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Elastix (image registration)

Elastix is an image registration toolbox built upon the Insight Segmentation and Registration Toolkit (ITK). It is entirely open-source and provides a wide range of algorithms employed in image registration problems. Its components are designed to be modular to ease a fast and reliable creation of various registration pipelines tailored for case-specific applications. It was first developed by Stefan Klein and Marius Staring under the supervision of Josien P.W. Pluim at Image Sciences Institute (ISI). Its first version was command-line based, allowing the final user to employ scripts to automatically process big data-sets and deploy multiple registration pipelines with few lines of code. Nowadays, to further widen its audience, a version called SimpleElastix is also available, developed by Kasper Marstal, which allows the integration of elastix with high level languages, such as Python, Java, and R. == Image registration fundamentals == Image registration is a well-known technique in digital image processing that searches for the geometric transformation that, applied to a moving image, obtains a one-to-one map with a target image. Generally, the images acquired from different sensors (multimodal), time instants (multitemporal), and points of view (multiview) should be correctly aligned to proceed with further processing and feature extraction. Even though there are a plethora of different approaches to image registration, the majority is composed of the same macro building blocks, namely the transformation, the interpolator, the metric, and the optimizer. Registering two or more images can be framed as an optimization problem that requires multiple iterations to converge to the best solution. Starting from an initial transformation computed from the image moments the optimization process searches for the best transformation parameters based on the value of the selected similarity metric. The figure on the right shows the high-level representation of the registration of two images, where the reference remains constant during the entire process, while the moving one will be transformed according to the transformation parameters. In other words, the registration ends when the similarity metric, which is a mathematical function with a certain number of parameters to be optimized, reaches the optimal value which is highly dependent on the specific application. == Main building blocks == Following the structure of the image registration workflow, the elastix toolbox proposes a modular solution that implements for each of the building blocks different algorithms, highly employed in medical image registration, and helps the final users to build their specific pipeline by selecting the most suitable algorithm for each of the main building blocks. Each block is easily configurable both by selecting pre-defined initialization values or by trying multiple sets of parameters and then choosing the most performing one. The registration is performed on images, and the elastix toolbox supports all the data formats supported by ITK, ranging from JPEG and PNG to medical standard formats such as DICOM and NIFTI. It also stores physical pixel spacing, the origin and the relative position to an external world reference system, when provided in the metadata, to facilitate the registration process, especially in medical field applications. === Transformation === The transformation is an essential building block, since it defines the allowable transformations. In image registration, the main distinction can be done between parallel-to-parallel and parallel-to-non parallel (deformable) line mapping transformations. In the elastix toolbox, the final users can select one transformation or compose more transformations either through addition or via composition. Below are reported the different transformation models in order of increasing flexibility, along with the corresponding elastix class names between brackets. Translation (TranslationTransform) allows only translations Rigid (EulerTransform) expands the translation adding rotations and the object is seen as a rigid body Similarity (SimilarityTransform) expands the rigid transformation by introducing isotropic scaling Affine (AffineTransform) expands the rigid transformation allowing both scaling and shear B-splines (BSplineTransform) is a deformable transformation usually preceded by a rigid or affine one Thin-plate splines (SplineKernelTransform) is a deformable transformation belonging to the class of kernel-based transformations that is a composition of and affine and a non-rigid part === Metric === The similarity metric is the mathematical function whose parameters should be optimized to reach the desired registration, and, during the process, it is computed multiple times. Below are reported the available metrics computed employing the reference and the transformed images and the corresponding elastix class names between brackets. Mean squared difference (AdvancedMeanSquares) to be used for mono-modal applications Normalized correlation coefficient (AdvancedNormalizedCorrelation) to be used for images that have an intensity linear relationship Mutual information (AdvancedMattesMutualInformation) to be used for both mono- and multi-modal applications and optimized to reach better performance compared to the normalized version Normalized mutual information (NormalizedMutualInformation) for both mono- and multi-modal applications Kappa statistic (AdvancedKappaStatistic) to be used only for binary images === Sampler === For the computation of the similarity metrics, it is not always necessary to consider all the voxels and, sometimes, it can be useful to use only a fraction of the voxels of the images, i.e. to reduce the execution time for big input images. Below are reported the available criteria for selecting a fraction of the voxels for the similarity metric computation and the corresponding elastix class names between brackets. Full (Full) to employ all the voxels Grid (Grid) to employ a regular grid defined by the user to downsample the image Random (Random) to randomly select a percentage of voxels defined by the users (all voxels have equal probability to be selected) Random coordinate (RandomCoordinate) like the random criterion, but in this case also off-grid positions can be selected to simplify the optimization process === Interpolator === After the application of the transformation, it may occur that the voxels used for the similarity metric computation are at non-voxel positions, so intensity interpolation should be performed to ensure the correctness of the computed values. Below are reported the implemented interpolators and the corresponding elastix class names between brackets. Nearest neighbor (NearestNeighborInterpolator) exploits little resources, but gives low quality results Linear (LinearInterpolator) is sufficient in general applications N-th order B-spline (BSplineInterpolator) can be used to increase the order N, increasing quality and computation time. N=0 and N=1 indicate the nearest neighbor and linear cases respectively. === Optimizer === The optimizer defines the strategy employed for searching the best transformation parameter to reach the correct registration, and it is commonly an iterative strategy. Below are reported some of the implemented optimization strategies. Gradient descent Robbins-Monro, similar to the gradient descent, but employing an approximation of the cost function derivatives A wider range of optimizers is also available, such as Quasi-Newton or evolutionary strategies. === Other features === The elastix software also offers other features that can be employed to speed up the registration procedure and to provide more advanced algorithms to the end-users. Some examples are the introduction of blur and Gaussian pyramid to reduce data complexity, and multi-image and multi-metric framework to deal with more complex applications. == Applications == Elastix has applications mainly in the medical field, where image registration is fundamental to get comprehensive information regarding the analysed anatomical region. It is widely employed in image-guided surgery, tumour monitoring, and treatment assessment. For example, in radiotherapy planning, image registration allows to correctly deliver the treatment and evaluate the obtained results. Thanks to the wide range of implemented algorithms, the use of the elastix software allows physicians and researchers to test different registration pipelines from the simplest to more complex ones, and to save the best one as a configuration file. This file and the fact that the software is completely open-source makes it easy to reproduce the work, that can help supporting the open science paradigm, and allows fast reuse on different patients data. In image-guided surgery, registration time and accuracy are critical points, considering that, during the registration, the patient is on the operating table, and the imag
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Babel Fish (website)

Yahoo! Babel Fish was a free Web-based machine translation service by Yahoo!. In May 2012 it was replaced by Bing Translator (now Microsoft Translator), to which queries were redirected. Although Yahoo! has transitioned its Babel Fish translation services to Bing Translator, it did not sell its translation application to Microsoft outright. As the oldest free online language translator, the service translated text or Web pages in 36 pairs between 13 languages, including English, Simplified Chinese, Traditional Chinese, Dutch, French, German, Greek, Italian, Japanese, Korean, Portuguese, Russian, and Spanish. The internet service derived its name from the Babel fish, a fictional species in Douglas Adams's book and radio series The Hitchhiker's Guide to the Galaxy that could instantly translate languages. In turn, the name of the fictional creature refers to the biblical account of the confusion of languages that arose in the city of Babel. == History == On December 9, 1997, Digital Equipment Corporation (DEC) and SYSTRAN S.A. launched AltaVista Translation Service at babelfish.altavista.com, which was developed by a team of researchers at DEC. In February 2003, AltaVista was bought by Overture Services, Inc. In July 2003, Overture, in turn, was taken over by Yahoo!. The web address for Babel Fish remained at babelfish.altavista.com until May 9, 2008, when the address changed to babelfish.yahoo.com. In 2012, the Web address changed again, this time redirecting babelfish.yahoo.com to www.microsofttranslator.com when Microsoft's Bing Translator replaced Yahoo Babel Fish. As of June 2013, babelfish.yahoo.com no longer redirects to the Microsoft Bing Translator. Instead, it refers directly back to the main Yahoo.com page.
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Emma Brunskill

Emma Patricia Brunskill is an American computer scientist. Her research combines machine learning with human–computer interaction by studying the effects of AI systems in human-centered applications including educational software and healthcare, and the theory of reinforcement learning in situations where mistakes impose high risks or costs. She is an associate professor of computer science at Stanford University, where she also holds a courtesy appointment in the Stanford Graduate School of Education and is an affiliate of the King Center on Global Development. == Education and career == Brunskill grew up in Seattle and Edmonds, Washington, and entered the University of Washington at age 15. She graduated magna cum laude in 2000, with a bachelor's degree in computer engineering and physics. A Rhodes Scholarship took her to Magdalen College, Oxford in England, where she received a master's degree in neuroscience in 2002. After a summer working in Rwanda, she became a graduate student of computer science at the Massachusetts Institute of Technology, where she completed her Ph.D. in 2009. Her doctoral dissertation, Compact parametric models for efficient sequential decision making in high-dimensional, uncertain domains, was supervised by Nicholas Roy. After working as an NSF Postdoctoral Research Fellow at the University of California, Berkeley, she joined Carnegie Mellon University (CMU) in 2011 as an assistant professor of computer science. She moved from CMU to Stanford University in 2017. == Recognition == Brunskill was a 2014 recipient of the National Science Foundation CAREER Award and a 2015 recipient of the Office of Naval Research Young Investigator Award. She was one of two alumni of the University of Washington's Paul G. Allen School of Computer Science and Engineering to be honored in 2020 by the school's Alumni Impact Awards. She was elected as a Fellow of the Association for the Advancement of Artificial Intelligence in 2025, "for significant contributions to the field of reinforcement learning, and applications for societal benefit, in particular AI for education".
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Oren Etzioni

Oren Etzioni (born 1964) is Professor Emeritus of Computer Science at the University of Washington, and founding CEO of the Allen Institute for Artificial Intelligence (AI2). Etzioni is a co-founder of Vercept, an AI startup, and founder and CEO of TrueMedia.org, a non-profit dedicated to fighting political deepfakes, which launched in April 2024. He is also the Founder and Technical Director of the AI2 Incubator and a venture partner at the Madrona Venture Group. == Early life and education == Etzioni is the son of Israeli-American intellectual Amitai Etzioni. He was the first student to major in computer science at Harvard University, where he earned a bachelor's degree in 1986. He earned a PhD from Carnegie Mellon University in January, 1991, supervised by Tom M. Mitchell. == University of Washington career == Etzioni joined the University of Washington faculty in 1991, immediately after receiving his PhD. He rose through the ranks to become the Washington Research Foundation Entrepreneurship Professor in Computer Science & Engineering. Etzioni's research has been focused on basic problems in the study of intelligence, machine reading, machine learning and web search. Past projects include Internet Softbots—the study of intelligent agents in the context of real-world software testbeds. In 2003, he started the KnowItAll project for acquiring massive amounts of information from the web. In 2005, he founded and became the director of the university's Turing Center. The center investigated problems in data mining, natural language processing, the Semantic Web and other web search topics. Etzioni coined the term machine reading and helped to create the first commercial comparison shopping agent. He has published over 200 technical papers, and his H-index exceeds 100. == Entrepreneurship == As a faculty member Etzioni was also an active entrepreneur, founding multiple companies and pioneering multiple technologies including MetaCrawler (bought by Infospace), Netbot (bought by Excite in 1997 for $35 million), and ClearForest (bought by Reuters). He founded Farecast, a travel metasearch and price prediction site, which was acquired by Microsoft in 2008 for $115 million. Before founding Farecast, he developed a program originally called Hamlet, that used algorithms to identify patterns in airfare data using data-mining techniques. He also co-founded Decide.com, a website to help consumers make buying decisions using previous price history and recommendations from other users. Decide.com was bought by eBay in September, 2013. Etzioni is also a venture partner at the Madrona Venture Group. He is founder and CEO of TrueMedia.org, a non-profit dedicated to fighting political deepfakes, which launched in April 2024. Etzioni is a co-founder of Vercept, an AI startup formed in 2025. == Founding CEO of AI2 == In September 2013 Etzioni was selected as the Founding CEO of the Allen Institute for Artificial Intelligence by philanthropist Paul G. Allen, and in January 2014 he took a leave of absence from the University of Washington to serve in that role. Etzioni's technical contributions continued at AI2; for example, in 2015, he helped to create the Semantic Scholar search engine. Under Etzioni’s leadership, AI2 grew from zero to over two hundred team members including notable researchers and engineers across several domains of AI. By 2021, its AI2 researchers had published near 700 papers in publications such as AAAI, ACL, CVPR, NeurIPS, and ICLR. Twenty-four of these papers had garnered special-recognition awards. AI2 also offered several key resources and tools to the AI community including the AllenNLP library, Semantic Scholar, and the conservation platforms EarthRanger and Skylight. Ed Lazowska, AI2 Board Member, has stated about Etzioni that he "took the collegial, collaborative culture that he absorbed in his 20+ years as a professor in UW's Allen School and mixed it with the singular focus that drives startups to create an elixir that AI2 folks have been drinking over the last eight years. The result is an exceptional organization of scientists, engineers, and entrepreneurs that's pursuing Paul Allen’s vision of ‘AI for the Common Good’ with extraordinary success.” == Popular press == In addition to his scientific publications, Etzioni has written commentary on AI for The New York Times, Wired, Nature, and other publications. After reading the idea in a book about AI by Brad Smith and Harry Shum, Etzioni has attempted to create an oath for AI practitioners. In 2018, he published what he called a "Hippocratic Oath for artificial intelligence practitioners" in TechCrunch. == Awards and recognition == In 1993, Etzioni received a National Young Investigator Award. In 2003, Etzioni was elected as AAAI Fellow. In 2005, Etzioni received an IJCAI Distinguished Paper Award for "A Probabilistic Model of Redundancy in Information Extraction". In 2007, he received the Robert S. Engelmore Memorial Award. In 2012 Etzioni was featured as GeekWire's "Geek of the Week". In 2013 Etzioni was voted "Geek of the Year" through GeekWire. In 2022, Etzioni received the 2012 ACL Test-of-Time Paper Award. In 2022, Etzioni, along with Ana-Maria Popescu and Henry Kautz, received the ACM Intelligent User Interfaces Most Impact Award for their 2003 paper, "Towards a Theory of Natural Language Interfaces to Databases". == Personal life == Etzioni has three children, and has said in interviews that family is his number one priority. He is married to Ivone Etzioni, and was previously married to Dr. Ruth Etzioni, a biostatistician at the Fred Hutchinson Cancer Center. Outside of his professional career, Etzioni has a wide range of personal interests. He has attended the Burning Man festival, which he described as a valuable way to step outside his comfort zone. His first computer was a TRS-80, and he has described his car’s GPS as his favorite gadget, joking that he has “no sense of direction.” == Selected publications == === Scholarly publications === Etzioni, Oren (July 1994). "A Softbot-based Interface to the Internet" (PDF). Communications of the ACM. Retrieved March 29, 2018. Etzioni, Oren (December 2008). "Open Information Extraction from the Web" (PDF). Communications of the ACM. Retrieved March 29, 2018. Zamir, Oren; Etzioni, Oren (1998). "Web document clustering". Proceedings of the 21st annual international ACM SIGIR conference on Research and development in information retrieval. ACM. pp. 46–54. doi:10.1145/290941.290956. ISBN 978-1-58113-015-7. S2CID 244069. Zamir, Oren; Etzioni, Oren (May 1999). "Grouper: a dynamic clustering interface to Web search results". Computer Networks. 31 (11–16): 1361–1374. CiteSeerX 10.1.1.31.8216. doi:10.1016/S1389-1286(99)00054-7. S2CID 206134308. Popescu, Ana-Maria; Etzioni, Oren (2005). "Extracting product features and opinions from reviews". Proceedings of the conference on Human Language Technology and Empirical Methods in Natural Language Processing - HLT '05. pp. 339–346. doi:10.3115/1220575.1220618. Etzioni, Oren; Cafarella, Michael; Downey, Doug; Popescu, Ana-Maria; Shaked, Tal; Sonderland, Stephen; Weld, Daniel; Yates, Alexander (June 2005). "Unsupervised named-entity extraction from the Web: An experimental study". Artificial Intelligence. 165 (1): 91–134. doi:10.1016/j.artint.2005.03.001. Downey, Doug; Etzioni, Oren; Sonderland, Stephen (July 2010). "Grouper: Analysis of a probabilistic model of redundancy in unsupervised information extraction". Artificial Intelligence. 174 (11): 726–748. CiteSeerX 10.1.1.174.2441. doi:10.1016/j.artint.2010.04.024. === Popular articles === Etzioni, Oren (August 4, 2011). "Web Search Needs a Shakeup" (PDF). Nature. Retrieved November 21, 2019. Etzioni, Oren (December 9, 2014). "AI Won't Exterminate Us – It Will Empower Us". Backchannel. Retrieved March 29, 2018. Etzioni, Oren (February 4, 2016). "To Keep AI Safe -- Use AI". Vox. Retrieved November 21, 2019. Etzioni, Oren (April 8, 2016). "Quora Session with Oren Etzioni". Quora. Retrieved March 29, 2018. Etzioni, Oren (June 15, 2016). "Deep Learning Isn't a Dangerous Magic Genie. It's Just Math". Wired. Retrieved March 29, 2018. Etzioni, Oren (September 20, 2016). "No, the Experts Don't Think Superintelligent AI is a Threat to Humanity". MIT Technology Review. Retrieved November 21, 2019. Etzioni, Oren (July 6, 2017). "Artificial intelligence: AI Zooms in on highly influential citations". Nature. Retrieved March 29, 2018. Etzioni, Oren (September 1, 2017). "How to Regulate Artificial Intelligence". The New York Times. Retrieved March 29, 2018. Etzioni, Oren (November 2, 2017). "Workers Displaced by Automation Should Try A New Job: Caregiver". Wired. Retrieved March 29, 2018. Etzioni, Oren (March 14, 2018). "A Hippocratic Oath for artificial intelligence practitioners". Tech Crunch. Retrieved March 29, 2018. Etzioni, Oren (March 7, 2018). "A 'Manhattan Project' for science research". The Hill. Retrieved November 21, 2019. Etzioni, Ore
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Outlook on the web

Outlook on the web (formerly Outlook Web App and Outlook Web Access) is a personal information manager web app from Microsoft. It is a web-based version of Microsoft Outlook, and is included in Exchange Server and Exchange Online (a component of Microsoft 365). It can be freely accessed from any web browser whether inside or outside an organization's network, and includes a web email client, a calendar tool, a contact manager, and a task manager. It also includes add-in integration, Skype on the web, and alerts as well as unified themes that span across all the web apps. == Purpose == Outlook on the web is available to Microsoft 365 (formerly Office 365) and Exchange Online subscribers, and is included with the on-premises Exchange Server, to enable users to connect to their email accounts via a web browser, without requiring the installation of Microsoft Outlook or other email clients. In case of Exchange Server, it is hosted on a local intranet and requires a network connection to the Exchange Server for users to work with e-mail, address book, calendars and task. The Exchange Online version, which can be bought either independently or through Office 365 licensing program, is hosted on Microsoft servers on the World Wide Web. == History == Outlook Web Access was created in 1995 by Microsoft Program Manager Thom McCann on the Exchange Server team. An early working version was demonstrated by Microsoft Vice President Paul Maritz at Microsoft's famous Internet summit in Seattle on December 27, 1995. The first customer version was shipped as part of the Exchange Server 5.0 release in early 1997. The first component to allow client-side scripts to issue HTTP requests (XMLHTTP) was originally written by the Outlook Web Access team. It soon became a part of Internet Explorer 5. Renamed XMLHttpRequest and standardized by the World Wide Web Consortium, it has since become one of the cornerstones of the Ajax technology used to build advanced web apps. Outlook Web Access was later renamed Outlook Web App in 2010. An update on August 4, 2015, renamed OWA to "Outlook on the web", often referred to in brief as simply "Outlook". == Components == === Mail === Mail is the webmail component of Outlook on the web. The default view is a three column view with folders and groups on the left, an email message list in the middle, and the selected message on the right. With the 2015 update, Microsoft introduced the ability to pin, sweep and archive messages, and undo the last action, as well as richer image editing features. It can connect to other services such as GitHub and Twitter through Office 365 Connectors. Actionable Messages in emails allows a user to complete a task from within the email, such as retweeting a Tweet on Twitter or setting a meeting date on a calendar. Outlook on the web supports S/MIME and includes features for managing calendars, contacts, tasks, documents (used with SharePoint or Office Web Apps), and other mailbox content. In the Exchange 2007 release, Outlook on the web (still called Outlook Web App at the time) also offers read-only access to documents stored in SharePoint sites and network UNC shares. === Calendar === Calendar is the calendaring component of Outlook on the web. With the update, Microsoft added a weather forecast directly in the Calendar, as well as icons (or "charms") as visual cues for an event. In addition, email reminders came to all events, and a special Birthday and Holiday event calendars are created automatically. Calendars can be shared and there are multiple views such as day, week, month, and today. Another view is work week which includes Mondays through Fridays in the calendar view. Calendar's "Board View" feature allows for a customizable calendar with widgets such as Goal, Calendar, Tasks and Tips. Calendar details can be added with HTML and rich-text editing, and files can be attached to calendar events and appointments. === People === People is the contact manager component of Outlook on the web. A user can search and edit existing contacts, as well as create new ones. Contacts can be placed into folders and duplicate contacts can be linked from multiple sources such as LinkedIn or Twitter. In Outlook Mail, a contact can be created by clicking on an email address sender, which pulls down a contact card with an add button to add to Outlook People. Contacts can be imported as well as placed into a list that can be utilized when composing an email in Outlook Mail. People can also sync with friends and connections lists on LinkedIn, Facebook, and Twitter. === To Do === To Do was originally launched as Tasks for Outlook Web App. Microsoft was slowly rolling out a preview of Tasks to its consumer-based Outlook.com service that in May 2015, was announced to be moving to the Office 365 infrastructure. It was initially a part of Calendar as a view. Microsoft has separated the services into its own web app in Outlook on the web. In a post on the Office Blogs in 2015, Microsoft announced that Outlook Web App would be renamed Outlook on the web and that Tasks would move under that brand. A user can create tasks, put them into categories, and move them to another folder. A feature added was the ability to set due days and sort and filter the tasks according to those criteria. The app provides the user with fields such as subject, start and end dates, percent complete, priority, and how much work was put into each task. Rich editing features like bold, italic, underline, numbering, and bullet points were also introduced. Tasks can be edited and categorized according to how the user wishes them to be sorted. == Removed features == Outlook on the web has had two interfaces available: one with a complete feature set (known as Premium) and one with reduced functionality (known as Light or sometimes Lite). Prior to Exchange 2010, the Premium client required Internet Explorer. Exchange 2000 and 2003 require Internet Explorer 5 and later, and Exchange 2007 requires Internet Explorer 6 and later. Exchange 2010 supports a wider range of web browsers: Internet Explorer 7 or later, Firefox 3.01 or later, Chrome, or Safari 3.1 or later. However, Exchange 2010 restricts its Firefox and Safari support to macOS and Linux. In Exchange 2013, these browser restrictions were lifted. In Exchange 2010 and earlier, the Light user interface is rendered for browsers other than Internet Explorer. The basic interface did not support search on Exchange Server 2003. In Exchange Server 2007, the Light interface supported searching mail items; managing contacts and the calendar was also improved. The 2010 version can connect to an external email account. The ability to add new accounts to Outlook on the web using the Connected accounts feature was removed in September 2018 and all connected accounts stopped synchronizing email the following month.
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Jacek M. Zurada

Jacek M. Zurada is a Polish-American computer scientist who is a Professor of the Electrical and Computer Engineering Department at the University of Louisville, Kentucky. His M.S. and Ph.D. degrees are from Politechnika Gdaṅska (Gdansk University of Technology, Poland). He has held visiting appointments at the Swiss Federal Institute of Technology, Zurich, Princeton, Northeastern, and Auburn, and at overseas universities in Australia, Chile, China, France, Germany, Hong Kong, Italy, Japan, Poland, Singapore, Spain, and South Africa. He is a life fellow of IEEE and a fellow of the International Neural Networks Society and Doctor Honoris Causa of Czestochowa Institute of Technology, Poland. == Research == Zurada's research covers neural networks, deep learning, data mining with emphasis on data and feature understanding, rule extraction from semantic and visual information, machine learning, decomposition methods for salient feature extraction, and lambda learning rule for neural networks. == Professional and editorial service == Zurada was the editor-in-chief of IEEE Transactions on Neural Networks (1998–2003), an associate editor of IEEE Transactions on Circuits and Systems, Pt. I and Pt. II, Action Editor in Neural Networks (Elsevier) and on the editorial board of the Proceedings of the IEEE. He is an associate editor of Neurocomputing, Schedae Informaticae, the International Journal of Applied Mathematics and Computer Science, and Editor of the Springer Natural Computing, Advances in Intelligent Systems and Computing and Studies in Computational Intelligence Book series or volumes. == Awards and honours == In 2003 he was given the title of Professor by the President of Poland. Since 2005 he has been an elected Foreign Member of the Polish Academy of Sciences. He also received five honorary professorships from foreign universities, including Sichuan University in Chengdu, China, and Obuda University in Budapest, Hungary.
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State complexity

State complexity is an area of theoretical computer science dealing with the size of abstract automata, such as different kinds of finite automata. The classical result in the area is that simulating an n {\displaystyle n} -state nondeterministic finite automaton by a deterministic finite automaton requires exactly 2 n {\displaystyle 2^{n}} states in the worst case. == Transformation between variants of finite automata == Finite automata can be deterministic and nondeterministic, one-way (DFA, NFA) and two-way (2DFA, 2NFA). Other related classes are unambiguous (UFA), self-verifying (SVFA) and alternating (AFA) finite automata. These automata can also be two-way (2UFA, 2SVFA, 2AFA). All these machines can accept exactly the regular languages. However, the size of different types of automata necessary to accept the same language (measured in the number of their states) may be different. For any two types of finite automata, the state complexity tradeoff between them is an integer function f {\displaystyle f} where f ( n ) {\displaystyle f(n)} is the least number of states in automata of the second type sufficient to recognize every language recognized by an n {\displaystyle n} -state automaton of the first type. The following results are known. NFA to DFA: 2 n {\displaystyle 2^{n}} states. This is the subset construction by Rabin and Scott, proved optimal by Lupanov. UFA to DFA: 2 n {\displaystyle 2^{n}} states, see Leung, An earlier lower bound by Schmidt was smaller. NFA to UFA: 2 n − 1 {\displaystyle 2^{n}-1} states, see Leung. There was an earlier smaller lower bound by Schmidt. SVFA to DFA: Θ ( 3 n / 3 ) {\displaystyle \Theta (3^{n/3})} states, see Jirásková and Pighizzini 2DFA to DFA: n ( n n − ( n − 1 ) n ) {\displaystyle n(n^{n}-(n-1)^{n})} states, see Kapoutsis. Earlier construction by Shepherdson used more states, and an earlier lower bound by Moore was smaller. 2DFA to NFA: ( 2 n n + 1 ) = O ( 4 n n ) {\displaystyle {\binom {2n}{n+1}}=O({\frac {4^{n}}{\sqrt {n}}})} , see Kapoutsis. Earlier construction by Birget used more states. 2NFA to NFA: ( 2 n n + 1 ) {\displaystyle {\binom {2n}{n+1}}} , see Kapoutsis. 2NFA to NFA accepting the complement: O ( 4 n ) {\displaystyle O(4^{n})} states, see Vardi. AFA to DFA: 2 2 n {\displaystyle 2^{2^{n}}} states, see Chandra, Kozen and Stockmeyer. AFA to NFA: 2 n {\displaystyle 2^{n}} states, see Fellah, Jürgensen and Yu. 2AFA to DFA: 2 n 2 n {\displaystyle 2^{n2^{n}}} , see Ladner, Lipton and Stockmeyer. 2AFA to NFA: 2 Θ ( n log ⁡ n ) {\displaystyle 2^{\Theta (n\log n)}} , see Geffert and Okhotin. === The 2DFA vs. 2NFA problem and logarithmic space === It is an open problem whether all 2NFAs can be converted to 2DFAs with polynomially many states, i.e. whether there is a polynomial p ( n ) {\displaystyle p(n)} such that for every n {\displaystyle n} -state 2NFA there exists a p ( n ) {\displaystyle p(n)} -state 2DFA. The problem was raised by Sakoda and Sipser, who compared it to the P vs. NP problem in the computational complexity theory. Berman and Lingas discovered a formal relation between this problem and the L vs. NL open problem. This relation was further elaborated by Kapoutsis. == State complexity of operations for finite automata == Given a binary regularity-preserving operation on languages ∘ {\displaystyle \circ } and a family of automata X (DFA, NFA, etc.), the state complexity of ∘ {\displaystyle \circ } is an integer function f ( m , n ) {\displaystyle f(m,n)} such that for each m-state X-automaton A and n-state X-automaton B there is an f ( m , n ) {\displaystyle f(m,n)} -state X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} , and for all integers m, n there is an m-state X-automaton A and an n-state X-automaton B such that every X-automaton for L ( A ) ∘ L ( B ) {\displaystyle L(A)\circ L(B)} must have at least f ( m , n ) {\displaystyle f(m,n)} states. Analogous definition applies for operations with any number of arguments. The first results on state complexity of operations for DFAs were published by Maslov and by Yu, Zhuang and Salomaa. Holzer and Kutrib pioneered the state complexity of operations on NFA. The known results for basic operations are listed below. === Union === If language L 1 {\displaystyle L_{1}} requires m states and language L 2 {\displaystyle L_{2}} requires n states, how many states does L 1 ∪ L 2 {\displaystyle L_{1}\cup L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n + 1 {\displaystyle m+n+1} states, see Holzer and Kutrib. UFA: at least min ( n , m ) Ω ( log ⁡ ( min ( n , m ) ) ) {\displaystyle \min(n,m)^{\Omega (\log(\min(n,m)))}} ; between m n + m + n {\displaystyle mn+m+n} and m + n m 2 0.79 m {\displaystyle m+nm2^{0.79m}} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and 4 m + n + 4 {\displaystyle 4m+n+4} states, see Kunc and Okhotin. 2NFA: m + n {\displaystyle m+n} states, see Kunc and Okhotin. === Intersection === How many states does L 1 ∩ L 2 {\displaystyle L_{1}\cap L_{2}} require? DFA: m n {\displaystyle mn} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m n {\displaystyle mn} states, see Holzer and Kutrib. UFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Šebej. SVFA: m n {\displaystyle mn} states, see Jirásek, Jirásková and Szabari. 2DFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. 2NFA: between m + n {\displaystyle m+n} and m + n + 1 {\displaystyle m+n+1} states, see Kunc and Okhotin. === Complementation === If language L requires n states then how many states does its complement require? DFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. NFA: 2 n {\displaystyle 2^{n}} states, see Birget. or Jirásková UFA: at least n Ω ~ ( log ⁡ n ) {\displaystyle n^{{\tilde {\Omega }}(\log n)}} states, see Göös, Kiefer and Yuan, (this follows an earlier bound by Raskin); and at most n + 1 ⋅ 2 0.5 n {\displaystyle {\sqrt {n+1}}\cdot 2^{0.5n}} states, see Indzhev and Kiefer. SVFA: n {\displaystyle n} states, by exchanging accepting and rejecting states. 2DFA: at least n {\displaystyle n} and at most 4 n {\displaystyle 4n} states, see Geffert, Mereghetti and Pighizzini. === Concatenation === How many states does L 1 L 2 = { w 1 w 2 ∣ w 1 ∈ L 1 , w 2 ∈ L 2 } {\displaystyle L_{1}L_{2}=\{w_{1}w_{2}\mid w_{1}\in L_{1},w_{2}\in L_{2}\}} require? DFA: m ⋅ 2 n − 2 n − 1 {\displaystyle m\cdot 2^{n}-2^{n-1}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: m + n {\displaystyle m+n} states, see Holzer and Kutrib. UFA: 3 4 2 m + n − 1 {\displaystyle {\frac {3}{4}}2^{m+n}-1} states, see Jirásek, Jirásková and Šebej. SVFA: Θ ( 3 n / 3 2 m ) {\displaystyle \Theta (3^{n/3}2^{m})} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 2 Ω ( n ) log ⁡ m {\displaystyle {\frac {2^{\Omega (n)}}{\log m}}} and at most 2 m m + 1 ⋅ 2 n n + 1 {\displaystyle 2m^{m+1}\cdot 2^{n^{n+1}}} states, see Jirásková and Okhotin. === Kleene star === DFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Maslov and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Šebej. SVFA: 3 4 2 n {\displaystyle {\frac {3}{4}}2^{n}} states, see Jirásek, Jirásková and Szabari. 2DFA: at least 1 n 2 n 2 − 1 {\displaystyle {\frac {1}{n}}2^{{\frac {n}{2}}-1}} and at most 2 O ( n n + 1 ) {\displaystyle 2^{O(n^{n+1})}} states, see Jirásková and Okhotin. === Reversal === DFA: 2 n {\displaystyle 2^{n}} states, see Mirkin, Leiss, and Yu, Zhuang and Salomaa. NFA: n + 1 {\displaystyle n+1} states, see Holzer and Kutrib. UFA: n {\displaystyle n} states. SVFA: 2 n + 1 {\displaystyle 2n+1} states, see Jirásek, Jirásková and Szabari. 2DFA: between n + 1 {\displaystyle n+1} and n + 2 {\displaystyle n+2} states, see Jirásková and Okhotin. == Finite automata over a unary alphabet == State complexity of finite automata with a one-letter (unary) alphabet, pioneered by Chrobak, is different from the multi-letter case. Let g ( n ) = e Θ ( n ln ⁡ n ) {\displaystyle g(n)=e^{\Theta ({\sqrt {n\ln n}})}} be Landau's function. === Transformation between models === For a one-letter alphabet, transformations between different types of finite automata are sometimes more efficient than in the general case. NFA to DFA: g ( n ) + O ( n 2 ) {\displaystyle g(n)+O(n^{2})} states, see Chrobak. 2DFA to DFA: g ( n ) + O ( n ) {\displaystyle g(n)+O(n)} states, see Chrobak and Kunc and Okhotin. 2NFA to DFA: O ( g ( n ) ) {\displaystyle O(g(n))} states, see Mereghetti and Pighizzini. and Geffert, Mereghetti and Pighizzini. NFA to 2DFA: at most O ( n 2 ) {\displaystyle O(n^{2})} states, see Chrobak. 2NFA to 2DFA: at most n O ( log ⁡ n ) {\displaystyle n^{O(\log n)}} states, proved by implementing the method of Savitch's theorem, see
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Top 10 AI Marketing Tools Compared (2026)

Comparing the best AI marketing tool? An AI marketing tool is software that uses machine learning to help you get more done — it lowers the barrier so anyone can produce professional output. Privacy matters too: check whether your data trains the model and whether a no-log or enterprise tier is available. Whether you are a beginner or a pro, the right AI marketing tool slots into your workflow and pays for itself fast. We tested the leading options and ranked them by quality, value, and ease of use.
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Closest point method

The closest point method (CPM) is an embedding method for solving partial differential equations on surfaces. The closest point method uses standard numerical approaches such as finite differences, finite element or spectral methods in order to solve the embedding partial differential equation (PDE) which is equal to the original PDE on the surface. The solution is computed in a band surrounding the surface in order to be computationally efficient. In order to extend the data off the surface, the closest point method uses a closest point representation. This representation extends function values to be constant along directions normal to the surface. == Definitions == Closest Point function: Given a surface S , c p ( x ) {\displaystyle {\mathcal {S}},cp(\mathbf {x} )} refers to a (possibly non-unique) point belonging to S {\displaystyle {\mathcal {S}}} , which is closest to x {\displaystyle \mathbf {x} } [SE]. Closest point extension: Let S {\displaystyle {\mathcal {S}}} , be a smooth surface in R d {\displaystyle \mathbb {R} ^{d}} . The closest point extension of a function u : S → R {\displaystyle u:{\mathcal {S}}\rightarrow \mathbb {R} } , to a neighborhood Ω {\displaystyle \Omega } of S {\displaystyle {\mathcal {S}}} , is the function v : Ω → R {\displaystyle v:\Omega \rightarrow \mathbb {R} } , defined by v ( x ) = u ( c p ( x ) ) {\displaystyle v(\mathbf {x} )=u(cp(\mathbf {x} ))} . == Closest point method == Initialization consists of these steps [EW]: If it is not already given, a closest point representation of the surface is constructed. A computational domain is chosen. Typically this is a band around the surface. Replace surface gradients by standard gradients in R 3 {\displaystyle \mathbb {R} ^{3}} . Solution is initialized by extending the initial surface data on to the computational domain using the closest point function. After initialization, alternate between the following two steps: Using the closest point function, extend the solution off the surface to the computational domain. Compute the solution to the embedding PDE on a Cartesian mesh in the computational domain for one time step. == Banding == The surface PDE is extended into R 3 {\displaystyle \mathbb {R} ^{3}} however it is only necessary to solve this new PDE near the surface. Hence, we solve the PDE in a band surrounding the surface for efficient computational purposes. Ω c x : ‖ x − c p ( x ) ‖ 2 ≤ λ {\displaystyle \Omega _{c}{x:\|x-cp(x)\|_{2}\leq \lambda }} where λ {\displaystyle \lambda } is the bandwidth. == Example: Heat equation on a circle == Using initial profile u S ( θ , t ) = sin ⁡ ( θ ) {\displaystyle u_{S}(\theta ,t)=\sin(\theta )} leads to the solution u S ( θ , t ) = exp ⁡ ( − t ) sin ⁡ ( θ ) {\displaystyle u_{S}(\theta ,t)=\exp(-t)\sin(\theta )} for the heat equation. Forward Euler time-stepping is used with relation Δ t = 0.1 Δ x 2 {\displaystyle \Delta t=0.1\Delta x^{2}} and degree-four interpolation polynomials for the interpolations. Second-order centered differences are used for the spatial discretization. The CPM results in the expected second order error in the solution u {\displaystyle u} . == Applications == The closest point method can be applied to various PDEs on surfaces. Reaction–diffusion problems on point clouds [RD], eigenvalue problems [EV], and level set equations [LS] are a few examples.
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Douwe Kiela

Douwe Kiela is a Dutch-American research scientist and entrepreneur working in the field of artificial intelligence with a focus on machine learning and natural language processing. He is a research scientist director at Google DeepMind. He previously co-founded and served as CEO of Contextual AI, an enterprise software company that provides a platform for building grounded AI agents for enterprise knowledge bases. He previously led the research team at Meta AI that introduced the RAG approach in 2020, co-authoring the foundational paper "Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks." Kiela also served as Head of Research at Hugging Face and is an adjunct professor in Symbolic Systems at Stanford University. == Early life and education == Douwe Kiela was born in Amsterdam, Netherlands, in 1986. He earned a Bachelor of Science degree in Liberal Arts and Sciences from Utrecht University, with a double major in Cognitive Artificial Intelligence and Philosophy. He then obtained an MSc in logic (cum laude) from the University of Amsterdam's Institute for Logic, Language and Computation (ILLC). Kiela received an MPhil and PhD in Computer Science from the University of Cambridge, specializing in natural language processing and machine learning. == Career == === Facebook AI Research (Meta) === In 2016, Kiela joined Facebook AI Research (FAIR) as a postdoctoral researcher, later becoming a research scientist in New York. While at Meta, he co-authored papers in natural language processing, with a focus on multimodal and grounded language learning. His projects included creating a virtual assistant bot that could navigate tourists around a city and leading the development of Dynabench, an interactive benchmarking platform released in 2020 that used human feedback to test and improve language models. In 2020, Kiela led the Meta AI research team that introduced Retrieval-Augmented Generation (RAG), co-authoring the influential paper "Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks," alongside Patrick Lewis, Ethan Perez, and other researchers. The RAG framework transformed how large language models access and incorporate external information by allowing them to retrieve relevant context from external knowledge bases at query time, rather than relying solely on pre-trained data. This approach addressed key limitations such as hallucination, outdated information, and lack of source attribution. The RAG technique has since become widely adopted in enterprise AI applications and knowledge-intensive natural language processing tasks. === Hugging Face === After leaving Meta, Kiela served as Head of Research at Hugging Face. === Contextual AI === In 2023, Kiela co-founded Contextual AI with Amanpreet Singh, another former researcher at Facebook AI Research and Hugging Face. The Mountain View-based company develops a platform for building grounded AI agents for enterprises, focusing on applications in technology, semiconductor, logistics, finance, and media sectors. Contextual AI raised $20 million in seed funding in June 2023, led by Bain Capital Ventures. In August 2024, the company completed an $80 million Series A funding round led by Greycroft, with participation from Bezos Expeditions, NVentures (Nvidia), HSBC Ventures, and Snowflake Ventures, among others. In May 2026, Kiela joined Google DeepMind as part of a licensing agreement between Google and Contextual AI under which more than 20 Contextual AI researchers joined DeepMind. Following his departure, Jay Chen became interim CEO of Contextual AI. === Academic roles === Douwe Kiela serves as an adjunct professor in Symbolic Systems at Stanford University. In a 2023 interview with the Stanford Daily, he commented on the development of Alpaca, a low-cost instruction-finetuned model based on Meta's LLaMA, and emphasized the importance of open academic research in large language models.
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AI Essay Writers: Free vs Paid (2026)

Looking for the best AI essay writer? An AI essay writer is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI essay writer slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Top 10 AI Video Editors Compared (2026)

Looking for the best AI video editor? An AI video editor is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI video editor slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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Scientific Working Group – Imaging Technology

The Scientific Working Group on Imaging Technology was convened by the Federal Bureau of Investigation in 1997 to provide guidance to law enforcement agencies and others in the criminal justice system regarding the best practices for photography, videography, and video and image analysis. This group was terminated in 2015. == History == As technology has advanced through the years, law enforcement has needed to stay abreast of emerging technological advances and use these in the investigation of crime. A factor that is considered when new technology is used in these investigations is the determination of whether the use of that new technology will be admissible in court. The judicial system in the United States currently has two standards used in the determination of admissibility of testimony regarding scientific evidence; the Daubert Standard and the Frye Standard. These standards guide the courts in the admissibility of testimony derived from the use of new technologies and scientific techniques. The Federal Bureau of Investigation (FBI), seeking to address possible admissibility issues with such testimony, established Scientific Working Groups starting with the Scientific Working Group on DNA Analysis and Methods (SWGDAM) in 1988. The goal of these groups is to open lines of communication between law enforcement agencies and forensic laboratories around the world while providing guidance on the use of new and innovative technologies and techniques. This guidance can lead to admissibility of evidence and/or testimony, provided proper methods in the collection of evidence and its analysis are employed. In 2009, the National Academy of Sciences released a report entitled, "Strengthening Forensic Science in the United States: A Path Forward." This report addresses many topics including challenges and disparities facing the forensic science community, standardization, certification of practitioners and accreditation of their respective entities, problems related to the interpretation of forensic evidence, the need for research, and the admission of forensic science evidence in litigation. This report mentions the Scientific Working Groups and their role in forensic science. The history of imaging technology (photography) can be said to extend back to the times of Chinese philosopher Mo-Ti (470-390 B.C.) who described the principles behind the precursor to the camera obscura. Since that time, advances in imaging technology include the discovery of chemical photographic processes in the 19th century and the use of electronic imaging technology that includes analog video cameras and digital video and still cameras. By the mid 1990s, it was apparent that technologically advanced camera systems such as these were being adopted for use in the criminal justice system. This led the FBI to convene a meeting of individuals working in the field of forensic imaging from federal, state, local, and foreign law enforcement, and the U.S. military, during the summer of 1997. As a result of this meeting, the Technical Working Group on Imaging Technology was formed from a core group of the meeting’s participants. This group later became the Scientific Working Group on Imaging Technology (SWGIT). Prior to the inception of SWGIT, some law enforcement agencies began adopting digital imaging technology. Due to the lack of guidelines or standards, some of these agencies attempted to replace all their film cameras with substandard digital cameras, only to find that the equipment they had purchased was not capable of accomplishing the mission for which they were intended. At that time only low resolution digital cameras were deemed affordable by some law enforcement agencies. Some of these agencies were forced to rethink their photography procedures and reverted to the use of film cameras or replaced their low-resolution digital cameras with higher quality, more expensive equipment. Also lacking at this early stage was guidance on how to store and archive digital image files. When SWGIT was formed, it was tasked with providing guidance to law enforcement and others in the criminal justice system by releasing documents that describe the best practices and guidelines for the use of imaging technology, to include these concerns and many others. This group was terminated in 2015. == SWGIT Function == During its existence, SWGIT provided information on the appropriate use of various imaging technologies including both established and new. This was accomplished through the release of documents such as the SWGIT Best Practices documents. As changes in technology occurred, these documents were updated. Over the course of its existence, SWGIT collaborated with other Scientific Working Groups to address imaging concerns within their respective disciplines. SWGIT published over 20 documents that dealt specifically with imaging technology. SWGIT also co-published documents with the Scientific Working Group on Digital Evidence (SWGDE) that had a component or components dealing with imaging technology. SWGIT also provided imaging technology guidance and input for documents from the Scientific Working Group on Friction Ridge Analysis, Study and Technology (SWGFAST), the Scientific Working Group for Forensic Document Examination (SWGDOC), and the Scientific Working Group on Shoeprint and Tire Tread Evidence (SWGTREAD). SWGIT assisted the American Society of Crime Lab Directors/Laboratory Accreditation Board (ASCLD/LAB) in the writing of definitions and standards for the accreditation of Digital and Multimedia Evidence sections of crime laboratories. In addition to releasing documents, SWGIT members disseminated best practices for law enforcement professionals where imaging technology was concerned. This was carried out by attending and lecturing at meetings and conferences of various forensic organizations that included: The American Academy of Forensic Sciences (AAFS) The International Association for Identification (IAI) The Law Enforcement and Emergency Services Video Association (LEVA) The American Society of Crime Lab Directors (ASCLD) The SWGIT membership consisted of approximately fifty scientists, photographers, instructors, and managers from more than two dozen federal, state, and local law enforcement agencies, as well as from the academic and research communities. The membership elected its officers from within. SWGIT was composed of the Executive Committee, four standing subcommittees, and ad hoc subcommittees appointed on an as-needed basis. The standing subcommittees were: Image Analysis, Forensic Photography, Video, and Outreach. This group was terminated in 2015. == Legal Proceedings == The following court cases have conducted Daubert v. Merrell Dow Pharm., Inc., 509 U.S. 579 (1993) hearings in which SWGIT best practice documents have been cited as accepted protocol, methodology, and as generally accepted techniques in the forensic community: U. S. v. Rudy Frabizio, U.S. District Court, Boston, MA, 2008 (Image Authentication) U.S. v. Nobumochi Furukawa, U.S. District Court, Minnesota, 2007 (Video Authentication) U.S. v. John Stroman, U.S. District Court, South Carolina, 2007 (Facial Comparison Analysis) State of Texas v. Daniel Day, Tarrant County Texas, 2005 (Camera Identification to Images) U.S. v. Marc Watzman, U.S. District Court, Northern Illinois, 2004 (Video Authentication) U.S. v. McKreith, U.S. District Court, Fort Lauderdale, FL, 2002 (Photo comparison of shirt) == Termination == This group was unfunded by the FBI in 2015.
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Markov partition

A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like. == Motivation == Let ( M , φ ) {\displaystyle (M,\varphi )} be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic representation: a faithful encoding of the points of M {\displaystyle M} by sequences of symbols such that the map φ {\displaystyle \varphi } becomes the shift map. Suppose that M {\displaystyle M} has been divided into a number of pieces E 1 , E 2 , … , E r {\displaystyle E_{1},E_{2},\ldots ,E_{r}} which are thought to be as small and localized, with virtually no overlaps. The behavior of a point x {\displaystyle x} under the iterates of φ {\displaystyle \varphi } can be tracked by recording, for each n {\displaystyle n} , the part E i {\displaystyle E_{i}} which contains φ n ( x ) {\displaystyle \varphi ^{n}(x)} . This results in an infinite sequence on the alphabet { 1 , 2 , … , r } {\displaystyle \{1,2,\ldots ,r\}} which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of M {\displaystyle M} becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system ( M , φ ) {\displaystyle (M,\varphi )} . == Formal definition == A Markov partition is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles { E 1 , E 2 , … , E r } {\displaystyle \{E_{1},E_{2},\ldots ,E_{r}\}} such that For any pair of points x , y ∈ E i {\displaystyle x,y\in E_{i}} , that W s ( x ) ∩ W u ( y ) ∈ E i {\displaystyle W_{s}(x)\cap W_{u}(y)\in E_{i}} Int ⁡ E i ∩ Int ⁡ E j = ∅ {\displaystyle \operatorname {Int} E_{i}\cap \operatorname {Int} E_{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} If x ∈ Int ⁡ E i {\displaystyle x\in \operatorname {Int} E_{i}} and φ ( x ) ∈ Int ⁡ E j {\displaystyle \varphi (x)\in \operatorname {Int} E_{j}} , then φ [ W u ( x ) ∩ E i ] ⊃ W u ( φ x ) ∩ E j {\displaystyle \varphi \left[W_{u}(x)\cap E_{i}\right]\supset W_{u}(\varphi x)\cap E_{j}} φ [ W s ( x ) ∩ E i ] ⊂ W s ( φ x ) ∩ E j {\displaystyle \varphi \left[W_{s}(x)\cap E_{i}\right]\subset W_{s}(\varphi x)\cap E_{j}} Here, W u ( x ) {\displaystyle W_{u}(x)} and W s ( x ) {\displaystyle W_{s}(x)} are the unstable and stable manifolds of x, respectively, and Int ⁡ E i {\displaystyle \operatorname {Int} E_{i}} simply denotes the interior of E i {\displaystyle E_{i}} . These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system. It is this property of the covering that merits the 'Markov' appellation. The resulting dynamics is that of a Markov shift; that this is indeed the case is due to theorems by Yakov Sinai (1968) and Rufus Bowen (1975), thus putting symbolic dynamics on a firm footing. Variants of the definition are found, corresponding to conditions on the geometry of the pieces E i {\displaystyle E_{i}} . == Examples == Markov partitions have been constructed in several situations. Anosov diffeomorphisms of the torus. Dynamical billiards, in which case the covering is countable. Markov partitions make homoclinic and heteroclinic orbits particularly easy to describe. The system ( [ 0 , 1 ) , x ↦ 2 x m o d 1 ) {\displaystyle ([0,1),x\mapsto 2x\ mod\ 1)} has the Markov partition E 0 = ( 0 , 1 / 2 ) , E 1 = ( 1 / 2 , 1 ) {\displaystyle E_{0}=(0,1/2),E_{1}=(1/2,1)} , and in this case the symbolic representation of a real number in [ 0 , 1 ) {\displaystyle [0,1)} is its binary expansion. For example: x ∈ E 0 , T x ∈ E 1 , T 2 x ∈ E 1 , T 3 x ∈ E 1 , T 4 x ∈ E 0 ⇒ x = ( 0.01110... ) 2 {\displaystyle x\in E_{0},Tx\in E_{1},T^{2}x\in E_{1},T^{3}x\in E_{1},T^{4}x\in E_{0}\Rightarrow x=(0.01110...)_{2}} . The assignment of points of [ 0 , 1 ) {\displaystyle [0,1)} to their sequences in the Markov partition is well defined except on the dyadic rationals - morally speaking, this is because ( 0.01111 … ) 2 = ( 0.10000 … ) 2 {\displaystyle (0.01111\dots )_{2}=(0.10000\dots )_{2}} , in the same way as 1 = 0.999 … {\displaystyle 1=0.999\dots } in decimal expansions.
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The Best Free AI Clip Maker for Beginners

Looking for the best AI clip maker? An AI clip maker is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI clip maker slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.
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