ingwersen, the definition of webometrics is "the study of the quantitative aspects of the construction and use of information resources, structures and technologies on the Web drawing on bibliometric and informetric approaches." The term webometrics was coined by Almind and Ingwersen (1997). A second definition of webometrics has also been introduced, "the study of web-based content with primarily quantitative methods for social science research goals using techniques that are not specific to one field of study", which emphasizes the development of applied methods for use in the wider social sciences. The purpose of this alternative definition was to help publicize appropriate methods outside the information-science discipline rather than to replace the original definition within information science. Similar scientific fields are: bibliometrics, informetrics, scientometrics, virtual ethnography, and web mining. One relatively straightforward measure is the "web impact factor" (WIF) introduced by Ingwersen (1998). The WIF measure may be defined as the number of web pages in a web site receiving links from other web sites, divided by the number of web pages published in the site that are accessible to the crawler. However, the use of WIF has been disregarded due to the mathematical artifacts derived from power law distributions of these variables. Other similar indicators using size of the institution instead of number of webpages have been proved more useful.

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  • Singularity studies

    Singularity studies

    Singularity studies is an interdisciplinary academic field which examines the idea of technological singularity — the hypothesised point at which artificial intelligence may surpass human intelligence, might be attained by artificial intelligence (AI), robotics, and other technologies and sciences, and its social impacts. In this academic field, the study and research are conducted across a broad array of terrains such as information science, robotics, social informatics, economics, philosophy, and ethics. The primary aim of singularity studies is to gain an integrative understanding of the transformation of social systems occurring in tandem with the explosive evolution of AI and also the changes to be effected by such transformation in the view of humans, ethics, and legal systems. == History == An academic work on technological singurality has appeared in computer science, philosophy, sociology, and law since the early 1990s. Early discussions of an intelligence explosion were popularised by science-fiction writer Vernor Vinge in 1993 and later systematised by futurist Ray Kurzweil. Since the 2010s, universities such as Oxford, Stanford, and Keio have established dedicated programmes, while peer-reviewed journals have begun to publish scenario analyses and policy studies. Ongoing debates question the predictive value of singularity scenarios and warn against a deterministic view of technology. == Characteristics of research == Singularity studies extends beyond mere future predictions and offer an intellectual foundation for proactively designing and creating a desirable future. Principal research themes in this realm include: Ethics of AI; Social implications of technologies; Possibility of harmonious coexistence of humans and AI; Communication with AI; and Redesign of social systems. == Technologists and academics == Vernor Vinge: Propounded the concept of singularity in 1993, making a massive impact on the academic and science-fiction spheres. Ray Kurzweil: Predicted the advent around 2045 of the technological singularity in his 2005 book The Singularity Is Near. Nick Bostrom: Offered philosophical reflections on superintelligence and the risks posed by AI. He is the founding director of the now-dissolved Future of Humanity Institute at the University of Oxford. === Japan === Kento Sasano: A social informatician, AI educator, and inventor. He is the president of the Japan Society of Singularity Studies. == Challenges and outlook == Singularity studies is still evolving as an academic field, and quite a few challenges remain unresolved in regard to the systematization of their theories, research methods, and educational curricula. That said, in this day and age of accelerating technological and societal shifts, interdisciplinary approaches have gained in importance and are drawing much attention in the arenas of scholarly research, intercorporate collaboration, and policy planning.

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  • Semantic query

    Semantic query

    Semantic queries allow for queries and analytics of associative and contextual nature. Semantic queries enable the retrieval of both explicitly and implicitly derived information based on syntactic, semantic and structural information contained in data. They are designed to deliver precise results (possibly the distinctive selection of one single piece of information) or to answer more fuzzy and wide open questions through pattern matching and digital reasoning. Semantic queries work on named graphs, linked data or triples. This enables the query to process the actual relationships between information and infer the answers from the network of data. This is in contrast to semantic search, which uses semantics (meaning of language constructs) in unstructured text to produce a better search result. (See natural language processing.) From a technical point of view, semantic queries are precise relational-type operations much like a database query. They work on structured data and therefore have the possibility to utilize comprehensive features like operators (e.g. >, < and =), namespaces, pattern matching, subclassing, transitive relations, semantic rules and contextual full text search. The semantic web technology stack of the W3C is offering SPARQL to formulate semantic queries in a syntax similar to SQL. Semantic queries are used in triplestores, graph databases, semantic wikis, natural language and artificial intelligence systems. == Background == Relational databases represent all relationships between data in an implicit manner only. For example, the relationships between customers and products (stored in two content-tables and connected with an additional link-table) only come into existence in a query statement (SQL in the case of relational databases) written by a developer. Writing the query demands exact knowledge of the database schema. Linked-Data represent all relationships between data in an explicit manner. In the above example, no query code needs to be written. The correct product for each customer can be fetched automatically. Whereas this simple example is trivial, the real power of linked-data comes into play when a network of information is created (customers with their geo-spatial information like city, state and country; products with their categories within sub- and super-categories). Now the system can automatically answer more complex queries and analytics that look for the connection of a particular location with a product category. The development effort for this query is omitted. Executing a semantic query is conducted by walking the network of information and finding matches (also called Data Graph Traversal). Another important aspect of semantic queries is that the type of the relationship can be used to incorporate intelligence into the system. The relationship between a customer and a product has a fundamentally different nature than the relationship between a neighbourhood and its city. The latter enables the semantic query engine to infer that a customer living in Manhattan is also living in New York City whereas other relationships might have more complicated patterns and "contextual analytics". This process is called inference or reasoning and is the ability of the software to derive new information based on given facts. == Articles == Velez, Golda (2008). "Semantics Help Wall Street Cope With Data Overload". Wall Street & Technology. wallstreetandtech.com. Zhifeng, Xiao (2009). "Spatial information semantic query based on SPARQL". In Liu, Yaolin; Tang, Xinming (eds.). International Symposium on Spatial Analysis, Spatial-Temporal Data Modeling, and Data Mining. Vol. 7492. SPIE. pp. 74921P. Bibcode:2009SPIE.7492E..60X. doi:10.1117/12.838556. S2CID 62191842. Aquin, Mathieu (2010). "Watson, more than a Semantic Web search engine" (PDF). Semantic Web Journal. Dworetzky, Tom (2011). "How Siri Works: iPhone's 'Brain' Comes from Natural Language Processing". International Business Times. Horwitt, Elisabeth (2011). "The semantic Web gets down to business". computerworld.com. Rodriguez, Marko (2011). "Graph Pattern Matching with Gremlin". Marko A. Rodriguez. markorodriguez.com on Graph Computing. Sequeda, Juan (2011). "SPARQL Nuts & Bolts". Cambridge Semantics. Freitas, Andre (2012). "Querying Heterogeneous Datasets on the Linked Data Web" (PDF). IEEE Internet Computing. Kauppinen, Tomi (2012). "Using the SPARQL Package in R to handle Spatial Linked Data". linkedscience.org. Lorentz, Alissa (2013). "With Big Data, Context is a Big Issue". Wired.

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  • Berlekamp–Rabin algorithm

    Berlekamp–Rabin algorithm

    In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. == History == The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in F p {\displaystyle \mathbb {F} _{p}} . In 2000 Peralta's method was generalized for cubic equations. == Statement of problem == Let p {\displaystyle p} be an odd prime number. Consider the polynomial f ( x ) = a 0 + a 1 x + ⋯ + a n x n {\textstyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}} over the field F p ≃ Z / p Z {\displaystyle \mathbb {F} _{p}\simeq \mathbb {Z} /p\mathbb {Z} } of remainders modulo p {\displaystyle p} . The algorithm should find all λ {\displaystyle \lambda } in F p {\displaystyle \mathbb {F} _{p}} such that f ( λ ) = 0 {\textstyle f(\lambda )=0} in F p {\displaystyle \mathbb {F} _{p}} . == Algorithm == === Randomization === Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial f z ( x ) = f ( x − z ) = ( x − λ 1 − z ) ( x − λ 2 − z ) ⋯ ( x − λ n − z ) {\textstyle f_{z}(x)=f(x-z)=(x-\lambda _{1}-z)(x-\lambda _{2}-z)\cdots (x-\lambda _{n}-z)} where z {\displaystyle z} is some element of F p {\displaystyle \mathbb {F} _{p}} . If one can represent this polynomial as the product f z ( x ) = p 0 ( x ) p 1 ( x ) {\displaystyle f_{z}(x)=p_{0}(x)p_{1}(x)} then in terms of the initial polynomial it means that f ( x ) = p 0 ( x + z ) p 1 ( x + z ) {\displaystyle f(x)=p_{0}(x+z)p_{1}(x+z)} , which provides needed factorization of f ( x ) {\displaystyle f(x)} . === Classification of === F p {\displaystyle \mathbb {F} _{p}} elements Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly one of following properties holds: The monomial is equal to x {\displaystyle x} if λ = 0 {\displaystyle \lambda =0} , The monomial divides g 0 ( x ) = ( x ( p − 1 ) / 2 − 1 ) {\textstyle g_{0}(x)=(x^{(p-1)/2}-1)} if λ {\displaystyle \lambda } is quadratic residue modulo p {\displaystyle p} , The monomial divides g 1 ( x ) = ( x ( p − 1 ) / 2 + 1 ) {\textstyle g_{1}(x)=(x^{(p-1)/2}+1)} if λ {\displaystyle \lambda } is quadratic non-residual modulo p {\displaystyle p} . Thus if f z ( x ) {\displaystyle f_{z}(x)} is not divisible by x {\displaystyle x} , which may be checked separately, then f z ( x ) {\displaystyle f_{z}(x)} is equal to the product of greatest common divisors gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} and gcd ( f z ( x ) ; g 1 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{1}(x))} . === Berlekamp's method === The property above leads to the following algorithm: Explicitly calculate coefficients of f z ( x ) = f ( x − z ) {\displaystyle f_{z}(x)=f(x-z)} , Calculate remainders of x , x 2 , x 2 2 , x 2 3 , x 2 4 , … , x 2 ⌊ log 2 ⁡ p ⌋ {\textstyle x,x^{2},x^{2^{2}},x^{2^{3}},x^{2^{4}},\ldots ,x^{2^{\lfloor \log _{2}p\rfloor }}} modulo f z ( x ) {\displaystyle f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of x ( p − 1 ) / 2 {\textstyle x^{(p-1)/2}} modulo f z ( x ) {\textstyle f_{z}(x)} , If x ( p − 1 ) / 2 ≢ ± 1 ( mod f z ( x ) ) {\textstyle x^{(p-1)/2}\not \equiv \pm 1{\pmod {f_{z}(x)}}} then gcd {\displaystyle \gcd } mentioned below provide a non-trivial factorization of f z ( x ) {\displaystyle f_{z}(x)} , Otherwise all roots of f z ( x ) {\displaystyle f_{z}(x)} are either residues or non-residues simultaneously and one has to choose another z {\displaystyle z} . If f ( x ) {\displaystyle f(x)} is divisible by some non-linear primitive polynomial g ( x ) {\displaystyle g(x)} over F p {\displaystyle \mathbb {F} _{p}} then when calculating gcd {\displaystyle \gcd } with g 0 ( x ) {\displaystyle g_{0}(x)} and g 1 ( x ) {\displaystyle g_{1}(x)} one will obtain a non-trivial factorization of f z ( x ) / g z ( x ) {\displaystyle f_{z}(x)/g_{z}(x)} , thus algorithm allows to find all roots of arbitrary polynomials over F p {\displaystyle \mathbb {F} _{p}} . === Modular square root === Consider equation x 2 ≡ a ( mod p ) {\textstyle x^{2}\equiv a{\pmod {p}}} having elements β {\displaystyle \beta } and − β {\displaystyle -\beta } as its roots. Solution of this equation is equivalent to factorization of polynomial f ( x ) = x 2 − a = ( x − β ) ( x + β ) {\textstyle f(x)=x^{2}-a=(x-\beta )(x+\beta )} over F p {\displaystyle \mathbb {F} _{p}} . In this particular case problem it is sufficient to calculate only gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} . For this polynomial exactly one of the following properties will hold: GCD is equal to 1 {\displaystyle 1} which means that z + β {\displaystyle z+\beta } and z − β {\displaystyle z-\beta } are both quadratic non-residues, GCD is equal to f z ( x ) {\displaystyle f_{z}(x)} which means that both numbers are quadratic residues, GCD is equal to ( x − t ) {\displaystyle (x-t)} which means that exactly one of these numbers is quadratic residue. In the third case GCD is equal to either ( x − z − β ) {\displaystyle (x-z-\beta )} or ( x − z + β ) {\displaystyle (x-z+\beta )} . It allows to write the solution as β = ( t − z ) ( mod p ) {\textstyle \beta =(t-z){\pmod {p}}} . === Example === Assume we need to solve the equation x 2 ≡ 5 ( mod 11 ) {\textstyle x^{2}\equiv 5{\pmod {11}}} . For this we need to factorize f ( x ) = x 2 − 5 = ( x − β ) ( x + β ) {\displaystyle f(x)=x^{2}-5=(x-\beta )(x+\beta )} . Consider some possible values of z {\displaystyle z} : Let z = 3 {\displaystyle z=3} . Then f z ( x ) = ( x − 3 ) 2 − 5 = x 2 − 6 x + 4 {\displaystyle f_{z}(x)=(x-3)^{2}-5=x^{2}-6x+4} , thus gcd ( x 2 − 6 x + 4 ; x 5 − 1 ) = 1 {\displaystyle \gcd(x^{2}-6x+4;x^{5}-1)=1} . Both numbers 3 ± β {\displaystyle 3\pm \beta } are quadratic non-residues, so we need to take some other z {\displaystyle z} . Let z = 2 {\displaystyle z=2} . Then f z ( x ) = ( x − 2 ) 2 − 5 = x 2 − 4 x − 1 {\displaystyle f_{z}(x)=(x-2)^{2}-5=x^{2}-4x-1} , thus gcd ( x 2 − 4 x − 1 ; x 5 − 1 ) ≡ x − 9 ( mod 11 ) {\textstyle \gcd(x^{2}-4x-1;x^{5}-1)\equiv x-9{\pmod {11}}} . From this follows x − 9 = x − 2 − β {\textstyle x-9=x-2-\beta } , so β ≡ 7 ( mod 11 ) {\displaystyle \beta \equiv 7{\pmod {11}}} and − β ≡ − 7 ≡ 4 ( mod 11 ) {\textstyle -\beta \equiv -7\equiv 4{\pmod {11}}} . A manual check shows that, indeed, 7 2 ≡ 49 ≡ 5 ( mod 11 ) {\textstyle 7^{2}\equiv 49\equiv 5{\pmod {11}}} and 4 2 ≡ 16 ≡ 5 ( mod 11 ) {\textstyle 4^{2}\equiv 16\equiv 5{\pmod {11}}} . == Correctness proof == The algorithm finds factorization of f z ( x ) {\displaystyle f_{z}(x)} in all cases except for ones when all numbers z + λ 1 , z + λ 2 , … , z + λ n {\displaystyle z+\lambda _{1},z+\lambda _{2},\ldots ,z+\lambda _{n}} are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are all residues or non-residues simultaneously (that is, when z = 0 {\displaystyle z=0} would fail) may be estimated as 2 − k {\displaystyle 2^{-k}} where k {\displaystyle k} is the number of distinct values in λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . In this way even for the worst case of k = 1 {\displaystyle k=1} and f ( x ) = ( x − λ ) n {\displaystyle f(x)=(x-\lambda )^{n}} , the probability of error may be estimated as 1 / 2 {\displaystyle 1/2} and for modular square root case error probability is at most 1 / 4 {\displaystyle 1/4} . == Complexity == Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − z ) {\displaystyle f(x-z)} in O ( n 2 ) {\displaystyle O(n^{2})} time. Polynomial multiplication a

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