The ISLRN or International Standard Language Resource Number is Persistent Unique Identifier for Language Resources. == Context == On November 18, 2013, 12 major organisations (see list below) from the fields Language Resources and Technologies, Computational Linguistics, and Digital Humanities held a cooperation meeting in Paris (France) and agreed to announce the establishment of the International Standard Language Resource Number (ISLRN), to be assigned to each Language Resource. Among the 12 organisations, 4 institutions constitute the ISLRN Steering Committee (ST) ADHO ACL Asian Federation of Natural Language Processing ST COCOSDA, International Committee for the Coordination & Standardisation of Speech Databases and Assessment Techniques ICCL (COLING) European Data Forum ELRA ST IAMT, International Association for Machine Translation Archived 2010-06-24 at the Wayback Machine ISCA LDC ST Oriental COCOSDA ST RMA, Language Resource Management Agency == Size and Content == The Joint Research Centre(JRC), the [European Commission]'s in-house science service, was the first organisation to adopt the ISLRN initiative and requested. 2500 resources and tools have already been allocated an ISLRN. These resources include written data (Annotated corpus, Annotated text, List of misspelled word, Terminological database, Treebank, Wordnet, etc.) and speech corpora (Synthesised Speech, Transcripts and Audiovisual Recordings, Conversational Speech, Folk Sayings, etc.) == Objectives == Providing Language Resources with unique names and identifiers using a standardized nomenclature ensures the identification of each Language Resources and streamlines the citation with proper references in activities within Human Language Technology as well as in documents and scientific publications. Such unique identifier also enhances the reproducibility, an essential feature of scientific work.
Augmented Analytics
Augmented Analytics is an approach of data analytics that employs the use of machine learning and natural language processing to automate analysis processes normally done by a specialist or data scientist. The term was introduced in 2017 by Rita Sallam, Cindi Howson, and Carlie Idoine in a Gartner research paper. Augmented analytics is based on business intelligence and analytics. In the graph extraction step, data from different sources are investigated. == Defining Augmented Analytics == Machine Learning – a systematic computing method that uses algorithms to sift through data to identify relationships, trends, and patterns. It is a process that allows algorithms to dynamically learn from data instead of having a set base of programmed rules. Natural language generation (NLG) – a software capability that takes unstructured data and translates it into plain-English, readable, language. Automating Insights – using machine learning algorithms to automate data analysis processes. Natural Language Query – enabling users to query data using business terms that are either typed onto a search box or spoken. == Data Democratization == Data Democratization is the democratizing data access in order to relieve data congestion and get rid of any sense of data "gatekeepers". This process must be implemented alongside a method for users to make sense of the data. This process is used in hopes of speeding up company decision making and uncovering opportunities hidden in data. There are three aspects to democratising data: Data Parameterisation and Characterisation. Data Decentralisation using an OS of blockchain and DLT technologies, as well as an independently governed secure data exchange to enable trust. Consent Market-driven Data Monetisation. When it comes to connecting assets, there are two features that will accelerate the adoption and usage of data democratisation: decentralized identity management and business data object monetization of data ownership. It enables multiple individuals and organizations to identify, authenticate, and authorize participants and organizations, enabling them to access services, data or systems across multiple networks, organizations, environments, and use cases. It empowers users and enables a personalized, self-service digital onboarding system so that users can self-authenticate without relying on a central administration function to process their information. Simultaneously, decentralized identity management ensures the user is authorized to perform actions subject to the system’s policies based on their attributes (role, department, organization, etc.) and/ or physical location. == Use cases == Agriculture – Farmers collect data on water use, soil temperature, moisture content and crop growth, augmented analytics can be used to make sense of this data and possibly identify insights that the user can then use to make business decisions. Smart Cities – Many cities across the United States, known as Smart Cities collect large amounts of data on a daily basis. Augmented analytics can be used to simplify this data in order to increase effectiveness in city management (transportation, natural disasters, etc.). Analytic Dashboards – Augmented analytics has the ability to take large data sets and create highly interactive and informative analytical dashboards that assist in many organizational decisions. Augmented Data Discovery – Using an augmented analytics process can assist organizations in automatically finding, visualizing and narrating potentially important data correlations and trends. Data Preparation – Augmented analytics platforms have the ability to take large amounts of data and organize and "clean" the data in order for it to be usable for future analyses. Business – Businesses collect large amounts of data, daily. Some examples of types of data collected in business operations include; sales data, consumer behavior data, distribution data. An augmented analytics platform provides access to analysis of this data, which could be used in making business decisions.
Thompson's construction
In computer science, Thompson's construction algorithm, also called the McNaughton–Yamada–Thompson algorithm, is a method of transforming a regular expression into an equivalent nondeterministic finite automaton (NFA). This NFA can be used to match strings against the regular expression. This algorithm is credited to Ken Thompson. Regular expressions and nondeterministic finite automata are two representations of formal languages. For instance, text processing utilities use regular expressions to describe advanced search patterns, but NFAs are better suited for execution on a computer. Hence, this algorithm is of practical interest, since it can compile regular expressions into NFAs. From a theoretical point of view, this algorithm is a part of the proof that they both accept exactly the same languages, that is, the regular languages. An NFA can be made deterministic by the powerset construction and then be minimized to get an optimal automaton corresponding to the given regular expression. However, an NFA may also be interpreted directly. To decide whether two given regular expressions describe the same language, each can be converted into an equivalent minimal deterministic finite automaton via Thompson's construction, powerset construction, and DFA minimization. If, and only if, the resulting automata agree up to renaming of states, the regular expressions' languages agree. == The algorithm == The algorithm works recursively by splitting an expression into its constituent subexpressions, from which the NFA will be constructed using a set of rules. More precisely, from a regular expression E, the obtained automaton A with the transition function Δ respects the following properties: A has exactly one initial state q0, which is not accessible from any other state. That is, for any state q and any letter a, Δ ( q , a ) {\displaystyle \Delta (q,a)} does not contain q0. A has exactly one final state qf, which is not co-accessible from any other state. That is, for any letter a, Δ ( q f , a ) = ∅ {\displaystyle \Delta (q_{f},a)=\emptyset } . Let c be the number of concatenation of the regular expression E and let s be the number of symbols apart from parentheses — that is, |, , a and ε. Then, the number of states of A is 2s − c (linear in the size of E). The number of transitions leaving any state is at most two. Since an NFA of m states and at most e transitions from each state can match a string of length n in time O(emn), a Thompson NFA can do pattern matching in linear time, assuming a fixed-size alphabet. === Rules === The following rules are depicted according to Aho et al. (2007), p. 122. In what follows, N(s) and N(t) are the NFA of the subexpressions s and t, respectively. The empty-expression ε is converted to A symbol a of the input alphabet is converted to The union expression s|t is converted to State q goes via ε either to the initial state of N(s) or N(t). Their final states become intermediate states of the whole NFA and merge via two ε-transitions into the final state of the NFA. The concatenation expression st is converted to The initial state of N(s) is the initial state of the whole NFA. The final state of N(s) becomes the initial state of N(t). The final state of N(t) is the final state of the whole NFA. The Kleene star expression s is converted to An ε-transition connects initial and final state of the NFA with the sub-NFA N(s) in between. Another ε-transition from the inner final to the inner initial state of N(s) allows for repetition of expression s according to the star operator. The parenthesized expression (s) is converted to N(s) itself. With these rules, using the empty expression and symbol rules as base cases, it is possible to prove with structural induction that any regular expression may be converted into an equivalent NFA. == Example == Two examples are now given, a small informal one with the result, and a bigger with a step by step application of the algorithm. === Small Example === The picture below shows the result of Thompson's construction on (ε|ab). The purple oval corresponds to a, the teal oval corresponds to a, the green oval corresponds to b, the orange oval corresponds to ab, and the blue oval corresponds to ε. === Application of the algorithm === As an example, the picture shows the result of Thompson's construction algorithm on the regular expression (0|(1(01(00)0)1)) that denotes the set of binary numbers that are multiples of 3: { ε, "0", "00", "11", "000", "011", "110", "0000", "0011", "0110", "1001", "1100", "1111", "00000", ... }. The upper right part shows the logical structure (syntax tree) of the expression, with "." denoting concatenation (assumed to have variable arity); subexpressions are named a-q for reference purposes. The left part shows the nondeterministic finite automaton resulting from Thompson's algorithm, with the entry and exit state of each subexpression colored in magenta and cyan, respectively. An ε as transition label is omitted for clarity — unlabelled transitions are in fact ε transitions. The entry and exit state corresponding to the root expression q is the start and accept state of the automaton, respectively. The algorithm's steps are as follows: An equivalent minimal deterministic automaton is shown below. == Relation to other algorithms == Thompson's is one of several algorithms for constructing NFAs from regular expressions; an earlier algorithm was given by McNaughton and Yamada. Converse to Thompson's construction, Kleene's algorithm transforms a finite automaton into a regular expression. Glushkov's construction algorithm is similar to Thompson's construction, once the ε-transitions are removed. == Use in string pattern matching == Regular expressions are often used to specify patterns that software is then asked to match. Generating an NFA by Thompson's construction, and using an appropriate algorithm to simulate it, it is possible to create pattern-matching software with performance that is O ( m n ) {\displaystyle O(mn)} , where m is the length of the regular expression and n is the length of the string being matched. This is much better than is achieved by many popular programming-language implementations; however, it is restricted to purely regular expressions and does not support patterns for non-regular languages like backreferences.
Michael I. Jordan
Michael Irwin Jordan (born February 25, 1956) is an American scientist, professor at the University of California, Berkeley, research scientist at the Inria Paris, and researcher in machine learning, statistics, and artificial intelligence. Jordan was elected a member of the National Academy of Engineering in 2010 for contributions to the foundations and applications of machine learning. He is one of the leading figures in machine learning, and in 2016 Science reported him as the world's most influential computer scientist. In 2022, Jordan won the inaugural World Laureates Association Prize in Computer Science or Mathematics, "for fundamental contributions to the foundations of machine learning and its application." == Education == Jordan received a Bachelor of Science magna cum laude in psychology from the Louisiana State University in 1978, a Master of Science in mathematics from Arizona State University in 1980, and a Doctor of Philosophy in cognitive science from the University of California, San Diego in 1985. At UC San Diego, Jordan was a student of David Rumelhart and a member of the Parallel Distributed Processing (PDP) Group in the 1980s. == Career and research == Jordan is the Pehong Chen Distinguished Professor at the University of California, Berkeley, where his appointment is split across EECS and Statistics. He was a professor at the Department of Brain and Cognitive Sciences at MIT from 1988 to 1998. In the 1980s Jordan started developing recurrent neural networks as a cognitive model. In recent years, his work is less driven from a cognitive perspective and more from the background of traditional statistics. Jordan popularised Bayesian networks in the machine learning community and is known for pointing out links between machine learning and statistics. He was also prominent in the formalisation of variational methods for approximate inference and the popularisation of the expectation–maximization algorithm in machine learning. === Resignation from Machine Learning === In 2001, Jordan and others resigned from the editorial board of the journal Machine Learning. In a public letter, they argued for less restrictive access and pledged support for a new open access journal, the Journal of Machine Learning Research, which was created by Leslie Kaelbling to support the evolution of the field of machine learning. === Honors and awards === Jordan has received numerous awards, including a best student paper award (with X. Nguyen and M. Wainwright) at the International Conference on Machine Learning (ICML 2004), a best paper award (with R. Jacobs) at the American Control Conference (ACC 1991), the ACM-AAAI Allen Newell Award, the IEEE Neural Networks Pioneer Award, and an NSF Presidential Young Investigator Award. In 2002 he was named an AAAI Fellow "for significant contributions to reasoning under uncertainty, machine learning, and human motor control." In 2004 he was named an IMS Fellow "for contributions to graphical models and machine learning." In 2005 he was named an IEEE Fellow "for contributions to probabilistic graphical models and neural information processing systems." In 2007 he was named an ASA Fellow. In 2010 he was named a Cognitive Science Society Fellow and named an ACM Fellow "for contributions to the theory and application of machine learning." In 2012 he was named a SIAM Fellow "for contributions to machine learning, in particular variational approaches to statistical inference." In 2014 he was named an International Society for Bayesian Analysis Fellow "for his outstanding research contributions at the interface of statistics, computer sciences and probability, for his leading role in promoting Bayesian methods in machine learning, engineering and other fields, and for his extensive service to ISBA in many roles." Jordan is a member of the National Academy of Sciences, a member of the National Academy of Engineering and a member of the American Academy of Arts and Sciences. He has been named a Neyman Lecturer and a Medallion Lecturer by the Institute of Mathematical Statistics. He received the David E. Rumelhart Prize in 2015 and the ACM/AAAI Allen Newell Award in 2009. He also won the 2020 IEEE John von Neumann Medal. In 2016, Jordan was identified as the "most influential computer scientist", based on an analysis of the published literature by the Semantic Scholar project. In 2019, Jordan argued that the artificial intelligence revolution hasn't happened yet and that the AI revolution required a blending of computer science with statistics. In 2022, Jordan was awarded the inaugural World Laureates Association Prize by non-governmental and non-profit international organization World Laureates Association, for fundamental contributions to the foundations of machine learning and its application. For 2024 he received the BBVA Foundation Frontiers of Knowledge Award in the category of "Information and Communication Technologies".
AI Video Generators: Free vs Paid (2026)
In search of the best AI video generator? An AI video generator is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI video generator slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
Wix.com
Wix.com Ltd. (Hebrew: וויקס.קום, romanized: wix.com) or simply Wix is an Israeli software company, publicly listed in the US, that provides cloud-based web development services. It offers tools for creating HTML5 websites for desktop and mobile platforms using online drag-and-drop editing. Along with its headquarters and other offices in Israel, Wix also has offices in Brazil, Canada, Germany, India, Ireland, Japan, Lithuania, Poland, the Netherlands, the United States, Ukraine, and Singapore. Users can add applications for social media, e-commerce, online marketing, contact forms, e-mail marketing, and community forums to their websites. The Wix website builder is built on a freemium business model, earning its revenues through premium upgrades. According to the W3Techs technology survey website, Wix was used by 2.5% of websites as of September 2023; at the end of May 2025, it was 3.8%. == History == === Corporate affairs === Wix was founded in 2006 by Israeli developers Avishai Abrahami, Nadav Abrahami, and Giora Kaplan. With its main offices in Tel Aviv, Wix was backed by investors Insight Venture Partners, Mangrove Capital Partners, Bessemer Venture Partners, DAG Ventures, and Benchmark Capital. By April 2010, Wix had 3.5 million users and raised US$10 million in Series C funding provided by Benchmark Capital and existing investors Bessemer Venture Partners and Mangrove Capital Partners. In March 2011, Wix had 8.5 million users and raised US$40 million in Series D funding, bringing its total funding to that date to US$61 million. By August 2013, the Wix platform had more than 34 million registered users. On 5 November 2013, Wix had an initial public offering on NASDAQ, raising about US$127 million for the company and some share holders. In 2016, Mark Tluszcz became the chair of the board of directors. In 2020, Wix's revenue increased to $989 million, a 30% rise year-on-year, primarily due to the shift of businesses online during the coronavirus pandemic. The company added over 31 million new registered users in 2020, reaching a total of 196.7 million by year's end. Wix added approximately 1 million net new premium subscriptions in 2020, surpassing $1 billion in annual collections for the first time. By the end of the year, there were 5.5 million premium subscriptions, a 22% increase compared to the end of 2019. As of its most recent reporting in June 2024, Wix has over 260 million users worldwide. === Product development === ==== 2000s ==== Wix entered an open beta phase in 2007 using a platform based on Adobe Flash. ==== 2010s ==== In June 2011, Wix launched the Facebook store module, making its first step into social commerce. In March 2012, Wix launched a new HTML5 site builder, replacing the Adobe Flash technology. In October 2012, Wix launched an app market for users to sell applications built with the company's automated web development technology. In August 2014, Wix launched Wix Hotels, a booking system for hotels, bed and breakfasts, and vacation rentals that use Wix websites. In June 2016, Wix introduced Wix ADI (Artificial Design Intelligence), a platform that uses artificial intelligence to design websites. ==== 2020s ==== In 2020, Wix launched an additional CMS, EditorX, which included additional CSS features to the original builder. In July 2023, Wix announced that it would be building on its ADI technology to create an AI powered website generator In October 2023, Wix launched the Wix Studio website builder. Co-founder and CEO, Avishai Abrahami described the platform as a “product for agencies”. In March 2024, the AI web builder, which uses a chatbot to help users create content was launched to the public. In March 2025, the digital publisher CNET has identified Wix as the "Best overall website builder overall." In August 2025, Wix announced it would launch banking services—including checking accounts and loans for small businesses—via a partnership with Israeli fintech Unit Finance, as it sought to diversify amid what it described as threats to its core website-building business from artificial intelligence. In January 2026, Wix launched Wix Harmony. Wix harmony is an AI website builder that uses agentic technology, generative design and vibe coding—with manual editing features for additional control. In May 2026, Wix announced layoffs affecting approximately 1,000 employees, or 20% of its workforce. CEO Avishai Abrahami cited two factors: the need to restructure around artificial intelligence and the appreciation of the Israeli shekel against the US dollar, which increased the cost of its Israel-based workforce relative to its dollar-denominated revenue. === Acquisitions === In April 2014, Wix announced the acquisition of Appixia, an Israeli startup for creating native mobile commerce (mCommerce) apps. In October 2014, Wix announced its acquisition of OpenRest, a developer of online ordering systems for restaurants. In April 2015, Wix acquired Moment.me, a mobile website builder for events and marketing tools for social lead generation. On 23 February 2017, Wix acquired the online art community DeviantArt for US$36 million. In January 2017, the company acquired Flok, a provider of customer loyalty programs tools. In February 2020, Wix acquired Inkfrog for eBay sellers, a web design company that provides customized business management software for eBay sellers. On 2 March 2021, Wix acquired SpeedETab, a Miami-based restaurant online technology provider. In May 2021, Wix acquired Rise.ai, a gift card and customer re-engagement package for online brands. A month later, Wix acquired Modalyst, a marketplace and drop-shipping platform. In May 2025, Wix acquired Hour One, a startup specializing in AI-powered video creation tools, to enhance its generative AI capabilities. In June 2025, the company acquired Base44, owned by independent entrepreneur Maor Shlomo, with the intention of integrating Base44's artificial intelligence capabilities and conversational interface into Wix's website and app building platform. == Description == Wix uses a freemium business model. Users can create websites for free then must purchase premium packages to connect their sites to their own domains, remove Wix ads, access the form builder, add e-commerce capabilities, or buy extra data storage and bandwidth. Wix provides customizable website templates and a drag-and-drop HTML5 website builder that includes apps, graphics, image galleries, fonts, vectors, animations, and other options. Users also may opt to create their web sites from scratch. In October 2013, Wix introduced a mobile editor for mobile viewing customization. Wix App Market offers both free and subscription-based applications, with a revenue split of 80% for the developer and 20% for Wix. Customers can integrate third-party applications into their own web sites, such as photograph feeds, blogging, music playlists, online community, e-mail marketing, and file management. Custom JavaScript code can be inserted into Wix webpages using the Velo API. == Controversies == === Use of WordPress code === In October 2016, there was a controversy over Wix's use of WordPress's GPL-licensed code. In response, Avishai Abrahami, Wix's CEO, published a response describing which open-source code was used and how Wix says it collaborates with the open-source community. However, it was subsequently noted that collaboration with the open-source community was not sufficient under the terms of the GPL license, which requires any code built on GPL-licensed code to be released under the same license. === Censorship === On 31 May 2021, 2021 Hong Kong Charter, a Wix-hosted website run by exiled Hong Kong activists, was shut down at the request of the Hong Kong Police. This was the first known case of Hong Kong's National Security Law being used to censor content on an overseas website. Wix later apologized for "mistakenly removing the website" and reinstated the website after it had been down for four days. In October 2023, Wix fired an employee in Dublin, Ireland, for having made social media posts critical of Israel. This incident led to criticism of Wix from members of the Oireachtas (the Irish parliament) and from the head of the Irish government, Taoiseach Leo Varadkar, who said it was "not okay to dismiss somebody because of their political views". Deputy head of government, Tánaiste Micheál Martin, also condemned their dismissal, stating "we tolerate debate with freedom of speech, freedom of opinion, and people have different opinions on these issues." The dismissed employee, Courtney Carey, successfully sued the company for unfair dismissal. Wix did not contest the charge, admitting liability. === Outreach abroad === In October 2023, The Irish Times reported that an Israeli advertising agency advised Wix staff how they can tailor posts for "outreach abroad". This included advice for Wix employees to “show Westernity” in social media posts supporting Israel, stating that “unlike the Gazans,
Models of DNA evolution
A number of different Markov models of DNA sequence evolution have been proposed. These substitution models differ in terms of the parameters used to describe the rates at which one nucleotide replaces another during evolution. These models are frequently used in molecular phylogenetic analyses. In particular, they are used during the calculation of likelihood of a tree (in Bayesian and maximum likelihood approaches to tree estimation) and they are used to estimate the evolutionary distance between sequences from the observed differences between the sequences. == Introduction == These models are phenomenological descriptions of the evolution of DNA as a string of four discrete states. These Markov models do not explicitly depict the mechanism of mutation nor the action of natural selection. Rather they describe the relative rates of different changes. For example, mutational biases and purifying selection favoring conservative changes are probably both responsible for the relatively high rate of transitions compared to transversions in evolving sequences. However, the Kimura (K80) model described below only attempts to capture the effect of both forces in a parameter that reflects the relative rate of transitions to transversions. Evolutionary analyses of sequences are conducted on a wide variety of time scales. Thus, it is convenient to express these models in terms of the instantaneous rates of change between different states (the Q matrices below). If we are given a starting (ancestral) state at one position, the model's Q matrix and a branch length expressing the expected number of changes to have occurred since the ancestor, then we can derive the probability of the descendant sequence having each of the four states. The mathematical details of this transformation from rate-matrix to probability matrix are described in the mathematics of substitution models section of the substitution model page. By expressing models in terms of the instantaneous rates of change we can avoid estimating a large numbers of parameters for each branch on a phylogenetic tree (or each comparison if the analysis involves many pairwise sequence comparisons). The models described on this page describe the evolution of a single site within a set of sequences. They are often used for analyzing the evolution of an entire locus by making the simplifying assumption that different sites evolve independently and are identically distributed. This assumption may be justifiable if the sites can be assumed to be evolving neutrally. If the primary effect of natural selection on the evolution of the sequences is to constrain some sites, then models of among-site rate-heterogeneity can be used. This approach allows one to estimate only one matrix of relative rates of substitution, and another set of parameters describing the variance in the total rate of substitution across sites. == DNA evolution as a continuous-time Markov chain == === Continuous-time Markov chains === Continuous-time Markov chains have the usual transition matrices which are, in addition, parameterized by time, t {\displaystyle t} . Specifically, if E 1 , E 2 , E 3 , E 4 {\displaystyle E_{1},E_{2},E_{3},E_{4}} are the states, then the transition matrix P ( t ) = ( P i j ( t ) ) {\displaystyle P(t)={\big (}P_{ij}(t){\big )}} where each individual entry, P i j ( t ) {\displaystyle P_{ij}(t)} refers to the probability that state E i {\displaystyle E_{i}} will change to state E j {\displaystyle E_{j}} in time t {\displaystyle t} . Example: We would like to model the substitution process in DNA sequences (i.e. Jukes–Cantor, Kimura, etc.) in a continuous-time fashion. The corresponding transition matrices will look like: P ( t ) = ( p A A ( t ) p A G ( t ) p A C ( t ) p A T ( t ) p G A ( t ) p G G ( t ) p G C ( t ) p G T ( t ) p C A ( t ) p C G ( t ) p C C ( t ) p C T ( t ) p T A ( t ) p T G ( t ) p T C ( t ) p T T ( t ) ) {\displaystyle P(t)={\begin{pmatrix}p_{\mathrm {AA} }(t)&p_{\mathrm {AG} }(t)&p_{\mathrm {AC} }(t)&p_{\mathrm {AT} }(t)\\p_{\mathrm {GA} }(t)&p_{\mathrm {GG} }(t)&p_{\mathrm {GC} }(t)&p_{\mathrm {GT} }(t)\\p_{\mathrm {CA} }(t)&p_{\mathrm {CG} }(t)&p_{\mathrm {CC} }(t)&p_{\mathrm {CT} }(t)\\p_{\mathrm {TA} }(t)&p_{\mathrm {TG} }(t)&p_{\mathrm {TC} }(t)&p_{\mathrm {TT} }(t)\end{pmatrix}}} where the top-left and bottom-right 2 × 2 blocks correspond to transition probabilities and the top-right and bottom-left 2 × 2 blocks corresponds to transversion probabilities. Assumption: If at some time t 0 {\displaystyle t_{0}} , the Markov chain is in state E i {\displaystyle E_{i}} , then the probability that at time t 0 + t {\displaystyle t_{0}+t} , it will be in state E j {\displaystyle E_{j}} depends only upon i {\displaystyle i} , j {\displaystyle j} and t {\displaystyle t} . This then allows us to write that probability as p i j ( t ) {\displaystyle p_{ij}(t)} . Theorem: Continuous-time transition matrices satisfy: P ( t + τ ) = P ( t ) P ( τ ) {\displaystyle P(t+\tau )=P(t)P(\tau )} Note: There is here a possible confusion between two meanings of the word transition. (i) In the context of Markov chains, transition is the general term for the change between two states. (ii) In the context of nucleotide changes in DNA sequences, transition is a specific term for the exchange between either the two purines (A ↔ G) or the two pyrimidines (C ↔ T) (for additional details, see the article about transitions in genetics). By contrast, an exchange between one purine and one pyrimidine is called a transversion. === Deriving the dynamics of substitution === Consider a DNA sequence of fixed length m evolving in time by base replacement. Assume that the processes followed by the m sites are Markovian independent, identically distributed and that the process is constant over time. For a particular site, let E = { A , G , C , T } {\displaystyle {\mathcal {E}}=\{A,\,G,\,C,\,T\}} be the set of possible states for the site, and p ( t ) = ( p A ( t ) , p G ( t ) , p C ( t ) , p T ( t ) ) {\displaystyle \mathbf {p} (t)=(p_{A}(t),\,p_{G}(t),\,p_{C}(t),\,p_{T}(t))} their respective probabilities at time t {\displaystyle t} . For two distinct x , y ∈ E {\displaystyle x,y\in {\mathcal {E}}} , let μ x y {\displaystyle \mu _{xy}\ } be the transition rate from state x {\displaystyle x} to state y {\displaystyle y} . Similarly, for any x {\displaystyle x} , let the total rate of change from x {\displaystyle x} be μ x = ∑ y ≠ x μ x y . {\displaystyle \mu _{x}=\sum _{y\neq x}\mu _{xy}\,.} The changes in the probability distribution p A ( t ) {\displaystyle p_{A}(t)} for small increments of time Δ t {\displaystyle \Delta t} are given by p A ( t + Δ t ) = p A ( t ) − p A ( t ) μ A Δ t + ∑ x ≠ A p x ( t ) μ x A Δ t . {\displaystyle p_{A}(t+\Delta t)=p_{A}(t)-p_{A}(t)\mu _{A}\Delta t+\sum _{x\neq A}p_{x}(t)\mu _{xA}\Delta t\,.} In other words, (in frequentist language), the frequency of A {\displaystyle A} 's at time t + Δ t {\displaystyle t+\Delta t} is equal to the frequency at time t {\displaystyle t} minus the frequency of the lost A {\displaystyle A} 's plus the frequency of the newly created A {\displaystyle A} 's. Similarly for the probabilities p G ( t ) {\displaystyle p_{G}(t)} , p C ( t ) {\displaystyle p_{C}(t)} and p T ( t ) {\displaystyle p_{T}(t)} . These equations can be written compactly as p ( t + Δ t ) = p ( t ) + p ( t ) Q Δ t , {\displaystyle \mathbf {p} (t+\Delta t)=\mathbf {p} (t)+\mathbf {p} (t)Q\Delta t\,,} where Q = ( − μ A μ A G μ A C μ A T μ G A − μ G μ G C μ G T μ C A μ C G − μ C μ C T μ T A μ T G μ T C − μ T ) {\displaystyle Q={\begin{pmatrix}-\mu _{A}&\mu _{AG}&\mu _{AC}&\mu _{AT}\\\mu _{GA}&-\mu _{G}&\mu _{GC}&\mu _{GT}\\\mu _{CA}&\mu _{CG}&-\mu _{C}&\mu _{CT}\\\mu _{TA}&\mu _{TG}&\mu _{TC}&-\mu _{T}\end{pmatrix}}} is known as the rate matrix. Note that, by definition, the sum of the entries in each row of Q {\displaystyle Q} is equal to zero. It follows that p ′ ( t ) = p ( t ) Q . {\displaystyle \mathbf {p} '(t)=\mathbf {p} (t)Q\,.} For a stationary process, where Q {\displaystyle Q} does not depend on time t, this differential equation can be solved. First, P ( t ) = exp ( t Q ) , {\displaystyle P(t)=\exp(tQ),} where exp ( t Q ) {\displaystyle \exp(tQ)} denotes the exponential of the matrix t Q {\displaystyle tQ} . As a result, p ( t ) = p ( 0 ) P ( t ) = p ( 0 ) exp ( t Q ) . {\displaystyle \mathbf {p} (t)=\mathbf {p} (0)P(t)=\mathbf {p} (0)\exp(tQ)\,.} === Ergodicity === If the Markov chain is irreducible, i.e. if it is always possible to go from a state x {\displaystyle x} to a state y {\displaystyle y} (possibly in several steps), then it is also ergodic. As a result, it has a unique stationary distribution π = { π x , x ∈ E } {\displaystyle {\boldsymbol {\pi }}=\{\pi _{x},\,x\in {\mathcal {E}}\}} , where π x {\displaystyle \pi _{x}} corresponds to the proportion of time spent in state x {\displaystyle x} after the Markov chain has run for an infinite amount of time. In DNA evo