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Problem solving

Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to get from point A to B) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices. Solutions require sufficient resources and knowledge to attain the goal. Professionals such as lawyers, doctors, programmers, and consultants are largely problem solvers for issues that require technical skills and knowledge beyond general competence. Many businesses have found profitable markets by recognizing a problem and creating a solution: the more widespread and inconvenient the problem, the greater the opportunity to develop a scalable solution. There are many specialized problem-solving techniques and methods in fields such as science, engineering, business, medicine, mathematics, computer science, philosophy, and social organization. The mental techniques to identify, analyze, and solve problems are studied in psychology and cognitive sciences. Also widely researched are the mental obstacles that prevent people from finding solutions; problem-solving impediments include confirmation bias, mental set, and functional fixedness. == Definition == The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. Well-defined problems have specific end goals and clearly expected solutions, while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves dealing with pragmatics (the way that context contributes to meaning) and semantics (the interpretation of the problem). The ability to understand what the end goal of the problem is, and what rules could be applied, represents the key to solving the problem. Sometimes a problem requires abstract thinking or coming up with a creative solution. Problem solving has two major domains: mathematical problem solving and personal problem solving. Each concerns some difficulty or barrier that is encountered. === Psychology === Problem solving in psychology refers to the process of finding solutions to problems encountered in life. Solutions to these problems are usually situation- or context-specific. The process starts with problem finding and problem shaping, in which the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached; how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis. Mental health professionals study the human problem-solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying people with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems. Interpersonal everyday problem solving is dependent upon personal motivational and contextual components. One such component is the emotional valence of "real-world" problems, which can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task, impede problem resolution, and lead to negative outcomes such as fatigue, depression, and inertia. In conceptualization,human problem solving consists of two related processes: problem orientation, and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. People's strategies cohere with their goals and stem from the process of comparing oneself with others. === Cognitive sciences === Among the first experimental psychologists to study problem solving were the Gestaltists in Germany, such as Karl Duncker in The Psychology of Productive Thinking (1935). Perhaps best known is the work of Allen Newell and Herbert A. Simon. Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. These simple problems, such as the Tower of Hanoi, admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process. Researchers assumed that these model problems would elicit the characteristic cognitive processes by which more complex "real world" problems are solved. An outstanding problem-solving technique found by this research is the principle of decomposition. === Computer science === Much of computer science and artificial intelligence involves designing automated systems to solve a specified type of problem: to accept input data and calculate a correct or adequate response, reasonably quickly. Algorithms are recipes or instructions that direct such systems, written into computer programs. Steps for designing such systems include problem determination, heuristics, root cause analysis, de-duplication, analysis, diagnosis, and repair. Analytic techniques include linear and nonlinear programming, queuing systems, and simulation. A large, perennial obstacle is to find and fix errors in computer programs: debugging. === Logic === Formal logic concerns issues like validity, truth, inference, argumentation, and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution. The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods such as the resolution principle developed by John Alan Robinson. In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. In 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, who used a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning. The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of that approach from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution, which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving and computational logic to improve human thinking. === Engineering === When products or processes fail, problem solving techniques can be used to develop corrective actions that can be taken to prevent further failures. Such techniques can also be applied to a product or process prior to an actual failure event—to predict, analyze, and mitigate a potential problem in advance. Techniques such as failure mode and effects analysis can proactively reduce the likelihood of problems. In either the reactive or the proactive case, it is necessary to build a causal explanation through a process of diagnosis. In deriving an explanation of effects in terms of causes, abduction generates new ideas or hypothes
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Machine translation of sign languages

The machine translation of sign languages has been possible, albeit in a limited fashion, since 1977. When a research project successfully matched English letters from a keyboard to ASL manual alphabet letters which were simulated on a robotic hand. These technologies translate signed languages into written or spoken language, and written or spoken language to sign language, without the use of a human interpreter. Sign languages possess different phonological features than spoken languages, which has created obstacles for developers. Developers use computer vision and machine learning to recognize specific phonological parameters and epentheses unique to sign languages, and speech recognition and natural language processing allow interactive communication between hearing and deaf people. == Limitations == Sign language translation technologies are limited in the same way as spoken language translation. None can translate with 100% accuracy. In fact, sign language translation technologies are far behind their spoken language counterparts. This is, in no trivial way, due to the fact that signed languages have multiple articulators. Where spoken languages are articulated through the vocal tract, signed languages are articulated through the hands, arms, head, shoulders, torso, and parts of the face. This multi-channel articulation makes translating sign languages very difficult. An additional challenge for sign language MT is the fact that there is no formal written format for signed languages. There are notations systems but no writing system has been adopted widely enough, by the international Deaf community, that it could be considered the 'written form' of a given sign language. Sign Languages then are recorded in various video formats. There is no gold standard parallel corpus that is large enough for SMT, for example. == History == The history of automatic sign language translation started with the development of hardware such as finger-spelling robotic hands. In 1977, a finger-spelling hand project called RALPH (short for "Robotic Alphabet") created a robotic hand that can translate alphabets into finger-spellings. Later, the use of gloves with motion sensors became the mainstream, and some projects such as the CyberGlove and VPL Data Glove were born. The wearable hardware made it possible to capture the signers' hand shapes and movements with the help of the computer software. However, with the development of computer vision, wearable devices were replaced by cameras due to their efficiency and fewer physical restrictions on signers. To process the data collected through the devices, researchers implemented neural networks such as the Stuttgart Neural Network Simulator for pattern recognition in projects such as the CyberGlove. Researchers also use many other approaches for sign recognition. For example, Hidden Markov Models are used to analyze data statistically, and GRASP and other machine learning programs use training sets to improve the accuracy of sign recognition. Fusion of non-wearable technologies such as cameras and Leap Motion controllers have shown to increase the ability of automatic sign language recognition and translation software. == Technologies == === VISICAST === http://www.visicast.cmp.uea.ac.uk/Visicast_index.html === eSIGN project === http://www.visicast.cmp.uea.ac.uk/eSIGN/index.html === The American Sign Language Avatar Project at DePaul University === http://asl.cs.depaul.edu/ === Spanish to LSE === López-Ludeña, Verónica; San-Segundo, Rubén; González, Carlos; López, Juan Carlos; Pardo, José M. (2012), Methodology for developing a Speech into Sign Language Translation System in a New Semantic Domain (PDF), CiteSeerX 10.1.1.1065.5265, S2CID 2724186 === SignAloud === SignAloud is a technology that incorporates a pair of gloves made by a group of students at University of Washington that transliterate American Sign Language (ASL) into English. In February 2015 Thomas Pryor, a hearing student from the University of Washington, created the first prototype for this device at Hack Arizona, a hackathon at the University of Arizona. Pryor continued to develop the invention and in October 2015, Pryor brought Navid Azodi onto the SignAloud project for marketing and help with public relations. Azodi has a rich background and involvement in business administration, while Pryor has a wealth of experience in engineering. In May 2016, the duo told NPR that they are working more closely with people who use ASL so that they can better understand their audience and tailor their product to the needs of these people rather than the assumed needs. However, no further versions have been released since then. The invention was one of seven to win the Lemelson-MIT Student Prize, which seeks to award and applaud young inventors. Their invention fell under the "Use it!" category of the award which includes technological advances to existing products. They were awarded $10,000. The gloves have sensors that track the users hand movements and then send the data to a computer system via Bluetooth. The computer system analyzes the data and matches it to English words, which are then spoken aloud by a digital voice. The gloves do not have capability for written English input to glove movement output or the ability to hear language and then sign it to a deaf person, which means they do not provide reciprocal communication. The device also does not incorporate facial expressions and other nonmanual markers of sign languages, which may alter the actual interpretation from ASL. === ProDeaf === ProDeaf (WebLibras) is a computer software that can translate both text and voice into Portuguese Libras (Portuguese Sign Language) "with the goal of improving communication between the deaf and hearing." There is currently a beta edition in production for American Sign Language as well. The original team began the project in 2010 with a combination of experts including linguists, designers, programmers, and translators, both hearing and deaf. The team originated at Federal University of Pernambuco (UFPE) from a group of students involved in a computer science project. The group had a deaf team member who had difficulty communicating with the rest of the group. In order to complete the project and help the teammate communicate, the group created Proativa Soluções and have been moving forward ever since. The current beta version in American Sign Language is very limited. For example, there is a dictionary section and the only word under the letter 'j' is 'jump'. If the device has not been programmed with the word, then the digital avatar must fingerspell the word. The last update of the app was in June 2016, but ProDeaf has been featured in over 400 stories across the country's most popular media outlets. The application cannot read sign language and turn it into word or text, so it only serves as a one-way communication. Additionally, the user cannot sign to the app and receive an English translation in any form, as English is still in the beta edition. === Kinect Sign Language Translator === Since 2012, researchers from the Chinese Academy of Sciences and specialists of deaf education from Beijing Union University in China have been collaborating with Microsoft Research Asian team to create Kinect Sign Language Translator. The translator consists of two modes: translator mode and communication mode. The translator mode is capable of translating single words from sign into written words and vice versa. The communication mode can translate full sentences and the conversation can be automatically translated with the use of the 3D avatar. The translator mode can also detect the postures and hand shapes of a signer as well as the movement trajectory using the technologies of machine learning, pattern recognition, and computer vision. The device also allows for reciprocal communication because the speech recognition technology allows the spoken language to be translated into the sign language and the 3D modeling avatar can sign back to the deaf people. The original project was started in China based on translating Chinese Sign Language. In 2013, the project was presented at Microsoft Research Faculty Summit and Microsoft company meeting. Currently, this project is also being worked by researchers in the United States to implement American Sign Language translation. As of now, the device is still a prototype, and the accuracy of translation in the communication mode is still not perfect. === SignAll === SignAll is an automatic sign language translation system provided by Dolphio Technologies in Hungary. The team is "pioneering the first automated sign language translation solution, based on computer vision and natural language processing (NLP), to enable everyday communication between individuals with hearing who use spoken English and deaf or hard of hearing individuals who use ASL." The system of SignAll uses Kinect from Microsoft and other web camera
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Top 10 AI Essay Writers Compared (2026)

Curious about the best AI essay writer? An AI essay writer is software that uses machine learning to help you get more done — it combines speed, accuracy, and an interface that just works. Hands-on testing shows real-world results vary, so a short free trial is the smartest way to decide. Whether you are a beginner or a pro, the right AI essay writer 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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Powerset construction

In the theory of computation and automata theory, the powerset construction or subset construction is a standard method for converting a nondeterministic finite automaton (NFA) into a deterministic finite automaton (DFA) that recognizes the same formal language. It is important in theory because it establishes that NFAs, despite their additional flexibility, are unable to recognize any language that cannot be recognized by some DFA. It is also important in practice for converting easier-to-construct NFAs into more efficiently executable DFAs. However, if the NFA has n states, the resulting DFA may have up to 2n states, an exponentially larger number, which sometimes makes the construction impractical for large NFAs. The construction, sometimes called the Rabin–Scott powerset construction (or subset construction) to distinguish it from similar constructions for other types of automata, was first published by Michael O. Rabin and Dana Scott in 1959. == Intuition == To simulate the operation of a DFA on a given input string, one needs to keep track of a single state at any time: the state that the automaton will reach after seeing a prefix of the input. In contrast, to simulate an NFA, one needs to keep track of a set of states: all of the states that the automaton could reach after seeing the same prefix of the input, according to the nondeterministic choices made by the automaton. If, after a certain prefix of the input, a set S of states can be reached, then after the next input symbol x the set of reachable states is a deterministic function of S and x. Therefore, the sets of reachable NFA states play the same role in the NFA simulation as single DFA states play in the DFA simulation, and in fact the sets of NFA states appearing in this simulation may be re-interpreted as being states of a DFA. == Construction == The powerset construction applies most directly to an NFA that does not allow state transformations without consuming input symbols (aka: "ε-moves"). Such an automaton may be defined as a 5-tuple (Q, Σ, T, q0, F), in which Q is the set of states, Σ is the set of input symbols, T is the transition function (mapping a state and an input symbol to a set of states), q0 is the initial state, and F is the set of accepting states. The corresponding DFA has states corresponding to subsets of Q. The initial state of the DFA is {q0}, the (one-element) set of initial states. The transition function of the DFA maps a state S (representing a subset of Q) and an input symbol x to the set T(S,x) = ∪{T(q,x) | q ∈ S}, the set of all states that can be reached by an x-transition from a state in S. A state S of the DFA is an accepting state if and only if at least one member of S is an accepting state of the NFA. In the simplest version of the powerset construction, the set of all states of the DFA is the powerset of Q, the set of all possible subsets of Q. However, many states of the resulting DFA may be useless as they may be unreachable from the initial state. An alternative version of the construction creates only the states that are actually reachable. === NFA with ε-moves === For an NFA with ε-moves (also called an ε-NFA), the construction must be modified to deal with these by computing the ε-closure of states: the set of all states reachable from some given state using only ε-moves. Van Noord recognizes three possible ways of incorporating this closure computation in the powerset construction: Compute the ε-closure of the entire automaton as a preprocessing step, producing an equivalent NFA without ε-moves, then apply the regular powerset construction. This version, also discussed by Hopcroft and Ullman, is straightforward to implement, but impractical for automata with large numbers of ε-moves, as commonly arise in natural language processing application. During the powerset computation, compute the ε-closure { q ′ | q → ε ∗ q ′ } {\displaystyle \{q'~|~q\to _{\varepsilon }^{}q'\}} of each state q that is considered by the algorithm (and cache the result). During the powerset computation, compute the ε-closure { q ′ | ∃ q ∈ Q ′ , q → ε ∗ q ′ } {\displaystyle \{q'~|~\exists q\in Q',q\to _{\varepsilon }^{}q'\}} of each subset of states Q' that is considered by the algorithm, and add its elements to Q'. === Multiple initial states === If NFAs are defined to allow for multiple initial states, the initial state of the corresponding DFA is the set of all initial states of the NFA, or (if the NFA also has ε-moves) the set of all states reachable from initial states by ε-moves. == Example == The NFA below has four states; state 1 is initial, and states 3 and 4 are accepting. Its alphabet consists of the two symbols 0 and 1, and it has ε-moves. The initial state of the DFA constructed from this NFA is the set of all NFA states that are reachable from state 1 by ε-moves; that is, it is the set {1,2,3}. A transition from {1,2,3} by input symbol 0 must follow either the arrow from state 1 to state 2, or the arrow from state 3 to state 4. Additionally, neither state 2 nor state 4 have outgoing ε-moves. Therefore, T({1,2,3},0) = {2,4}, and by the same reasoning the full DFA constructed from the NFA is as shown below. As can be seen in this example, there are five states reachable from the start state of the DFA; the remaining 11 sets in the powerset of the set of NFA states are not reachable. == Complexity == Because the DFA states consist of sets of NFA states, an n-state NFA may be converted to a DFA with at most 2n states. For every n, there exist n-state NFAs such that every subset of states is reachable from the initial subset, so that the converted DFA has exactly 2n states, giving Θ(2n) worst-case time complexity. A simple example requiring nearly this many states is the language of strings over the alphabet {0,1} in which there are at least n characters, the nth from last of which is 1. It can be represented by an (n + 1)-state NFA, but it requires 2n DFA states, one for each n-character suffix of the input; cf. picture for n=4. == Applications == Brzozowski's algorithm for DFA minimization uses the powerset construction, twice. It converts the input DFA into an NFA for the reverse language, by reversing all its arrows and exchanging the roles of initial and accepting states, converts the NFA back into a DFA using the powerset construction, and then repeats its process. Its worst-case complexity is exponential, unlike some other known DFA minimization algorithms, but in many examples it performs more quickly than its worst-case complexity would suggest. Safra's construction, which converts a non-deterministic Büchi automaton with n states into a deterministic Muller automaton or into a deterministic Rabin automaton with 2O(n log n) states, uses the powerset construction as part of its machinery.
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Sentence extraction

Sentence extraction is a technique used for automatic summarization of a text. In this shallow approach, statistical heuristics are used to identify the most salient sentences of a text. Sentence extraction is a low-cost approach compared to more knowledge-intensive deeper approaches which require additional knowledge bases such as ontologies or linguistic knowledge. In short, sentence extraction works as a filter that allows only meaningful sentences to pass. The major downside of applying sentence-extraction techniques to the task of summarization is the loss of coherence in the resulting summary. Nevertheless, sentence extraction summaries can give valuable clues to the main points of a document and are frequently sufficiently intelligible to human readers. == Procedure == Usually, a combination of heuristics is used to determine the most important sentences within the document. Each heuristic assigns a (positive or negative) score to the sentence. After all heuristics have been applied, the highest-scoring sentences are included in the summary. The individual heuristics are weighted according to their importance. === Early approaches and some sample heuristics === Seminal papers which laid the foundations for many techniques used today have been published by Hans Peter Luhn in 1958 and H. P Edmundson in 1969. Luhn proposed to assign more weight to sentences at the beginning of the document or a paragraph. Edmundson stressed the importance of title-words for summarization and was the first to employ stop-lists in order to filter uninformative words of low semantic content (e.g. most grammatical words such as of, the, a). He also distinguished between bonus words and stigma words, i.e. words that probably occur together with important (e.g. the word form significant) or unimportant information. His idea of using key-words, i.e. words which occur significantly frequently in the document, is still one of the core heuristics of today's summarizers. With large linguistic corpora available today, the tf–idf value which originated in information retrieval, can be successfully applied to identify the key words of a text: If for example the word cat occurs significantly more often in the text to be summarized (TF = "term frequency") than in the corpus (IDF means "inverse document frequency"; here the corpus is meant by document), then cat is likely to be an important word of the text; the text may in fact be a text about cats.
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Imitation learning

Imitation learning is a paradigm in reinforcement learning, where an agent learns to perform a task by supervised learning from expert demonstrations . It is also called learning from demonstration and apprenticeship learning. It has been applied to underactuated robotics, self-driving cars, quadcopter navigation, helicopter aerobatics, and locomotion. == Approaches == Expert demonstrations are recordings of an expert performing the desired task, often collected as state-action pairs ( o t ∗ , a t ∗ ) {\displaystyle (o_{t}^{},a_{t}^{})} . === Behavior Cloning === Behavior Cloning (BC) is the most basic form of imitation learning. Essentially, it uses supervised learning to train a policy π θ {\displaystyle \pi _{\theta }} such that, given an observation o t {\displaystyle o_{t}} , it would output an action distribution π θ ( ⋅ | o t ) {\displaystyle \pi _{\theta }(\cdot |o_{t})} that is approximately the same as the action distribution of the experts. BC is susceptible to distribution shift. Specifically, if the trained policy differs from the expert policy, it might find itself straying from expert trajectory into observations that would have never occurred in expert trajectories. This was already noted by ALVINN, where they trained a neural network to drive a van using human demonstrations. They noticed that because a human driver never strays far from the path, the network would never be trained on what action to take if it ever finds itself straying far from the path. === DAgger === DAgger (Dataset Aggregation) improves on behavior cloning by iteratively training on a dataset of expert demonstrations. In each iteration, the algorithm first collects data by rolling out the learned policy π θ {\displaystyle \pi _{\theta }} . Then, it queries the expert for the optimal action a t ∗ {\displaystyle a_{t}^{}} on each observation o t {\displaystyle o_{t}} encountered during the rollout. Finally, it aggregates the new data into the dataset D ← D ∪ { ( o 1 , a 1 ∗ ) , ( o 2 , a 2 ∗ ) , . . . , ( o T , a T ∗ ) } {\displaystyle D\leftarrow D\cup \{(o_{1},a_{1}^{}),(o_{2},a_{2}^{}),...,(o_{T},a_{T}^{})\}} and trains a new policy on the aggregated dataset. === Decision transformer === The Decision Transformer approach models reinforcement learning as a sequence modelling problem. Similar to Behavior Cloning, it trains a sequence model, such as a Transformer, that models rollout sequences ( R 1 , o 1 , a 1 ) , ( R 2 , o 2 , a 2 ) , … , ( R t , o t , a t ) , {\displaystyle (R_{1},o_{1},a_{1}),(R_{2},o_{2},a_{2}),\dots ,(R_{t},o_{t},a_{t}),} where R t = r t + r t + 1 + ⋯ + r T {\displaystyle R_{t}=r_{t}+r_{t+1}+\dots +r_{T}} is the sum of future reward in the rollout. During training time, the sequence model is trained to predict each action a t {\displaystyle a_{t}} , given the previous rollout as context: ( R 1 , o 1 , a 1 ) , ( R 2 , o 2 , a 2 ) , … , ( R t , o t ) {\displaystyle (R_{1},o_{1},a_{1}),(R_{2},o_{2},a_{2}),\dots ,(R_{t},o_{t})} During inference time, to use the sequence model as an effective controller, it is simply given a very high reward prediction R {\displaystyle R} , and it would generalize by predicting an action that would result in the high reward. This was shown to scale predictably to a Transformer with 1 billion parameters that is superhuman on 41 Atari games. === Other approaches === See for more examples. == Related approaches == Inverse Reinforcement Learning (IRL) learns a reward function that explains the expert's behavior and then uses reinforcement learning to find a policy that maximizes this reward. Recent works have also explored multi-agent extensions of IRL in networked systems. Generative Adversarial Imitation Learning (GAIL) uses generative adversarial networks (GANs) to match the distribution of agent behavior to the distribution of expert demonstrations. It extends a previous approach using game theory.
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Markov chain Monte Carlo

In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it, i.e. the Markov chain's equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution. Markov chain Monte Carlo methods are used to study probability distributions that are too complex or too high dimensional to study with analytic techniques alone. Various algorithms exist for constructing such Markov chains, including the Metropolis–Hastings algorithm. == General explanation == Markov chain Monte Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to evaluate an integral over that variable, as its expected value or variance. Practically, an ensemble of chains is generally developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other. These chains are stochastic processes of "walkers" which move around randomly according to an algorithm that looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities. Random walk Monte Carlo methods are a kind of random simulation or Monte Carlo method. However, whereas the random samples of the integrand used in a conventional Monte Carlo integration are statistically independent, those used in MCMC are autocorrelated. Correlations of samples introduces the need to use the Markov chain central limit theorem when estimating the error of mean values. These algorithms create Markov chains such that they have an equilibrium distribution which is proportional to the function given. == History == The development of MCMC methods is deeply rooted in the early exploration of Monte Carlo (MC) techniques in the mid-20th century, particularly in physics. These developments were marked by the Metropolis algorithm proposed by Nicholas Metropolis, Arianna W. Rosenbluth, Marshall Rosenbluth, Augusta H. Teller, and Edward Teller in 1953, which was designed to tackle high-dimensional integration problems using early computers. Then in 1970, W. K. Hastings generalized this algorithm and inadvertently introduced the component-wise updating idea, later known as Gibbs sampling. Simultaneously, the theoretical foundations for Gibbs sampling were being developed, such as the Hammersley–Clifford theorem from Julian Besag's 1974 paper. Although the seeds of MCMC were sown earlier, including the formal naming of Gibbs sampling in image processing by Stuart Geman and Donald Geman (1984) and the data augmentation method by Martin A. Tanner and Wing Hung Wong (1987), its "revolution" in mainstream statistics largely followed demonstrations of the universality and ease of implementation of sampling methods (especially Gibbs sampling) for complex statistical (particularly Bayesian) problems, spurred by increasing computational power and software like BUGS. This transformation was accompanied by significant theoretical advancements, such as Luke Tierney's (1994) rigorous treatment of MCMC convergence, and Jun S. Liu, Wong, and Augustine Kong's (1994, 1995) analysis of Gibbs sampler structure. Subsequent developments further expanded the MCMC toolkit, including particle filters (Sequential Monte Carlo) for sequential problems, Perfect sampling aiming for exact simulation (Jim Propp and David B. Wilson, 1996), RJMCMC (Peter J. Green, 1995) for handling variable-dimension models, and deeper investigations into convergence diagnostics and the central limit theorem. Overall, the evolution of MCMC represents a paradigm shift in statistical computation, enabling the analysis of numerous previously intractable complex models and continually expanding the scope and impact of statistics. == Mathematical setting == Suppose (Xn) is a Markov Chain in the general state space X {\displaystyle {\mathcal {X}}} with specific properties. We are interested in the limiting behavior of the partial sums: S n ( h ) = 1 n ∑ i = 1 n h ( X i ) {\displaystyle S_{n}(h)={\dfrac {1}{n}}\sum _{i=1}^{n}h(X_{i})} as n goes to infinity. Particularly, we hope to establish the Law of Large Numbers and the Central Limit Theorem for MCMC. In the following, we state some definitions and theorems necessary for the important convergence results. In short, we need the existence of invariant measure and Harris recurrent to establish the Law of Large Numbers of MCMC (Ergodic Theorem). And we need aperiodicity, irreducibility and extra conditions such as reversibility to ensure the Central Limit Theorem holds in MCMC. === Irreducibility and aperiodicity === Recall that in the discrete setting, a Markov chain is said to be irreducible if it is possible to reach any state from any other state in a finite number of steps with positive probability. However, in the continuous setting, point-to-point transitions have zero probability. In this case, φ-irreducibility generalizes irreducibility by using a reference measure φ on the measurable space ( X , B ( X ) ) {\displaystyle ({\mathcal {X}},{\mathcal {B}}({\mathcal {X}}))} . Definition (φ-irreducibility) Given a measure φ {\displaystyle \varphi } defined on ( X , B ( X ) ) {\displaystyle ({\mathcal {X}},{\mathcal {B}}({\mathcal {X}}))} , the Markov chain ( X n ) {\displaystyle (X_{n})} with transition kernel K ( x , y ) {\displaystyle K(x,y)} is φ-irreducible if, for every A ∈ B ( X ) {\displaystyle A\in {\mathcal {B}}({\mathcal {X}})} with φ ( A ) > 0 {\displaystyle \varphi (A)>0} , there exists n {\displaystyle n} such that K n ( x , A ) > 0 {\displaystyle K^{n}(x,A)>0} for all x ∈ X {\displaystyle x\in {\mathcal {X}}} (Equivalently, P x ( τ A < ∞ ) > 0 {\displaystyle P_{x}(\tau _{A}<\infty )>0} , here τ A = inf { n ≥ 1 ; X n ∈ A } {\displaystyle \tau _{A}=\inf\{n\geq 1;X_{n}\in A\}} is the first n {\displaystyle n} for which the chain enters the set A {\displaystyle A} ). This is a more general definition for irreducibility of a Markov chain in non-discrete state space. In the discrete case, an irreducible Markov chain is said to be aperiodic if it has period 1. Formally, the period of a state ω ∈ X {\displaystyle \omega \in {\mathcal {X}}} is defined as: d ( ω ) := g c d { m ≥ 1 ; K m ( ω , ω ) > 0 } {\displaystyle d(\omega ):=\mathrm {gcd} \{m\geq 1\,;\,K^{m}(\omega ,\omega )>0\}} For the general (non-discrete) case, we define aperiodicity in terms of small sets: Definition (Cycle length and small sets) A φ-irreducible Markov chain ( X n ) {\displaystyle (X_{n})} has a cycle of length d if there exists a small set C {\displaystyle C} , an associated integer M {\displaystyle M} , and a probability distribution ν M {\displaystyle \nu _{M}} such that d is the greatest common divisor of: { m ≥ 1 ; ∃ δ m > 0 such that C is small for ν m ≥ δ m ν M } . {\displaystyle \{m\geq 1\,;\,\exists \,\delta _{m}>0{\text{ such that }}C{\text{ is small for }}\nu _{m}\geq \delta _{m}\nu _{M}\}.} A set C {\displaystyle C} is called small if there exists m ∈ N ∗ {\displaystyle m\in \mathbb {N} ^{}} and a nonzero measure ν m {\displaystyle \nu _{m}} such that: K m ( x , A ) ≥ ν m ( A ) , ∀ x ∈ C , ∀ A ∈ B ( X ) . {\displaystyle K^{m}(x,A)\geq \nu _{m}(A),\quad \forall x\in C,\,\forall A\in {\mathcal {B}}({\mathcal {X}}).} === Harris recurrent === Definition (Harris recurrence) A set A {\displaystyle A} is Harris recurrent if P x ( η A = ∞ ) = 1 {\displaystyle P_{x}(\eta _{A}=\infty )=1} for all x ∈ A {\displaystyle x\in A} , where η A = ∑ n = 1 ∞ I A ( X n ) {\displaystyle \eta _{A}=\sum _{n=1}^{\infty }\mathbb {I} _{A}(X_{n})} is the number of visits of the chain ( X n ) {\displaystyle (X_{n})} to the set A {\displaystyle A} . The chain ( X n ) {\displaystyle (X_{n})} is said to be Harris recurrent if there exists a measure ψ {\displaystyle \psi } such that the chain is ψ {\displaystyle \psi } -irreducible and every measurable set A {\displaystyle A} with ψ ( A ) > 0 {\displaystyle \psi (A)>0} is Harris recurrent. A useful criterion for verifying Harris recurrence is the following: Proposition If for every A ∈ B ( X ) {\displaystyle A\in {\mathcal {B}}({\mathcal {X}})} , we have P x ( τ A < ∞ ) = 1 {\displaystyle P_{x}(\tau _{A}<\infty )=1} for every x ∈ A {\displaystyle x\in A} , then P x ( η A = ∞ ) = 1 {\displaystyle P_{x}(\eta _{A}=\infty )=1} for all x ∈ X {\displaystyle x\in {\mathcal {X}}} , and the chain ( X n ) {\displaystyle (X_{n})} is Harris recurrent. This definition is only needed when the state space X {\displaystyle {\mathcal {X}}} is uncountable. In the countable case, recurrence corresponds to E x [ η x ] = ∞ {\displaystyle \mathbb {E} _{x}[\eta _{x}]=\infty } , which is equivalent to P x ( τ x < ∞ ) = 1 {\displaystyle P_{x}(\tau _{x}<\infty )=1} for all x ∈ X {\displaystyle x\i
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Frank Hutter

Frank Hutter is a German computer scientist recognized for his contributions to machine learning, particularly in the areas of automated machine learning (AutoML), hyperparameter optimization, meta-learning and tabular machine learning. He is currently a Hector-Endowed Fellow and PI at the ELLIS Institute Tübingen and a Full Professor (W3) for Machine Learning at the Department of Computer Science, University of Freiburg. Hutter is known for his role in establishing AutoML as a key area in artificial intelligence research. == Education and academic career == Frank Hutter received his academic training in computer science at Darmstadt University of Technology, where he completed his Vordiplom (comparable to a BSc) and Hauptdiplom (equivalent to MSc) by 2004. He later pursued his PhD at the University of British Columbia, under the supervision of Profs. Holger Hoos, Kevin Leyton-Brown and Kevin Murphy, where his doctoral thesis, titled "Automated Configuration of Algorithms for Solving Hard Computational Problems," was awarded the CAIAC Doctoral Dissertation Award for the best thesis in Artificial Intelligence completed at a Canadian university in 2009. Hutter did his postdoctoral research at the University of British Columbia, where he worked from 2009 to 2013. In 2013, he moved to the University of Freiburg, initially leading an Emmy Noether Research Group, and in 2017, he was appointed as a Full Professor. His contributions to machine learning have been recognized globally, particularly his work in AutoML and hyperparameter optimization. Overall, Hutter has authored over 180 peer-reviewed publications, which have garnered more than 89,000 citations, reflecting the high impact of his work. == Contributions in AutoML == Hutter's early research laid the groundwork for the field of Automated Machine Learning (AutoML). He has been a key figure in establishing AutoML as a distinct research area. Along with various colleagues, he organized the AutoML workshops from 2014 to 2021, wrote the first book on AutoML and taught the first MOOC on AutoML. He also co-founded the AutoML conference in 2022 and served as its general chair the first two years. He also published prominent works in various subfields of AutoML, such as hyperparameter optimization, neural architecture search, meta-Learning and AutoML systems. He is currently the most highly cited researcher in AutoML. == Contributions in machine learning for tabular data == Hutter has also made many contributions to machine learning for tabular data. He led the development of the first widely adopted AutoML system for tabular data, AutoWEKA, which was published at KDD 2013 and received the test of time award at KDD (2023). Subsequently, he led the development of Auto-sklearn, the first highly used AutoML system for tabular data in Python, and with it, won the first international AutoML challenge and the subsequent second international AutoML challenge, both of which only included tabular data. More recently, he focused on tabular foundation models, including TabPFN, which was published in Nature magazine. In 2024, he also co-founded Prior Labs, the first company focusing on tabular foundation models. == Awards and honors == Hutter has received numerous awards throughout his career. In 2023, he won the KDD Test of Time Award for Research together with Chris Thornton, Holger H. Hoos, and Kevin Leyton-Brown. He has received three grants from the ERC, including the ERC Starting Grant (2016) and ERC Consolidator Grant (2022), as well as an ERC Proof of Concept Grant (2020). In 2021, he became an ELLIS Unit Director and was also recognized as a EurAI Fellow, in addition to receiving the AIJ Prominent Paper Award. Earlier, he was a recipient of the Google Faculty Research Award in 2018. His groundbreaking research was acknowledged early in his career with the IJCAI Distinguished Paper Award in 2013 and the IJCAI/JAIR Best Paper Prize in 2010. == Representative publications == Hutter, F. Kotthoff, L. and Vanschoren, J., editors. Automated machine learning: methods, systems, challenges, Springer Nature, 2019. www.automl.org/book. Feurer, M., Klein, A., Eggensperger, K., Springenberg, T., Blum, M., Hutter, F. Efficient and Robust Automated Machine Learning. In NeurIPS 2015. Loshchilov, I., and Hutter, F. Decoupled weight decay regularization. In ICLR 2018. Zela, A., Elsken, T. ,Saikia, T. ,Marrakschi, Y. ,Brox, T. and Hutter. ,F.Understanding and Robustifying Differentiable Architecture Search. In ICLR 2020. Hollmann, N., Müller, S., Eggensperger, K. and Hutter, F. TabPFN: A Transformer That Solves Small Tabular Classification Problems in a Second, In ICLR 2023.
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Deep Instinct

Deep Instinct is a cybersecurity company that applies deep learning to cybersecurity. The company implements artificial intelligence to the task of preventing and detecting malware. The company was the recipient of the Technology Pioneer by The World Economic Forum in 2017. Lane Bess has been CEO of the company since 2022. == Overview == In 2015, Deep Instinct was founded by Guy Caspi, Dr. Eli David, and Nadav Maman. The headquarters of the company is located in New York City. In July 2017, NVIDIA became an investor. According to Tom's Hardware, NVIDIA’s investment enabled access to a GPU-based neural network and CUDA platform, which they were using to achieve maximum vulnerability detection rates. As of February 2020, the company had raised $43 million in Series C funding round. In April 2021, Deep Instinct raised $100 million in Series D funding to accelerate growth. == Partnerships == In April 2019, Deep Instinct partnered with Chinese artist, Guo O. Dong on an art project titled, The Persistence of Chaos, consisting of a laptop infected with 6 pieces of malware that represented $95 billion in damages. The art was auctioned with a final bid of $1,345,000. In the same year, Globes reported that, HP Inc partnered with Deep Instinct to launch their security solution HP SureSense, which has been applied to the EliteBook and Zbook devices.
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Machine translation in China

Machine translation in China is the history of machine translation systems developed in China. China became the fourth country that began machine translation (MT) research following USA, UK, and the Soviet Union. In 1957, the Language Institute of Chinese Academy of Sciences took the initiative in Russian-Chinese MT research program and set up an MT research group. From then on the research activities were directed and applied for academic purposes in Universities. The turning point of MT systems launching initiatives in market began from 1990s. MT systems went into blossom into the market. Among these systems, there were commercialized MT systems. To be more specific, Transtar was the first commercialized MT system and has been constantly upgraded. What's more, IMC/EC MT system which was developed by Computer Institute of Chinese Academy of Sciences has further made great advancement. Meanwhile, the practical MT system MT-IT-EC specific to communication domain was also striking to notice, for it has greatly improved the efficiency and productivity in the issue of publications. Government funding is a critical component and support in the development of market-oriented machine translation in China. It is evident to see that since Chinese opened up to the outside world and joined the WTO, the vigorous import and export trade generate opportunities for machine translation to transfer technical terms of products into the readable target information. Facing the increasing demand of sophisticated state-of -the -art translation technology, the academic area including research institute and universities are even launching bachelors’ and master's programs regarding machine translation. Thus, strong evidence illustrates the promising field of machine translation in the future market of China.
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Bibliotheca Polyglotta

The Bibliotheca Polyglotta is a Norwegian database for Multilingualism project, lingua franca and science per global history at the University of Oslo. The aim of the project is according to pages is "producing a web corpus of Buddhist texts for using in multilingual lexicography. More generally, will the texts used for the study Sanskrit, Chinese and Tibetan."
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Eric Xing

Eric Poe Xing (Chinese: 邢波) is an American computer scientist who has been serving as president of Mohamed bin Zayed University of Artificial Intelligence (MBZUAI) since January 2021. He is also a professor in the Carnegie Mellon University School of Computer Science where he founded the SAILING Lab in 2004, and is the co-founder of the AI companies Petuum and GenBio AI. Xing's research focuses on statistical machine learning, probabilistic graphical models, and systems for distributed machine learning. He was elected a Fellow of the Institute of Electrical and Electronics Engineers in 2019 for "contributions to machine learning algorithms and systems" and a Fellow of the Association for Computing Machinery in 2022 for "contributions to algorithms, architectures, and applications in machine learning." == Education == Xing earned a B.Sc. in physics from Tsinghua University in 1993, and an M.Sc. in computer science from Rutgers University in 1998. He earned a Ph.D. in molecular biology and biochemistry from Rutgers in 1999, supervised by molecular cancer researcher Chung S. Yang. His dissertation examined the inactivation of the Rb and p53 pathways in human esophageal squamous cell carcinoma. He earned a second Ph.D. in computer science from the University of California, Berkeley in 2004, supervised by Richard Karp, Michael I. Jordan, and Stuart J. Russell. His thesis applied probabilistic graphical models to motif identification and haplotype inference in genomic data. == Career == Xing joined Carnegie Mellon University (CMU) as a faculty member in 2004, where he created the Statistical Artificial Intelligence and Integrative Genomics (SAILING) Lab. He held visiting appointments from 2010 to 2011, serving as a visiting research professor at Facebook Inc. and as a visiting associate professor in the Department of Statistics at Stanford University. He served as co-Program Chair of the International Conference on Machine Learning (ICML) in 2014 and General Chair in 2019. Xing served as the founding director of CMU’s Center for Machine Learning and Health, established in 2015 as part of the Pittsburgh Health Data Alliance, a collaboration between CMU, the University of Pittsburgh, and the University of Pittsburgh Medical Center. In 2016, Xing co-founded Petuum Inc., a US-based startup. In 2017, Petuum raised $93 million in a round of venture funding from SoftBank. In 2018 Petuum was named a World Economic Forum Technology Pioneer. In 2019, Xing received the Carnegie Science Award for Startup Entrepreneurs in recognition of his leadership of Petuum. On 29 November 2020, Xing was appointed president of the Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), with the appointment taking effect in January 2021. In 2024, Xing co-founded GenBio AI where he is chief scientist. The US-based startup, which he co-founded with David Baker, Ziv Bar-Joseph, Emma Lundberg, Le Song and Fred Hu, aims to create AI-driven digital organisms (AIDO) for the purposes of modeling medical treatments. Xing has overseen the launch of the MBZUAI Institute of Foundation Models (IFM), which focuses on research and development of large-scale foundation models. In 2025–2026, IFM released the open-source reasoning model K2 Think, which was covered internationally as part of the UAE’s push to develop domestically controlled (“sovereign”) AI capabilities. IFM presented PAN as a “world model” research project and demonstrated related systems publicly. MBZUAI also collaborated with G42 and Cerebras Systems on the Jais language model, an open-source Arabic–English large language model released in 2023, according to Reuters. == Awards and honors == Xing is a recipient of the National Science Foundation (NSF) Career Award and the Alfred P. Sloan Research Fellowship. Xing is an elected Fellow of the following institutes and associations: Association for the Advancement of Artificial Intelligence (AAAI) 2016 Institute of Electrical and Electronics Engineers (IEEE) 2019 for "contributions to machine learning algorithms and systems" American Statistical Association (ASA) 2022 Association for Computing Machinery (ACM) 2022 for "contributions to algorithms, architectures, and applications in machine learning" Institute of Mathematical Statistics (IMS) 2023 International Society for Computational Biology (ISCB) 2026 == Selected publications == Eric P. Xing; Michael I. Jordan; Stuart J. Russell; Andrew Y. Ng (2003). "Distance Metric Learning with Application to Clustering with Side-Information" (PDF). Advances in Neural Information Processing Systems 15. Advances in Neural Information Processing Systems. Wikidata Q77691192. Edoardo M. Airoldi; David M. Blei; Stephen E Fienberg; Eric P Xing (1 September 2008). "Mixed Membership Stochastic Blockmodels". Journal of Machine Learning Research. 9: 1981–2014. ISSN 1533-7928. PMC 3119541. PMID 21701698. Wikidata Q35058357. Eric P. Xing; Michael I. Jordan; Richard M. Karp (28 June 2001), Feature selection for high-dimensional genomic microarray data, vol. 18, pp. 601–608, Wikidata Q138678867 Xing EP; Karp RM (1 January 2001). "CLIFF: clustering of high-dimensional microarray data via iterative feature filtering using normalized cuts". Bioinformatics. 17 Suppl 1: S306-15. doi:10.1093/BIOINFORMATICS/17.SUPPL_1.S306. ISSN 1367-4803. PMID 11473022. Wikidata Q30657299.
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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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Top 10 AI Chatbots Compared (2026)

Shopping for the best AI chatbot? An AI chatbot 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 chatbot slots into your workflow and pays for itself fast. Below we compare features, pricing, and real output so you can choose with confidence.
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Interacting particle system

In probability theory, an interacting particle system (IPS) is a stochastic process ( X ( t ) ) t ∈ R + {\displaystyle (X(t))_{t\in \mathbb {R} ^{+}}} on some configuration space Ω = S G {\displaystyle \Omega =S^{G}} given by a site space, a countably-infinite-order graph G {\displaystyle G} and a local state space, a compact metric space S {\displaystyle S} . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata. Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model. IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates c Λ ( η , ξ ) > 0 {\displaystyle c_{\Lambda }(\eta ,\xi )>0} where Λ ⊂ G {\displaystyle \Lambda \subset G} is a finite set of sites and η , ξ ∈ Ω {\displaystyle \eta ,\xi \in \Omega } with η i = ξ i {\displaystyle \eta _{i}=\xi _{i}} for all i ∉ Λ {\displaystyle i\notin \Lambda } . The rates describe exponential waiting times of the process to jump from configuration η {\displaystyle \eta } into configuration ξ {\displaystyle \xi } . More generally the transition rates are given in form of a finite measure c Λ ( η , d ξ ) {\displaystyle c_{\Lambda }(\eta ,d\xi )} on S Λ {\displaystyle S^{\Lambda }} . The generator L {\displaystyle L} of an IPS has the following form. First, the domain of L {\displaystyle L} is a subset of the space of "observables", that is, the set of real valued continuous functions on the configuration space Ω {\displaystyle \Omega } . Then for any observable f {\displaystyle f} in the domain of L {\displaystyle L} , one has L f ( η ) = ∑ Λ ∫ ξ : ξ Λ c = η Λ c c Λ ( η , d ξ ) [ f ( ξ ) − f ( η ) ] {\displaystyle Lf(\eta )=\sum _{\Lambda }\int _{\xi :\xi _{\Lambda ^{c}}=\eta _{\Lambda ^{c}}}c_{\Lambda }(\eta ,d\xi )[f(\xi )-f(\eta )]} . For example, for the stochastic Ising model we have G = Z d {\displaystyle G=\mathbb {Z} ^{d}} , S = { − 1 , + 1 } {\displaystyle S=\{-1,+1\}} , c Λ = 0 {\displaystyle c_{\Lambda }=0} if Λ ≠ { i } {\displaystyle \Lambda \neq \{i\}} for some i ∈ G {\displaystyle i\in G} and c i ( η , η i ) = exp ⁡ [ − β ∑ j : | j − i | = 1 η i η j ] {\displaystyle c_{i}(\eta ,\eta ^{i})=\exp[-\beta \sum _{j:|j-i|=1}\eta _{i}\eta _{j}]} where η i {\displaystyle \eta ^{i}} is the configuration equal to η {\displaystyle \eta } except it is flipped at site i {\displaystyle i} . β {\displaystyle \beta } is a new parameter modeling the inverse temperature. == The Voter model == The voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process η ( x ) {\displaystyle \eta (x)} is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform. === Discrete time process === In the discrete time voter model in one dimension, ξ t ( x ) : Z → { 0 , 1 } {\displaystyle \xi _{t}(x):\mathbb {Z} \to \{0,1\}} represents the state of particle x {\displaystyle x} at time t {\displaystyle t} . Informally each individual is arranged on a line and can "see" other individuals that are within a radius, r {\displaystyle r} . If more than a certain proportion, θ {\displaystyle \theta } of these people disagree then the individual changes her attitude, otherwise she keeps it the same. Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value θ c {\displaystyle \theta _{c}} such that if θ > θ c {\displaystyle \theta >\theta _{c}} most individuals never change, and for θ ∈ ( 1 / 2 , θ c ) {\displaystyle \theta \in (1/2,\theta _{c})} in the limit most sites agree. (Both of these results assume the probability of ξ 0 ( x ) = 1 {\displaystyle \xi _{0}(x)=1} is one half.) This process has a natural generalization to more dimensions, some results for this are discussed in Durrett and Steif (1993). === Continuous time process === The continuous time process is similar in that it imagines each individual has a belief at a time and changes it based on the attitudes of its neighbors. The process is described informally by Liggett (1985, 226), "Periodically (i.e., at independent exponential times), an individual reassesses his view in a rather simple way: he chooses a 'friend' at random with certain probabilities and adopts his position." A model was constructed with this interpretation by Holley and Liggett (1975). This process is equivalent to a process first suggested by Clifford and Sudbury (1973) where animals are in conflict over territory and are equally matched. A site is selected to be invaded by a neighbor at a given time.
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