AI Analytical Thinking

AI Analytical Thinking — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  KidDesk

  KidDesk is an alternative desktop software application. The early childhood learning company Hatch Early Childhood created KidDesk; it subsequently went to Edmark, which was bought by IBM then sold to Riverdeep (now Houghton Mifflin Harcourt Learning Technology). KidDesk is compatible with Microsoft Windows 95 and newer, as well as Apple System 7 and newer. KidDesk can be set to start when the computer starts up, and can only be exited through password entry. Adults choose what programs are included for the child to use, what icon represented the desk, and customize the software programs available for use. == History == Edmark first started shipping KidDesk in 1992. In 1993, Edmark updated KidDesk with KidDesk Family Edition for Macintosh and DOS, adding more desk accessories and desk styles (Sometimes included as a free exclusive offer with the Early Learning House and Thinkin' Things Series). In 1995, KidDesk Family Edition was enhanced for Windows 95, and released one month after the new operating system shipped. In 1998, Edmark developed KidDesk Internet Safe. The Internet Safe edition was written for Windows 95, Windows 98, and Macintosh (including OS8). In 2008, HMH ported KidDesk Family Edition was to run on Windows Vista and in 2011 version 3.07 of KidDesk Family Edition was released as part of the 'Young Explorer' suite which is fully supported on Windows XP, Windows Vista and Windows 7. == Features == A picture editor incorporated into the desk. Used both in the Adult settings menu and in the desk itself. KidDesk users can edit their user logo with a pixel grid paint program. A calendar incorporated into the desk. This allows the user to set dates that the user finds important, and allows the date to be marked with a picture or text. A password exit feature. For security reasons, the adult can set a password so that KidDesk can only be exited if it is entered. As an extra security measure, the password exit function could only be accessed if the user pressed the ctrl + alt + A keyboard buttons simultaneously. A skin changer with several themes - farm, princess, sports, ocean, etc. These themes can be changed. The e-mail and voicemail features are customizable depending on the KidDesk installation. The ability to add websites that can be accessed on KidDesk, and the ability to block hyperlinks, JavaScript, data entry, etc., on said sites was an added for the 'Internet Safe' edition released in 1998. KidDesk Internet Safe edition is available in Spanish and Brazilian-Portuguese versions. == Reception == KidDesk was given a platinum award at the 1994 Oppenheim Toy Portfolio Awards. The judges praised the program's security features allowing "configur[ation] so that kids never have access to the possibly destructive DOS prompt", and concluded that "[i]f you and your kids share a computer, you need to install Kiddesk immediately!" === Awards === Since 1992, KidDesk has won 15 major awards.

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  Top 10 AI Bug Finders Compared (2026)

  Trying to pick the best AI bug finder? An AI bug finder is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI bug finder slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Paul Christiano

  Paul Christiano is an American researcher in the field of artificial intelligence (AI), with a specific focus on AI alignment, which is the subfield of AI safety research that aims to steer AI systems toward human interests. He serves as the Head of Safety for the Center for AI Standards and Innovation inside NIST. He formerly led the language model alignment team at OpenAI and became founder and head of the non-profit Alignment Research Center (ARC), which works on theoretical AI alignment and evaluations of machine learning models. In 2023, Christiano was named as one of the TIME 100 Most Influential People in AI (TIME100 AI). In September 2023, Christiano was appointed to the UK government's Frontier AI Taskforce advisory board. Before working at the Center for AI Standards and Innovation, he was an initial trustee on Anthropic's Long-Term Benefit Trust. == Education == Christiano attended the Harker School in San Jose, California. He competed on the U.S. team and won a silver medal at the 49th International Mathematics Olympiad (IMO) in 2008. In 2012, Christiano graduated from the Massachusetts Institute of Technology (MIT) with a degree in mathematics. At MIT, he researched data structures, quantum cryptography, and combinatorial optimization. He then went on to complete a PhD at the University of California, Berkeley. While at Berkeley, Christiano collaborated with researcher Katja Grace on AI Impacts, co-developing a preliminary methodology for comparing supercomputers to brains, using traversed edges per second (TEPS). He also experimented with putting Carl Shulman's donor lottery theory into practice, raising nearly $50,000 in a pool to be donated to a single charity. == Career == At OpenAI, Christiano co-authored the paper "Deep Reinforcement Learning from Human Preferences" (2017) and other works developing reinforcement learning from human feedback (RLHF). He is considered one of the principal architects of RLHF, which in 2017 was "considered a notable step forward in AI safety research", according to The New York Times. Other works such as "AI safety via debate" (2018) focus on the problem of scalable oversight – supervising AIs in domains where humans would have difficulty judging output quality. Christiano left OpenAI in 2021 to work on more conceptual and theoretical issues in AI alignment and subsequently founded the Alignment Research Center to focus on this area. One subject of study is the problem of eliciting latent knowledge from advanced machine learning models. ARC also develops techniques to identify and test whether an AI model is potentially dangerous. In April 2023, Christiano told The Economist that ARC was considering developing an industry standard for AI safety. As of April 2024, Christiano was listed as the head of AI safety for the US AI Safety Institute at NIST. One month earlier in March 2024, staff members and scientists at the institute threatened to resign upon being informed of Christiano's pending appointment to the role, stating that his ties to the effective altruism movement may jeopardize the AI Safety Institute's objectivity and integrity. === Views on AI risks === He is known for his views on the potential risks of advanced AI. In 2017, Wired magazine stated that Christiano and his colleagues at OpenAI weren't worried about the destruction of the human race by "evil robots", explaining that "[t]hey’re more concerned that, as AI progresses beyond human comprehension, the technology’s behavior may diverge from our intended goals." However, in a widely quoted interview with Business Insider in 2023, Christiano said that there is a “10–20% chance of AI takeover, [with] many [or] most humans dead.” He also conjectured a “50/50 chance of doom shortly after you have AI systems that are human level.” == Personal life == Christiano is married to Ajeya Cotra, a member of METR's technical staff.

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  How to Choose an AI Logo Maker

  Trying to pick the best AI logo maker? An AI logo maker is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI logo maker slots into your workflow and pays for itself fast. This guide breaks down the top picks, their pros and cons, and who each one is best for.

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  Embodied agent

  In artificial intelligence, an embodied agent, also sometimes referred to as an interface agent, is an intelligent agent that interacts with the environment through a physical body within that environment. Agents that are represented graphically with a body, for example a human or a cartoon animal, are also called embodied agents, although they have only virtual, not physical, embodiment. A branch of artificial intelligence focuses on empowering such agents to interact autonomously with human beings and the environment. Mobile robots are one example of physically embodied agents; Ananova and Microsoft Agent are examples of graphically embodied agents. Embodied conversational agents are embodied agents (usually with a graphical front-end as opposed to a robotic body) that are capable of engaging in conversation with one another and with humans employing the same verbal and nonverbal means that humans do (such as gesture, facial expression, and so forth). == Embodied conversational agents == Embodied conversational agents are a form of intelligent user interface. Graphically embodied agents aim to unite gesture, facial expression and speech to enable face-to-face communication with users, providing a powerful means of human-computer interaction. == Advantages == Face-to-face communication allows communication protocols that give a much richer communication channel than other means of communicating. It enables pragmatic communication acts such as conversational turn-taking, facial expression of emotions, information structure and emphasis, visualization and iconic gestures, and orientation in a three-dimensional environment. This communication takes place through both verbal and non-verbal channels such as gaze, gesture, spoken intonation and body posture. Research has found that users prefer a non-verbal visual indication of an embodied system's internal state to a verbal indication, demonstrating the value of additional non-verbal communication channels. As well as this, the face-to-face communication involved in interacting with an embodied agent can be conducted alongside another task without distracting the human participants, instead improving the enjoyment of such an interaction. Furthermore, the use of an embodied presentation agent results in improved recall of the presented information. Embodied agents also provide a social dimension to the interaction. Humans willingly ascribe social awareness to computers, and thus interaction with embodied agents follows social conventions, similar to human to human interactions. This social interaction both raises the believably and perceived trustworthiness of agents, and increases the user's engagement with the system. Rickenberg and Reeves found that the presence of an embodied agent on a website increased the level of user trust in that website, but also increased users' anxiety and affected their performance, as if they were being watched by a real human. Another effect of the social aspect of agents is that presentations given by an embodied agent are perceived as being more entertaining and less difficult than similar presentations given without an agent. Research shows that perceived enjoyment, followed by perceived usefulness and ease of use, is the major factor influencing user adoption of embodied agents. A study in January 2004 by Byron Reeves at Stanford demonstrated how digital characters could "enhance online experiences" through explaining how virtual characters essentially add a sense of familiarity to the user experience and make it more approachable. This increase in likability in turn helps make the products better, which benefits both the end users and those creating the product. === Applications === The rich style of communication that characterizes human conversation makes conversational interaction with embodied conversational agents ideal for many non-traditional interaction tasks. A familiar application of graphically embodied agents is computer games; embodied agents are ideal for this setting because the richer communication style makes interacting with the agent enjoyable. Embodied conversational agents have also been used in virtual training environments, portable personal navigation guides, interactive fiction and storytelling systems, interactive online characters and automated presenters and commentators. Major virtual assistants like Siri, Amazon Alexa and Google Assistant do not come with any visual embodied representation, which is believed to limit the sense of human presence by users. The U.S. Department of Defense utilizes a software agent called SGT STAR on U.S. Army-run Web sites and Web applications for site navigation, recruitment and propaganda purposes. Sgt. Star is run by the Army Marketing and Research Group, a division operated directly from The Pentagon. Sgt. Star is based upon the ActiveSentry technology developed by Next IT, a Washington-based information technology services company. Other such bots in the Sgt. Star "family" are utilized by the Federal Bureau of Investigation and the Central Intelligence Agency for intelligence gathering purposes.

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  SNNS

  SNNS (Stuttgart Neural Network Simulator) is a neural network simulator originally developed at the University of Stuttgart. While it was originally built for X11 under Unix, there are Windows ports. Its successor JavaNNS never reached the same popularity. == Features == SNNS is written around a simulation kernel to which user written activation functions, learning procedures and output functions can be added. It has support for arbitrary network topologies and the standard release contains support for a number of standard neural network architectures and training algorithms. == Status == There is currently no ongoing active development of SNNS. In July 2008 the license was changed to the GNU LGPL.

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  Anna Korhonen

  Anna-Leena Korhonen is a Finnish computer scientist who works in England as professor of natural language processing at the University of Cambridge, where she is co-director of the Language Technology Lab and the Institute for Technology and Humanity, fellow of the Alan Turing Institute, director of the Centre for Human Inspired Artificial Intelligence, fellow of the European Laboratory for Learning and Intelligent Systems, and a senior research fellow of Churchill College, Cambridge. Her research interests include natural language processing, the applications of natural language processing in health, and the social consequences of AI-based language tools. == Education and career == Korhonen studied linguistics as an undergraduate at the University of Helsinki. After a master's degree in linguistics at the University of Reading, she completed a Ph.D. in computer science at the University of Cambridge. Her 2002 doctoral dissertation, Subcategorization acquisition, was supervised by Ted Briscoe. After postdoctoral research at the University of Pennsylvania and at the National Institute of Informatics in Japan, she returned to Cambridge in 2005 as a senior research associate and Royal Society University Research Fellow. She became a reader in computational linguistics in 2014, professor of natural language processing in 2017, director of the Centre for Human Inspired Artificial Intelligence in 2022, and co-director of the Institute for Technology and Humanity in 2024. == Recognition == Korhonen was named as a Fellow of the Association for Computational Linguistics in 2023, "for significant contributions to lexical acquisition, multilingual and low resource NLP, socially beneficial language applications, and services to the ACL community". She was elected to the Academia Europaea in 2025.

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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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  DevOps toolchain

  A DevOps toolchain is a set or combination of tools that aid in the delivery, development, and management of software applications throughout the systems development life cycle, as coordinated by an organization that uses DevOps practices. Generally, DevOps tools fit into one or more activities, which supports specific DevOps initiatives: Plan, Create, Verify, Package, Release, Configure, Monitor, and Version Control. == Toolchains == In software, a toolchain is the set of programming tools that is used to perform a complex software development task or to create a software product, which is typically another computer program or a set of related programs. In general, the tools forming a toolchain are executed consecutively so the output or resulting environment state of each tool becomes the input or starting environment for the next one, but the term is also used when referring to a set of related tools that are not necessarily executed consecutively. As DevOps is a set of practices that emphasizes the collaboration and communication of both software developers and other information technology (IT) professionals, while automating the process of software delivery and infrastructure changes, its implementation can include the definition of the series of tools used at various stages of the lifecycle; because DevOps is a cultural shift and collaboration between development and operations, there is no one product that can be considered a single DevOps tool. Instead a collection of tools, potentially from a variety of vendors, are used in one or more stages of the lifecycle. == Stages of DevOps == === Plan === Plan consists of two elements: "define" and "plan". This activity refers to the business value and application requirements. Specifically "Plan" activities include: Production metrics, objects and feedback Requirements Business metrics Update release metrics Release plan, timing and business case Security policy and requirement A combination of the IT personnel will be involved in these activities: business application owners, software development, software architects, continual release management, security officers and the organization responsible for managing the production of IT infrastructure. === Create === Create consists of the building, coding, and configuring of the software development process. The specific activities are: Design of the software and configuration Coding including code quality and performance Software build and build performance Release candidate Tools and vendors in this category often overlap with other categories. Because DevOps is about breaking down silos, this is reflective in the activities and product solutions. === Verify === Verify is directly associated with ensuring the quality of the software release; activities designed to ensure code quality is maintained and the highest quality is deployed to production. The main activities in this are: Acceptance testing Regression testing Security and vulnerability analysis Performance Configuration testing Solutions for verify-related activities generally fall under four main categories: Test automation, Static analysis, Test Lab, and Security. === Package === Package refers to the activities involved once the release is ready for deployment, often also referred to as staging or Preproduction / "preprod". This often includes tasks and activities such as: Approval/preapprovals Package configuration Triggered releases Release staging and holding === Release === Release related activities include schedule, orchestration, provisioning and deploying software into production and targeted environment. The specific Release activities include: Release coordination Deploying and promoting applications Fallbacks and recovery Scheduled/timed releases Solutions that cover this aspect of the toolchain include application release automation, deployment automation and release management. === Configure === Configure activities fall under the operation side of DevOps. Once software is deployed, there may be additional IT infrastructure provisioning and configuration activities required. Specific activities including: Infrastructure storage, database and network provisioning and configuring Application provision and configuration. The main types of solutions that facilitate these activities are continuous configuration automation, configuration management, and infrastructure as code tools. === Monitor === Monitoring is an important link in a DevOps toolchain. It allows IT organization to identify specific issues of specific releases and to understand the impact on end-users. A summary of Monitor related activities are: Performance of IT infrastructure End-user response and experience Production metrics and statistics Information from monitoring activities often impacts Plan activities required for changes and for new release cycles. === Version Control === Version Control is an important link in a DevOps toolchain and a component of software configuration management. Version Control is the management of changes to documents, computer programs, large web sites, and other collections of information. A summary of Version Control related activities are: Non-linear development Distributed development Compatibility with existent systems and protocols Toolkit-based design Information from Version Control often supports Release activities required for changes and for new release cycles.

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  Trevor Hastie

  Trevor John Hastie (born 27 June 1953) is an American statistician and computer scientist. He is currently serving as the John A. Overdeck Professor of Mathematical Sciences and Professor of Statistics at Stanford University. Hastie is known for his contributions to applied statistics, especially in the field of machine learning, data mining, and bioinformatics. He has authored several popular books in statistical learning, including The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Hastie has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge. He also contributed to the development of S. == Education and career == Hastie was born on 27 June 1953 in South Africa. He received his B.S. in statistics from the Rhodes University in 1976 and master's degree from University of Cape Town in 1979. Hastie joined the doctoral program at Stanford University in 1980 and received his Ph.D. in 1984 under the supervision of Werner Stuetzle. His dissertation was "Principal Curves and Surfaces". Hastie began his professional career in 1977 with the South African Medical Research Council. After receiving his master's degree in 1979, he spent a year interning at the London School of Hygiene & Tropical Medicine, the Johnson Space Center in Houston, and the Biomath department at Oxford University. After receiving his doctoral degree from Stanford, Hastie returned to South Africa to work with his former employer South African Medical Research Council. He returned to United States in 1986 and joined the AT&T Bell Laboratories in Murray Hill, New Jersey and remained there for nine years. Working with John Chambers, he co-directed the development of the S programming language. He joined Stanford University in 1994 as Associate Professor in Statistics and Biostatistics. He was promoted to full Professor in 1999. During the period 2006–2009, he was the chair of the Department of Statistics at Stanford University. In 2013 he was named the John A. Overdeck Professor of Mathematical Sciences. == Awards and honors == Hastie is a Fellow of the Royal Statistical Society since 1979. He is also an elected Fellow of several professional and scholarly societies, including the Institute of Mathematical Statistics, the American Statistical Association, and the South African Statistical Society. He is a recipient of 'Myrto Lefkopolou Distinguished Lectureship' award of Biostatistics Department at the Harvard School of Public Health. In 2018, he was elected a member of the National Academy of Sciences. In 2019 Hastie became a foreign member of the Royal Netherlands Academy of Arts and Sciences. Hastie was named for the C.R. and Bhargavi Rao Prize in 2025. Hastie and Hui Zou received the 2025 Founders of Statistics prize for their elastic net paper. == Publications == Hastie is a prolific author of scientific works on numerous topics in applied statistics, including statistical learning, data mining, statistical computing, and bioinformatics. He along with his collaborators has authored about 125 scientific articles. Many of Hastie's scientific articles were coauthored by his longtime collaborator, Robert Tibshirani. Hastie has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge. He has coauthored the following books: T. Hastie and R. Tibshirani, Generalized Additive Models, Chapman and Hall, 1990. J. Chambers and T. Hastie, Statistical Models in S, Wadsworth/Brooks Cole, 1991. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning: Prediction, Inference and Data Mining, Second Edition, Springer Verlag, 2009 (available for free from the author's website). G. James, D. Witten, T. Hastie, R. Tibshirani, An Introduction to Statistical Learning with Applications in R, Springer Verlag, 2013 (available for free from the co-author's website). T. Hastie, R. Tibshirani, M. Wainwright, Statistical Learning with Sparsity: the Lasso and Generalizations, CRC Press, 2015 (available for free from the author's website). Bradley Efron; Trevor Hastie (2016). Computer Age Statistical Inference. Cambridge University Press. ISBN 9781107149892.

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  Ann Copestake

  Ann Alicia Copestake is professor of computational linguistics and head of the Department of Computer Science and Technology at the University of Cambridge and a fellow of Wolfson College, Cambridge. == Education == Copestake was educated at the University of Cambridge where she was awarded a Bachelor of Arts degree in Natural Sciences. After two years working for Unilever Research she completed the Cambridge Diploma in Computer Science. She went on to study at the University of Sussex where she was awarded a PhD in 1992 for research on lexical semantics supervised by Gerald Gazdar. == Career and research == Copestake started doing research in Natural language processing and Computational Linguistics at the University of Cambridge in 1985. Since then she has been a visiting researcher at Xerox PARC (1993/4) and the University of Stuttgart (1994/5). From July 1994 to October 2000 she worked at the Center for the Study of Language and Information (CSLI) at Stanford University, as a Senior Researcher. Copestake was appointed a University Lecturer at Cambridge in October 2000. In the UK, her research has been funded by the Engineering and Physical Sciences Research Council (EPSRC) and Arts and Humanities Research Council (AHRC). According to Google Scholar and Scopus her most cited publications include papers on minimal recursion semantics, multiword expressions, polysemy, named-entity recognition and feature structure grammars.

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  Levenshtein automaton

  In computer science, a Levenshtein automaton for a string w and a number n is a finite-state automaton that can recognize the set of all strings whose Levenshtein distance from w is at most n. That is, a string x is in the formal language recognized by the Levenshtein automaton if and only if x can be transformed into w by at most n single-character insertions, deletions, and substitutions. == Applications == Levenshtein automata may be used for spelling correction, by finding words in a given dictionary that are close to a misspelled word. In this application, once a word is identified as being misspelled, its Levenshtein automaton may be constructed, and then applied to all of the words in the dictionary to determine which ones are close to the misspelled word. If the dictionary is stored in compressed form as a trie, the time for this algorithm (after the automaton has been constructed) is proportional to the number of nodes in the trie, significantly faster than using dynamic programming to compute the Levenshtein distance separately for each dictionary word. It is also possible to find words in a regular language, rather than a finite dictionary, that are close to a given target word, by computing the Levenshtein automaton for the word, and then using a Cartesian product construction to combine it with an automaton for the regular language, giving an automaton for the intersection language. Alternatively, rather than using the product construction, both the Levenshtein automaton and the automaton for the given regular language may be traversed simultaneously using a backtracking algorithm. Levenshtein automata are used in Lucene for full-text searches that can return relevant documents even if the query is misspelled. == Construction == For any fixed constant n, the Levenshtein automaton for w and n may be constructed in time O(|w|). Mitankin studies a variant of this construction called the universal Levenshtein automaton, determined only by a numeric parameter n, that can recognize pairs of words (encoded in a certain way by bitvectors) that are within Levenshtein distance n of each other. Touzet proposed an effective algorithm to build this automaton. Yet a third finite automaton construction of Levenshtein (or Damerau–Levenshtein) distance are the Levenshtein transducers of Hassan et al., who show finite state transducers implementing edit distance one, then compose these to implement edit distances up to some constant.

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  Splitwise

  Splitwise is an online expense-splitting application software accessible via web browser and mobile app. The app facilitates repayments of shared bills by calculating what each person in a group owes. The primary competitor to the app is Venmo, which only operates in the U.S. Splitwise allows users to create groups with friends to determine what each person owes. All expenses and allocations are added to the app, and Splitwise simplifies the transaction history to determine exactly what payments need to be made to whom to settle outstanding balances. Splitwise stores user information via cloud storage. It was developed and is owned by Splitwise Inc., based in Providence, Rhode Island, United States. == History == The app was launched in February 2011 as SplitTheRent, intended to be used for rent splitting, by Ryan Laughlin, Jon Bittner and Marshall Weir. In September 2013, Splitwise was integrated with Venmo to allow users to settle payments via Venmo. In April 2024, Splitwise partnered with Tink, a Visa payment services company, to incorporate a bank transfer feature directly in the Splitwise app. === Financing === In December 2014, the company raised $1.4 million. In October 2016, the company raised $5 million. In April 2021, Splitwise raised $20 million in funding from series A round run by Insight Partners. == Reception == A 2022 opinion piece in The Guardian by London journalist Imogen West-Knights shared the negative effects of exactly splitting bills among friends and family members. West-Knights argued that Splitwise and similar apps can "turn people into those true enemies of all that is fun and joyful in the world: accountants." However, she said the app does work better when used by couples rather than friend groups. Other reviews noted that the app makes people petty. In contrast, an article published by Condé Nast Traveler describes how Splitwise eliminated stress caused by complicated offline bill splitting, saying it "fixed such a pervasive obstacle in group travel." Coverage by The Wall Street Journal lands somewhere in between the two contrasting views, saying Splitwise and similar apps are helpful, but users need to be prepared for difficult money-related conversations that may arise. An etiquette advisor at Debrett's, said, "The less talk you can have about money on any of these occasions, the better." An editor suggested conversations as simple as asking, "We’re splitting this evenly, right?" before a meal.

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  AI Text-to-image Tools: Free vs Paid (2026)

  Shopping for the best AI text-to-image tool? An AI text-to-image tool is software that uses machine learning to help you get more done — it keeps getting smarter as the underlying models improve. Pricing, accuracy, and the size of the model behind the tool are the three factors that most affect daily usefulness. Whether you are a beginner or a pro, the right AI text-to-image 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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  Markov property

  In probability theory and statistics, the Markov property is the memoryless property of a stochastic process, which means that its future evolution is independent of its history. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time. The term Markov assumption is used to describe a model where the Markov property is assumed to hold, such as a hidden Markov model. A Markov random field extends this property to two or more dimensions or to random variables defined for an interconnected network of items. An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the Markov property is known as a Markov chain. == Introduction == A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past. A process with this property is said to be Markov or Markovian and known as a Markov process. Two famous classes of Markov process are the Markov chain and Brownian motion. Note that there is a subtle, often overlooked and very important point that is often missed in the plain English statement of the definition: the statespace of the process is constant through time. The conditional description involves a fixed "bandwidth". For example, without this restriction we could augment any process to one which includes the complete history from a given initial condition and it would be made to be Markovian. But the state space would be of increasing dimensionality over time and does not meet the definition. == History == == Definition == Let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space with a filtration ( F s , s ∈ I ) {\displaystyle ({\mathcal {F}}_{s},\ s\in I)} , for some (totally ordered) index set I {\displaystyle I} ; and let ( S , Σ ) {\displaystyle (S,\Sigma )} be a measurable space. An ( S , Σ ) {\displaystyle (S,\Sigma )} -valued stochastic process X = { X t : Ω → S } t ∈ I {\displaystyle X=\{X_{t}:\Omega \to S\}_{t\in I}} adapted to the filtration is said to possess the Markov property if, for each A ∈ Σ {\displaystyle A\in \Sigma } and each s , t ∈ I {\displaystyle s,t\in I} with s < t {\displaystyle s Read more →