AI Video Tools

Explore the best AI Video Tools — independent reviews, comparisons, pricing and step-by-step how-to guides, curated by Aizhi.

  • Gas (app)

    Gas (app)

    Gas (sometimes stylized in all caps), formerly known as Melt as well as Crush, was an American anonymous social media app. Launched in August 2022, the app is oriented towards high schoolers. The app was developed by Nikita Bier, Isaiah Turner, and former Facebook engineer Dave Schatz. Gas was largely based upon the prior tbh app developed by co-founder Nikita Bier, along with Erik Hazzard, Kyle Zaragoza, and Nicolas Ducdodon in September 2017. tbh was acquired by Facebook inc. (now Meta Platforms) on October 16, 2017, and nearly a year later in July 2018 was dissolved, owing to low usage. Gas follows a similar purpose to tbh in being a social media app oriented towards high schoolers. In the app, users participate in anonymous polls regarding pre-written complimentary statements to their peers, such as "I'd say yes if (blank) asked me out on a date," "I think (blank) is the coolest kid in school," or "would make an ugly face and still look pretty." Winners of said polls receive a "flame." The name of the app is derived from this, with "gassing someone up" being Gen Z slang for complimenting someone. Users can pay a $6.99 subscription that enables "God Mode," which shows hints regarding who voted for them in a poll. Gas overtook TikTok and BeReal as the most downloaded app on the Apple App Store in October 2022 (the app is currently not available for Android). The app has over 5.1 million downloads as of early November 2022, over a million active users and 300 thousand daily downloads as of October 2022. Currently, the app is available in Canada and the majority of the United States. On January 17, 2023, Gas was acquired by Discord, however it would remain a standalone app and its developers became Discord staff members. On October 18, 2023, Discord announced that service for Gas would be permanently ending effective November 7, 2023, due to a steep decline in users. Effective November 7, the app became completely unusable. == Controversy regarding human-trafficking == Beginning in October 2022, rumors spread largely throughout TikTok and Snapchat alleged that the app was linked to human trafficking (in particular sex trafficking). According to Bier, the rumor originated with a single user review from China on October 5, and then was disseminated through TikTok accounts with "few to no US teen followers." Although largely dismissed as a hoax by experts, who cite how the app doesn't log user locations and general anonymity, the hoax became pervasive to the extent that various police departments, school systems, and local news outlets began issuing warnings regarding the app. For instance, on October 31, 2022, the police department of Piedmont, Oklahoma issued a warning to parents, encouraging them to check their children's phones, while on November 3, the Oklahoma Oktaha Public School system stated in a Facebook post that "Children are being kidnapped in other towns and this new app is thought to be the source of predators finding their location." (both statements have since been retracted by Police Chief Scott Singer and Superintendent Jerry Needham respectively). Additionally, local medial outlets such as KOCO in Oklahoma City ran stories making similar statements. The rumor had a negative impact on the app, with downloads plateauing for a two-week period in late October and with 3% of users in a single day reportedly uninstalling the app. Revenue and ratings have also reportedly dropped and the company's social media accounts have been bombarded with comments labeling them as sex-traffickers. Additionally, the four-person development team has reportedly been bombarded with various death threats as a result.

    Read more →
  • Evolvability (computer science)

    Evolvability (computer science)

    The term evolvability is a framework of computational learning introduced by Leslie Valiant in his paper of the same name. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries. == General framework == Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely approximates f {\displaystyle f\,} . This closeness is measured by the performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} of r {\displaystyle r\,} with respect to f {\displaystyle f\,} . As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some r , r ′ ∈ R n {\displaystyle r,r'\in R_{n}} , with r ≠ r ′ {\displaystyle r\neq r'\,} , still r ( x ) = r ′ ( x ) {\displaystyle r(x)=r'(x)\,} for all x ∈ X n {\displaystyle x\in X_{n}} . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the set of possible mutations of r {\displaystyle r\,} . For simplicity, consider Boolean functions on X n = { − 1 , 1 } n {\displaystyle X_{n}=\{-1,1\}^{n}\,} , and let D n {\displaystyle D_{n}\,} be a probability distribution on X n {\displaystyle X_{n}\,} . Define the performance in terms of this. Specifically, Perf ⁡ ( f , r ) = ∑ x ∈ X n f ( x ) r ( x ) D n ( x ) . {\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ⁡ ( f , r ) = Prob ⁡ ( f ( x ) = r ( x ) ) − Prob ⁡ ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship. Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by Perf s ⁡ ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\displaystyle D_{n}\,} . If s {\displaystyle s\,} is large enough, evidently Perf s ⁡ ( f , r ) {\displaystyle \operatorname {Perf} _{s}(f,r)} will be close to the actual performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} . Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , initial representation r ∈ R n {\displaystyle r\in R_{n}} , sample size s {\displaystyle s\,} , and tolerance t {\displaystyle t\,} , the mutator Mut ⁡ ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is a random variable defined as follows. Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)\geq t} ; r ′ {\displaystyle r'\,} is a neutral mutation if − t < Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) < t {\displaystyle -t<\operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r) 0 {\displaystyle \epsilon >0\,} , for all ideal functions f ∈ F n {\displaystyle f\in F_{n}} and representations r 0 ∈ R n {\displaystyle r_{0}\in R_{n}} , with probability at least 1 − ϵ {\displaystyle 1-\epsilon \,} , Perf ⁡ ( f , r g ( n , 1 / ϵ ) ) ≥ 1 − ϵ , {\displaystyle \operatorname {Perf} (f,r_{g(n,1/\epsilon )})\geq 1-\epsilon ,} where the sizes of neighborhoods N ( r ) {\displaystyle N(r)\,} for r ∈ R n {\displaystyle r\in R_{n}\,} are at most p ( n , 1 / ϵ ) {\displaystyle p(n,1/\epsilon )\,} , the sample size is s ( n , 1 / ϵ ) {\displaystyle s(n,1/\epsilon )\,} , the tolerance is t ( 1 / n , ϵ ) {\displaystyle t(1/n,\epsilon )\,} , and the generation size is g ( n , 1 / ϵ ) {\displaystyle g(n,1/\epsilon )\,} . F {\displaystyle F\,} is evolvable over D {\displaystyle D\,} if it is evolvable by some R {\displaystyle R\,} over D {\displaystyle D\,} . F {\displaystyle F\,} is evolvable if it is evolvable over all distributions D {\displaystyle D\,} . == Results == The class of conjunctions and the class of disjunctions are evolvable over the uniform distribution for short conjunctions and disjunctions, respectively. The class of parity functions (which evaluate to the parity of the number of true literals in a given subset of literals) are not evolvable, even for the uniform distribution. Evolvability implies PAC learnability.

    Read more →
  • CrewAI

    CrewAI

    CrewAI is an open-source software framework and platform for building AI agents and multi-agent systems. Written primarily in Python, it is used to define artificial-intelligence agents, assign tasks to them, and coordinate their work through agent teams and workflows. The framework is associated with CrewAI Inc., a startup developing enterprise tools for automating business workflows with large language model-based agents. == History == CrewAI was first released on the Python Package Index in December 2023. The project was created by João Moura and later developed by CrewAI Inc. and open-source contributors. In October 2024, TechCrunch reported that CrewAI had raised $18 million across seed and Series A funding rounds from investors including Boldstart Ventures, Craft Ventures, Earl Grey Capital, and Insight Partners. The report also stated that Andrew Ng and HubSpot co-founder Dharmesh Shah had invested in the company. SiliconANGLE described the company as the developer of an open-source framework for building artificial-intelligence agents and reported that the funding consisted of a seed round led by Boldstart Ventures and a Series A led by Insight Partners. By late 2024, CrewAI had introduced commercial enterprise products built on top of its open-source components. TechCrunch reported that the company's enterprise offering added access controls, analytics, support, and templates for workflow automation. == Features == CrewAI is designed around groups of agents, sometimes called "crews", that can be assigned roles, goals, and tasks. The framework supports agent collaboration, task delegation, tool use, memory, and knowledge sources for retrieval-augmented generation workflows. The project describes two main building blocks: "Crews", which are used for autonomous agent collaboration, and "Flows", which are used for more controlled event-driven workflows. The framework is independent of LangChain and is released under the MIT License. It can be installed as a Python package and is commonly used with external large language model APIs or local models, depending on the developer's configuration. == Business model == CrewAI combines an open-source framework with commercial enterprise products. Its enterprise products are intended for organizations that need to build, monitor, and manage agent-based automations with additional security, observability, and administrative controls.

    Read more →
  • AI Security Institute

    AI Security Institute

    The AI Security Institute (AISI) is a research organisation under the Department for Science, Innovation and Technology, UK, that aims "to equip governments with a scientific understanding of the risks posed by advanced AI". It conducts research and develop and test mitigations. Previously, it was known as the AI Safety Institute. Its creation followed world's first major AI Safety Summit that was held in Bletchley Park in 2023. The institute's professed goal is "building the world's leading understanding of advanced AI risks and solutions, to inform governments so they can keep the public safe". It is designed like a startup in the government "combining the authority of government with the expertise and agility of the private sector". AISI has made access agreements with Anthropic, Google and OpenAI to test their models before release. It has an open source platform called Inspect that permits companies, governments and academics to run standardised safety tests for AI usage. Among the works AISI has done is the reported detection of multiple serious vulnerabilities that could enable development of biological weapons; the vulnerabilities were fixed before the model was launched. It conducts research on diverse fields of AI application. One study by AISI found that LLMs post-trained for political persuasiveness became systematically less accurate and up to 51% more persuasive on political issues. AISI has also worked on the usage of AI for emotional needs. It found that nearly 10 percent of UK citizens used systems like chatbots for emotional purposes on a weekly basis. It found that "systems are now outperforming PhD-level researchers on scientific knowledge tests and helping non-experts succeed at lab work that would previously have been out of reach" in a report published in December 2025. Former chief AI officer of GCHQ Adam Beaumont is the institution's interim director. UK prime minister's AI advisor Jade Leung is the chief technology officer.

    Read more →
  • Drops (app)

    Drops (app)

    Drops is a language learning app that was created in Estonia by Daniel Farkas and Mark Szulyovszky in 2015. It is the second product from the company, after their first app, LearnInvisible, had issues in retaining a user's engagement over the required time period. The languages available include Native Hawaiian and Māori, and was classified as one of the fifty "Most Innovative Companies" for 2019 by Fast Company. The company partnered with Global Eagle Entertainment to include Travel Talk, a feature intended to focus on words and phrases frequently used by travelers. At the beginning of the COVID-19 pandemic in March 2020, the number of users increased by 55 percent in the United States and 92 percent in the United Kingdom. Droplets, a language app for children, includes profiles for multiple teachers working with remote students. The company also produces an app called Scripts, intended to help users learn to write alphabets. The app was purchased by the Norwegian company Kahoot! on 24 November 2020.

    Read more →
  • Exploration–exploitation dilemma

    Exploration–exploitation dilemma

    The exploration–exploitation dilemma, also known as the explore–exploit tradeoff, is a fundamental concept in decision-making that arises in many domains. It is depicted as the balancing act between two opposing strategies. Exploitation involves choosing the best option based on current knowledge of the system (which may be incomplete or misleading), while exploration involves trying out new options that may lead to better outcomes in the future at the expense of an exploitation opportunity. Finding the optimal balance between these two strategies is a crucial challenge in many decision-making problems whose goal is to maximize long-term benefits. == Application in machine learning == In the context of machine learning, the exploration–exploitation tradeoff is fundamental in reinforcement learning (RL), a type of machine learning that involves training agents to make decisions based on feedback from the environment. Crucially, this feedback may be incomplete or delayed. The agent must decide whether to exploit the current best-known policy or explore new policies to improve its performance. === Multi-armed bandit methods === The multi-armed bandit (MAB) problem was a classic example of the tradeoff, and many methods were developed for it, such as epsilon-greedy, Thompson sampling, and the upper confidence bound (UCB). See the page on MAB for details. In more complex RL situations than the MAB problem, the agent can treat each choice as a MAB, where the payoff is the expected future reward. For example, if the agent performs an epsilon-greedy method, then the agent will often "pull the best lever" by picking the action that had the best predicted expected reward (exploit). However, it would pick a random action with probability epsilon (explore). Monte Carlo tree search, for example, uses a variant of the UCB method. === Exploration problems === There are some problems that make exploration difficult. Sparse reward. If rewards occur only once a long while, then the agent might not persist in exploring. Furthermore, if the space of actions is large, then the sparse reward would mean the agent would not be guided by the reward to find a good direction for deeper exploration. A standard example is Montezuma's Revenge. Deceptive reward. If some early actions give immediate small reward, but other actions give later large reward, then the agent might be lured away from exploring the other actions. Noisy TV problem. If certain observations are irreducibly noisy (such as a television showing random images), then the agent might be trapped exploring those observations (watching the television). === Exploration reward === This section based on. The exploration reward (also called exploration bonus) methods convert the exploration-exploitation dilemma into a balance of exploitations. That is, instead of trying to get the agent to balance exploration and exploitation, exploration is simply treated as another form of exploitation, and the agent simply attempts to maximize the sum of rewards from exploration and exploitation. The exploration reward can be treated as a form of intrinsic reward. We write these as r t i , r t e {\displaystyle r_{t}^{i},r_{t}^{e}} , meaning the intrinsic and extrinsic rewards at time step t {\displaystyle t} . However, exploration reward is different from exploitation in two regards: The reward of exploitation is not freely chosen, but given by the environment, but the reward of exploration may be picked freely. Indeed, there are many different ways to design r t i {\displaystyle r_{t}^{i}} described below. The reward of exploitation is usually stationary (i.e. the same action in the same state gives the same reward), but the reward of exploration is non-stationary (i.e. the same action in the same state should give less and less reward). Count-based exploration uses N n ( s ) {\displaystyle N_{n}(s)} , the number of visits to a state s {\displaystyle s} during the time-steps 1 : n {\displaystyle 1:n} , to calculate the exploration reward. This is only possible in small and discrete state space. Density-based exploration extends count-based exploration by using a density model ρ n ( s ) {\displaystyle \rho _{n}(s)} . The idea is that, if a state has been visited, then nearby states are also partly-visited. In maximum entropy exploration, the entropy of the agent's policy π {\displaystyle \pi } is included as a term in the intrinsic reward. That is, r t i = − ∑ a π ( a | s t ) ln ⁡ π ( a | s t ) + ⋯ {\displaystyle r_{t}^{i}=-\sum _{a}\pi (a|s_{t})\ln \pi (a|s_{t})+\cdots } . === Prediction-based === This section based on. The forward dynamics model is a function for predicting the next state based on the current state and the current action: f : ( s t , a t ) ↦ s t + 1 {\displaystyle f:(s_{t},a_{t})\mapsto s_{t+1}} . The forward dynamics model is trained as the agent plays. The model becomes better at predicting state transition for state-action pairs that had been done many times. A forward dynamics model can define an exploration reward by r t i = ‖ f ( s t , a t ) − s t + 1 ‖ 2 2 {\displaystyle r_{t}^{i}=\|f(s_{t},a_{t})-s_{t+1}\|_{2}^{2}} . That is, the reward is the squared-error of the prediction compared to reality. This rewards the agent to perform state-action pairs that had not been done many times. This is however susceptible to the noisy TV problem. Dynamics model can be run in latent space. That is, r t i = ‖ f ( s t , a t ) − ϕ ( s t + 1 ) ‖ 2 2 {\displaystyle r_{t}^{i}=\|f(s_{t},a_{t})-\phi (s_{t+1})\|_{2}^{2}} for some featurizer ϕ {\displaystyle \phi } . The featurizer can be the identity function (i.e. ϕ ( x ) = x {\displaystyle \phi (x)=x} ), randomly generated, the encoder-half of a variational autoencoder, etc. A good featurizer improves forward dynamics exploration. The Intrinsic Curiosity Module (ICM) method trains simultaneously a forward dynamics model and a featurizer. The featurizer is trained by an inverse dynamics model, which is a function for predicting the current action based on the features of the current and the next state: g : ( ϕ ( s t ) , ϕ ( s t + 1 ) ) ↦ a t {\displaystyle g:(\phi (s_{t}),\phi (s_{t+1}))\mapsto a_{t}} . By optimizing the inverse dynamics, both the inverse dynamics model and the featurizer are improved. Then, the improved featurizer improves the forward dynamics model, which improves the exploration of the agent. Random Network Distillation (RND) method attempts to solve this problem by teacher–student distillation. Instead of a forward dynamics model, it has two models f , f ′ {\displaystyle f,f'} . The f ′ {\displaystyle f'} teacher model is fixed, and the f {\displaystyle f} student model is trained to minimize ‖ f ( s ) − f ′ ( s ) ‖ 2 2 {\displaystyle \|f(s)-f'(s)\|_{2}^{2}} on states s {\displaystyle s} . As a state is visited more and more, the student network becomes better at predicting the teacher. Meanwhile, the prediction error is also an exploration reward for the agent, and so the agent learns to perform actions that result in higher prediction error. Thus, we have a student network attempting to minimize the prediction error, while the agent attempting to maximize it, resulting in exploration. The states are normalized by subtracting a running average and dividing a running variance, which is necessary since the teacher model is frozen. The rewards are normalized by dividing with a running variance. Exploration by disagreement trains an ensemble of forward dynamics models, each on a random subset of all ( s t , a t , s t + 1 ) {\displaystyle (s_{t},a_{t},s_{t+1})} tuples. The exploration reward is the variance of the models' predictions. === Noise === For neural network–based agents, the NoisyNet method changes some of its neural network modules by noisy versions. That is, some network parameters are random variables from a probability distribution. The parameters of the distribution are themselves learnable. For example, in a linear layer y = W x + b {\displaystyle y=Wx+b} , both W , b {\displaystyle W,b} are sampled from Gaussian distributions N ( μ W , Σ W ) , N ( μ b , Σ b ) {\displaystyle {\mathcal {N}}(\mu _{W},\Sigma _{W}),{\mathcal {N}}(\mu _{b},\Sigma _{b})} at every step, and the parameters μ W , Σ W , μ b , Σ b {\displaystyle \mu _{W},\Sigma _{W},\mu _{b},\Sigma _{b}} are learned via the reparameterization trick.

    Read more →
  • Knowledge integration

    Knowledge integration

    Knowledge integration is the process of synthesizing multiple knowledge models (or representations) into a common model (representation). Compared to information integration, which involves merging information having different schemas and representation models, knowledge integration focuses more on synthesizing the understanding of a given subject from different perspectives. For example, multiple interpretations are possible of a set of student grades, typically each from a certain perspective. An overall, integrated view and understanding of this information can be achieved if these interpretations can be put under a common model, say, a student performance index. The Web-based Inquiry Science Environment (WISE), from the University of California at Berkeley has been developed along the lines of knowledge integration theory. Knowledge integration has also been studied as the process of incorporating new information into a body of existing knowledge with an interdisciplinary approach. This process involves determining how the new information and the existing knowledge interact, how existing knowledge should be modified to accommodate the new information, and how the new information should be modified in light of the existing knowledge. A learning agent that actively investigates the consequences of new information can detect and exploit a variety of learning opportunities; e.g., to resolve knowledge conflicts and to fill knowledge gaps. By exploiting these learning opportunities the learning agent is able to learn beyond the explicit content of the new information. The machine learning program KI, developed by Murray and Porter at the University of Texas at Austin, was created to study the use of automated and semi-automated knowledge integration to assist knowledge engineers constructing a large knowledge base. A possible technique which can be used is semantic matching. More recently, a technique useful to minimize the effort in mapping validation and visualization has been presented which is based on Minimal Mappings. Minimal mappings are high quality mappings such that i) all the other mappings can be computed from them in time linear in the size of the input graphs, and ii) none of them can be dropped without losing property i). The University of Waterloo operates a Bachelor of Knowledge Integration undergraduate degree program as an academic major or minor. The program started in 2008.

    Read more →
  • Alexander Y. Tetelbaum

    Alexander Y. Tetelbaum

    Alexander Y. Tetelbaum (born August 16, 1948) is a Ukrainian American computer scientist, inventor, and academic who has contributed to electronic design automation (EDA) and artificial intelligence (AI) since the late 1960s; and holds 46 U.S. patents in EDA and related fields. Tetelbaum is the founding president of International Solomon University, the first Jewish university in Ukraine, established during a period of renewed efforts to address antisemitism in Ukraine. == Early life and education == He graduated from a Kyiv mathematical high school with a silver medal in 1966. Tetelbaum enrolled at the Kyiv Polytechnic Institute (KPI), now National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" in 1966, graduating in 1972 with an MS in Electronics with honors. He earned his PhD in Electrical and Computer Engineering from KPI in 1975, with a dissertation on electronic design automation, and his Doctor of Engineering Science in 1986. == Academic career == Tetelbaum began his academic career at KPI in 1973 as a junior scientist, becoming a professor in the Computer and Electrical Engineering Department in 1980. Later, he founded and served as president of International Solomon University in Kyiv from 1991 to 1996, the first Jewish university in Ukraine. The university became a major academic center for computer science and Jewish studies in the post-Soviet era. He was a visiting and adjunct professor at Michigan State University from 1993 to 1996. == Professional career == Tetelbaum worked as an engineer at the Kiev Institute of Cybernetics from 1972 to 1973, and later, he led the Design Automation Lab at Kyiv Polytechnic Institute from 1975 to 1987. In the United States, he served as EDA manager at Silicon Graphics Corporation from 1996 to 1998 and principal engineer at LSI Corporation from 1998 to 2012. He founded and served as CEO of Abelite Design Automation, Inc., from 2012 to 2022. == Contributions in computer science == Tetelbaum has contributed to electronic design automation (EDA) and artificial intelligence (AI) since the 1960s. His early work included methods for EDA, particularly physical design automation and mathematical optimization; and he developed force-directed placement and topological routing methods. Tetelbaum generalized Rent's rule for hierarchical systems and large blocks, proposing a graph-based framework that extends applicability to arbitrary partition sizes with improved accuracy. Additional IEEE and related conference contributions from the mid-1990s include: "Path Search for Complicated Function", 1995 IEEE International Symposium on Circuits and Systems "A Performance-driven Placement Approach of Standard Cells" (International Conference on Intelligent Systems, 1995) "Framework of a New Methodology for Behavioral to Physical Design Linkage" (38th Midwest Symposium on Circuits and Systems, 1996) Statistical timing design and variations Test Methodologies These and other works and patents contributed to timing-driven placement, crosstalk reduction, clock tree synthesis, and interconnect optimization in VLSI design. == Patents == Tetelbaum holds 46 U.S. patents in EDA and related fields. Notable examples include: For the full list of patents, see Justia Patents or Google Patents. == Publications == === Early publications in the Soviet Union === Before the appearance of American books on electronic design automation (EDA), Tetelbaum published several scientific books and monographs on the subject in Russian/Ukrainian. Electronic Design Automation, Kiev: Znanie Publisher, 1975. Planar Design of Electronic Circuits, Kiev: Znanie Publisher, 1977. Formal Design of Computer Systems, Moscow: Sovetskoe Radio, 1979. CAD of Electronic Equipment: Topological Approach, Kiev: Vyssha Shkola, 1980; 2nd ed. 1981. Automated Design of Electronic Circuits (1981) CAD of VLSI Circuits, Kiev: Vyssha Shkola, 1983. Topological Algorithms of Multilayer Printed Circuit Boards Routing, Moscow: Radio i Svyaz, 1983. CAD of VLSI Circuits on Master Slice Chips, Moscow: Radio i Svyaz, 1988. Increasing the Effectiveness of CAD Systems, Kiev: UMKVO, 1991. === Scientific Monographs (English) === Minimum Number of Timing Signoff Corners (2022) Interviewing AI (2026) The AI Debate (2026) New Nostradamus Predictions: 2026: The Next Decade & Beyond (2035–2050+) (2026) For a consolidated record of Tetelbaum's publications, see Alexander Y. Tetelbaum, Wikidata Q4720205. === Other publications === Tetelbaum also published educational books on problem-solving methods: Yes-No Puzzles-Games Puzzle Games for Kids Solving Non-Standard Problems Solving Non-Standard Very Hard Problems Additionally, Tetelbaum published three thrillers: Omerta Operations Executive Director Eruption Yacht Finally, he published his memoir and an entertaining book: Unfinished Equations Artificially Intelligent Humor

    Read more →
  • Automated essay scoring

    Automated essay scoring

    Automated essay scoring (AES) is the use of specialized computer programs to assign grades to essays written in an educational setting. It is a form of educational assessment and an application of natural language processing. Its objective is to classify a large set of textual entities into a small number of discrete categories, corresponding to the possible grades, for example, the numbers 1 to 6. Therefore, it can be considered a problem of statistical classification. Several factors have contributed to a growing interest in AES. Among them are cost, accountability, standards, and technology. Rising education costs have led to pressure to hold the educational system accountable for results by imposing standards. The advance of information technology promises to measure educational achievement at reduced cost. The use of AES for high-stakes testing in education has generated significant backlash, with opponents pointing to research that computers cannot yet grade writing accurately and arguing that their use for such purposes promotes teaching writing in reductive ways (i.e. teaching to the test). == History == Most historical summaries of AES trace the origins of the field to the work of Ellis Batten Page. In 1966, he argued for the possibility of scoring essays by computer, and in 1968 he published his successful work with a program called Project Essay Grade (PEG). Using the technology of that time, computerized essay scoring would not have been cost-effective, so Page abated his efforts for about two decades. Eventually, Page sold PEG to Measurement Incorporated. By 1990, desktop computers had become so powerful and so widespread that AES was a practical possibility. As early as 1982, a UNIX program called Writer's Workbench was able to offer punctuation, spelling and grammar advice. In collaboration with several companies (notably Educational Testing Service), Page updated PEG and ran some successful trials in the early 1990s. Peter Foltz and Thomas Landauer developed a system using a scoring engine called the Intelligent Essay Assessor (IEA). IEA was first used to score essays in 1997 for their undergraduate courses. It is now a product from Pearson Educational Technologies and used for scoring within a number of commercial products and state and national exams. IntelliMetric is Vantage Learning's AES engine. Its development began in 1996. It was first used commercially to score essays in 1998. Educational Testing Service offers "e-rater", an automated essay scoring program. It was first used commercially in February 1999. Jill Burstein was the team leader in its development. ETS's Criterion Online Writing Evaluation Service uses the e-rater engine to provide both scores and targeted feedback. Lawrence Rudner has done some work with Bayesian scoring, and developed a system called BETSY (Bayesian Essay Test Scoring sYstem). Some of his results have been published in print or online, but no commercial system incorporates BETSY as yet. Under the leadership of Howard Mitzel and Sue Lottridge, Pacific Metrics developed a constructed response automated scoring engine, CRASE. Currently utilized by several state departments of education and in a U.S. Department of Education-funded Enhanced Assessment Grant, Pacific Metrics’ technology has been used in large-scale formative and summative assessment environments since 2007. Measurement Inc. acquired the rights to PEG in 2002 and has continued to develop it. In 2012, the Hewlett Foundation sponsored a competition on Kaggle called the Automated Student Assessment Prize (ASAP). 201 challenge participants attempted to predict, using AES, the scores that human raters would give to thousands of essays written to eight different prompts. The intent was to demonstrate that AES can be as reliable as human raters, or more so. The competition also hosted a separate demonstration among nine AES vendors on a subset of the ASAP data. Although the investigators reported that the automated essay scoring was as reliable as human scoring, this claim was not substantiated by any statistical tests because some of the vendors required that no such tests be performed as a precondition for their participation. Moreover, the claim that the Hewlett Study demonstrated that AES can be as reliable as human raters has since been strongly contested, including by Randy E. Bennett, the Norman O. Frederiksen Chair in Assessment Innovation at the Educational Testing Service. Some of the major criticisms of the study have been that five of the eight datasets consisted of paragraphs rather than essays, four of the eight data sets were graded by human readers for content only rather than for writing ability, and that rather than measuring human readers and the AES machines against the "true score", the average of the two readers' scores, the study employed an artificial construct, the "resolved score", which in four datasets consisted of the higher of the two human scores if there was a disagreement. This last practice, in particular, gave the machines an unfair advantage by allowing them to round up for these datasets. In 1966, Page hypothesized that, in the future, the computer-based judge will be better correlated with each human judge than the other human judges are. Despite criticizing the applicability of this approach to essay marking in general, this hypothesis was supported for marking free text answers to short questions, such as those typical of the British GCSE system. Results of supervised learning demonstrate that the automatic systems perform well when marking by different human teachers is in good agreement. Unsupervised clustering of answers showed that excellent papers and weak papers formed well-defined clusters, and the automated marking rule for these clusters worked well, whereas marks given by human teachers for the third cluster ('mixed') can be controversial, and the reliability of any assessment of works from the 'mixed' cluster can often be questioned (both human and computer-based). == Different dimensions of essay quality == According to a recent survey, modern AES systems try to score different dimensions of an essay's quality in order to provide feedback to users. These dimensions include the following items: Grammaticality: following grammar rules Usage: using of prepositions, word usage Mechanics: following rules for spelling, punctuation, capitalization Style: word choice, sentence structure variety Relevance: how relevant of the content to the prompt Organization: how well the essay is structured Development: development of ideas with examples Cohesion: appropriate use of transition phrases Coherence: appropriate transitions between ideas Thesis Clarity: clarity of the thesis Persuasiveness: convincingness of the major argument == Procedure == From the beginning, the basic procedure for AES has been to start with a training set of essays that have been carefully hand-scored. The program evaluates surface features of the text of each essay, such as the total number of words, the number of subordinate clauses, or the ratio of uppercase to lowercase letters—quantities that can be measured without any human insight. It then constructs a mathematical model that relates these quantities to the scores that the essays received. The same model is then applied to calculate scores of new essays. Recently, one such mathematical model was created by Isaac Persing and Vincent Ng. which not only evaluates essays on the above features, but also on their argument strength. It evaluates various features of the essay, such as the agreement level of the author and reasons for the same, adherence to the prompt's topic, locations of argument components (major claim, claim, premise), errors in the arguments, cohesion in the arguments among various other features. In contrast to the other models mentioned above, this model is closer in duplicating human insight while grading essays. Due to the growing popularity of deep neural networks, deep learning approaches have been adopted for automated essay scoring, generally obtaining superior results, often surpassing inter-human agreement levels. The various AES programs differ in what specific surface features they measure, how many essays are required in the training set, and most significantly in the mathematical modeling technique. Early attempts used linear regression. Modern systems may use linear regression or other machine learning techniques often in combination with other statistical techniques such as latent semantic analysis and Bayesian inference. The automated essay scoring task has also been studied in the cross-domain setting using machine learning models, where the models are trained on essays written for one prompt (topic) and tested on essays written for another prompt. Successful approaches in the cross-domain scenario are based on deep neural networks or models that combine deep and shallow features. == Criteria for success == Any method of a

    Read more →
  • Symbol level

    Symbol level

    In knowledge-based systems, agents choose actions based on the principle of rationality to move closer to a desired goal. The agent is able to make decisions based on knowledge it has about the world (see knowledge level). But for the agent to actually change its state, it must use whatever means it has available. This level of description for the agent's behavior is the symbol level. The term was coined by Allen Newell in 1982. For example, in a computer program, the knowledge level consists of the information contained in its data structures that it uses to perform certain actions. The symbol level consists of the program's algorithms, the data structures themselves, and so on.

    Read more →
  • Emergent algorithm

    Emergent algorithm

    An emergent algorithm is an algorithm that exhibits emergent behavior. In essence an emergent algorithm implements a set of simple building block behaviors that when combined exhibit more complex behaviors. One example of this is the implementation of fuzzy motion controllers used to adapt robot movement in response to environmental obstacles. An emergent algorithm has the following characteristics: it achieves predictable global effects it does not require global visibility it does not assume any kind of centralized control it is self-stabilizing Other examples of emergent algorithms and models include cellular automata, artificial neural networks and swarm intelligence systems (ant colony optimization, bees algorithm, etc.).

    Read more →
  • Nouvelle AI

    Nouvelle AI

    Nouvelle artificial intelligence (Nouvelle AI) is an approach to artificial intelligence pioneered in the 1980s by Rodney Brooks, who was then part of MIT artificial intelligence laboratory. Nouvelle AI differs from classical AI by aiming to produce robots with intelligence levels similar to insects. Researchers believe that intelligence can emerge organically from simple behaviors as these intelligences interacted with the "real world", instead of using the constructed worlds which symbolic AIs typically needed to have programmed into them. == Motivation == The differences between nouvelle AI and symbolic AI are apparent in early robots Shakey and Freddy. These robots contained an internal model (or "representation") of their micro-worlds consisting of symbolic descriptions. As a result, this structure of symbols had to be renewed as the robot moved or the world changed. Shakey's planning programs assessed the program structure and broke it down into the necessary steps to complete the desired action. This level of computation required a large amount time to process, so Shakey typically performed its tasks very slowly. Symbolic AI researchers had long been plagued by the problem of updating, searching, and otherwise manipulating the symbolic worlds inside their AIs. A nouvelle system refers continuously to its sensors rather than to an internal model of the world. It processes the external world information it needs from the senses when it is required. As Brooks puts it, "the world is its own best model--always exactly up to date and complete in every detail." A central idea of nouvelle AI is that simple behaviors combine to form more complex behaviors over time. For example, simple behaviors can include elements like "move forward" and "avoid obstacles." A robot using nouvelle AI with simple behaviors like collision avoidance and moving toward a moving object could possibly come together to produce a more complex behavior like chasing a moving object. === The frame problem === The frame problem describes an issue with using first-order logic (FOL) to express facts about a robot in the world. Representing the state of a robot with traditional FOL requires the use of many axioms (symbolic language) to imply that things about an environment do not change arbitrarily. Nouvelle AI seeks to sidestep the frame problem by dispensing with filling the AI or robot with volumes of symbolic language and instead letting more complex behaviors emerge by combining simpler behavioral elements. === Embodiment === The goal of traditional AI was to build intelligences without bodies, which would only have been able to interact with the world via keyboard, screen, or printer. However, nouvelle AI attempts to build embodied intelligence situated in the real world. Brooks quotes approvingly from the brief sketches that Turing gave in 1948 and 1950 of the "situated" approach. Turing wrote of equipping a machine "with the best sense organs that money can buy" and teaching it "to understand and speak English" by a process that would "follow the normal teaching of a child." This approach was contrasted to the others where they focused on abstract activities such as playing chess. == Brooks' robots == === Insectoid robots === Brooks focused on building robots that acted like simple insects while simultaneously working to remove some traditional AI characteristics. He created insect-like robots, named Allen and Herbert after cognitive science and AI pioneers Allen Newell and Herbert A. Simon. Brooks's insectoid robots contained no internal models of the world. Herbert, for example, discarded a high volume of the information received from its sensors and never stored information for more than two seconds. ==== Allen ==== Allen had a ring of twelve ultrasonic sonars as its primary sensors and three independent behavior-producing modules. These modules were programmed to avoid both stationary and moving objects. With only this module activated, Allen stayed in the middle of a room until an object approached and then it ran away while avoiding obstacles in its way. ==== Herbert ==== Herbert used infrared sensors to avoid obstacles and a laser system to collect 3D data over a distance of about 12 feet. Herbert also carried a number of simple sensors in its "hand." The robot's testing ground was the real world environment of the busy offices and workspaces of the MIT AI lab where it searched for empty soda cans and carried them away, a seemingly goal-oriented activity that emerged as a result of 15 simple behavior units combining. As a parallel, Simon noted that an ant's complicated path is due to the structure of its environment rather than the depth of its thought processes. ==== Other insectoid robots ==== Other robots by Brooks' team were Genghis and Squirt. Genghis had six legs and was able to walk over rough terrain and follow a human. Squirt's behavior modules had it stay in dark corners until it heard a noise, then it would begin to follow the source of the noise. Brooks agreed that the level of nouvelle AI had come near the complexity of a real insect, which raised a question about whether or not insect level-behavior was and is a reasonable goal for nouvelle AI. === Humanoid robots === Brooks' own recent work has taken the opposite direction to that proposed by Von Neumann in the quotations "theorists who select the human nervous system as their model are unrealistically picking 'the most complicated object under the sun,' and that there is little advantage in selecting instead the ant, since any nervous system at all exhibits exceptional complexity." ==== Cog ==== In the 1990s, Brooks decided to pursue the goal of human-level intelligence and, with Lynn Andrea Stein, built a humanoid robot called Cog. Cog is a robot with an extensive collection of sensors, a face, and arms (among other features) that allow it to interact with the world and gather information and experience so as to assemble intelligence organically in the manner described above by Turing. The team believed that Cog would be able to learn and able to find a correlation between the sensory information it received and its actions, and to learn common sense knowledge on its own. As of 2003, all development of the project had ceased.

    Read more →
  • Timeline of artificial intelligence risks in global finance

    Timeline of artificial intelligence risks in global finance

    The following article is a broad timeline of the course of events related to artificial intelligence risks in global finance. The AI boom has led to concerns including the existential risk from artificial intelligence, as the uptake on applications of artificial intelligence increases. By late 2025, global finance and artificial intelligence were "deeply intertwined". A June 2025 Menlo Ventures report raised concerns about the sustainability of future revenue and long-term profitability of AI, given the relatively low rate of consumer monetization. == 2017 == 30 NovemberThe New York Times said that new AI reports by McKinsey & Company, the National Bureau of Economic Research, and an AI Index created by university researchers, indicated an early AI boom. The Index built on a project—"The One Hundred Year Study on Artificial Intelligence" launched in 2014. == 2018 == 2018 was a year of incremental AI growth in finance. == 2022 == The release of ChatGPT by OpenAI became the catalyst for an artificial intelligence boom that continues to remake the global economy. According to a European Central Bank report, public interest in AI increased rapidly as evidenced with rising Google searches, AI jobs, models, patents, and innovations since late 2022. At that time Europe led the US in the size of its AI workforce. == 2023 == The regulatory body, the International Monetary Fund (IMF), published their report, "Generative Artificial Intelligence in Finance: Risk Considerations", drawing attention to oversight gaps and the need for regulations. The report explores the risks posed by using generative artificial intelligence (GenAI) systems in the financial sector including "broader risks to financial stability." == 2024 == January 12 In January 2024 Bloomberg's published its list of the "Magnificent Seven" Big Tech companies on the stock market based on their strength, size and market capitalization:Apple, Microsoft, Alphabet (Google), Amazon, Meta Platforms (Facebook), Nvidia, and Tesla. 21 June During the AI boom, Nvidia became the world's most valuable company, surpassing Microsoft, as its value increased to over US$4 trillion. In 2023 and 2024, the "Magnificent Seven" stocks were the primary drivers behind the increase in equity indexes, according to Reuters. == 2025 == === January === 23 January President Donald Trump's AI policy was announced calling for United States global leadership in artificial intelligence. The Economist noted that this politic shift in which the United States seeks "global dominance" in AI includes trimming regulations and assisting in expansion of infrastructure and increase in number of AI workers. Governments of Gulf nations were also investing trillions of dollars in AI. 27 January Against the backdrop of a tech war between China and the United States over AI dominance, within days of the launch of China's free DeepSeek App, it was the most downloaded app in the United States, rising to the first place in the Apple app store. President Trump responded immediately, saying this "sudden rise" should be a "wake-up" call to the United States, and called on US companies to be more competitive. === June === 26 June In their June 2025 report, Menlo Ventures estimated that only about 3% of consumers paid for artificial intelligence-related services, representing about $USD12 billion in annual spending. This is relatively low in contrast to the massive capital expenditure by AI infrastructure companies, which raises concerns about revenue sustainability and long-term profitability. === July === 23 July The Trump administration launched the US AI Action Plan, positioning the United States in a high-stakes technological race with China for global dominance in artificial intelligence, emphasizing that neither nation can afford to fall behind due to the exponential nature of AI advancement. The plan, a new government website and policy speech called for accelerated AI adoption across federal agencies, and a number of initiatives to make is easier for AI infrastructure expansion, and other measures to ensure American leadership in AI standards. Some leading experts warned that the administration failed to provide sufficient regulations and safeguards for AI safety. Concerns were raised about the negative impacts of cuts to research funding and tightened visa policies for scientists, potentially undermining public trust and America's ability to compete internationally. === September === 7 September The Economist cautioned that AI revenues are relatively modest compared to the high cost and investments in the creation of new data centers. Even Sam Altman, OpenAI CEO and one of the leading figures of the AI boom,, raised concerns about investors' outsized hopes for financial returns. At the same time, history has shown that new technologies, like railways and electricity, endured and spread after the initial hype faded. 12 September Economists warn that U.S. households' direct and indirect investments—mutual funds or retirement plans—in the stock market reached an unprecedented historically high level, now representing 45% of all financial assets, or about $USD51.2 trillion. Compared to the Dot-com bubble this represents a sharp increase in exposure. This makes U.S. households vulnerable to market downturns which in turn would result in decreasing consumer spending. U.S. household net worth rose to a record $176.3 trillion in the second quarter, an increase of $7.3 trillion since early 2025 and about $46 trillion higher than before the pandemic. Federal Reserve data attribute the surge primarily to gains in stock markets and housing values. However, the rise in wealth on paper coincided with increased household borrowing and growing government debt. 18 September Questions were being raised about how quickly the data centers, chips, servers, and GPUs assets of major AI companies will depreciate in value. Comparisons have been made to the Railway Mania in the aftermath of the stock market bubble where a valuable physical infrastructure remained standing, and the telecoms crash after the dot-com bubble which left fiber networks. 28 September There were warnings that record-high American stock ownership during the AI-fueled market boom is a red flag for systemic risk, as the current concentration in equities exceeds levels seen before the dot-com bubble burst in 2000, and could amplify the impact of any future stock market correction. === October === 3 October In 2025 alone, venture capitalists invested almost $USD200 billion in the artificial intelligence sector. 29 October Nvidia was the first company in the world to be valued at US$5 trillion, largely due to AI demand and strategic partnerships with leading technology and AI firms. Nvidia's increase in value was "meteoric". === November === 2 November Forbes reported that, since April, the 'Magnificent Seven' tech giants together contributed over 40% of the S&P 500's return, highlighting their outsized influence and the growing impact of AI on market valuations. CNN warned that while there is a current benefit to investors, with such a high concentration in the S&P 500, they are highly exposed to the fate of the Mag Seven. 2 November Globally there are 11,000 datacentres—huge campuses for AI infrastructure, including thousands of chips, GPUS, and servers. This represents a 500% increase over the last two decades. It is anticipated that $3USDtn more will be spent on increasing that number over the next two or three years. 5 November Concerns about the potential for a market bubble were raised as six of the AI-related Big Tech "Magnificent Seven"—that contribute to the AI boom—reported losing ground in the stock market. Global markets and artificial intelligence have become "deeply intertwined", according to a Reuters report. As of November 2025, more than 50% of the 20 largest S&P firms were deeply exposed to AI. In contrast, in 2000, the 20 S&P 500 firms represented 39% of its total value only 11 of these companies were exposed to the internet. If AI fails to deliver strong returns on their investments, these top S&P firms would be significantly impacted, according to the Economist. Analysts suggest that the AI market in 2025 may not behave like a traditional one, as investors are simultaneously aware of the risks and driven by the potential for outsized rewards. Leading AI labs may believe that the first company to achieve artificial general intelligence (AGI), when an AI system surpasses all human cognitive abilities and becomes capable of self-improvement—could dominate the future of technology and finance. While some have estimated that the potential value of such a breakthrough could be as high as $1.46 quadrillion, this figure is speculative and widely debated. 5 November Bloomberg described Nvidia's H100 Hopper-Blackwell AI chips as the "King of AI chips". Nvidia dominates the AI chip market with over 78% of the market share because of both speed and cost. According to B

    Read more →
  • Grammar systems theory

    Grammar systems theory

    Grammar systems theory is a field of theoretical computer science that studies systems of finite collections of formal grammars generating a formal language. Each grammar works on a string, a so-called sequential form that represents an environment. Grammar systems can thus be used as a formalization of decentralized or distributed systems of agents in artificial intelligence. Let A {\displaystyle \mathbb {A} } be a simple reactive agent moving on the table and trying not to fall down from the table with two reactions, t for turning and ƒ for moving forward. The set of possible behaviors of A {\displaystyle \mathbb {A} } can then be described as formal language L A = { ( f m t n f r ) + : 1 ≤ m ≤ k ; 1 ≤ n ≤ ℓ ; 1 ≤ r ≤ k } , {\displaystyle \mathbb {L_{A}} =\{(f^{m}t^{n}f^{r})^{+}:1\leq m\leq k;1\leq n\leq \ell ;1\leq r\leq k\},} where ƒ can be done maximally k times and t can be done maximally ℓ times considering the dimensions of the table. Let G A {\displaystyle \mathbb {G_{A}} } be a formal grammar which generates language L A {\displaystyle \mathbb {L_{A}} } . The behavior of A {\displaystyle \mathbb {A} } is then described by this grammar. Suppose the A {\displaystyle \mathbb {A} } has a subsumption architecture; each component of this architecture can be then represented as a formal grammar, too, and the final behavior of the agent is then described by this system of grammars. The schema on the right describes such a system of grammars which shares a common string representing an environment. The shared sequential form is sequentially rewritten by each grammar, which can represent either a component or generally an agent. If grammars communicate together and work on a shared sequential form, it is called a Cooperating Distributed (DC) grammar system. Shared sequential form is a similar concept to the blackboard approach in AI, which is inspired by an idea of experts solving some problem together while they share their proposals and ideas on a shared blackboard. Each grammar in a grammar system can also work on its own string and communicate with other grammars in a system by sending their sequential forms on request. Such a grammar system is then called a Parallel Communicating (PC) grammar system. PC and DC are inspired by distributed AI. If there is no communication between grammars, the system is close to the decentralized approaches in AI. These kinds of grammar systems are sometimes called colonies or Eco-Grammar systems, depending (besides others) on whether the environment is changing on its own (Eco-Grammar system) or not (colonies).

    Read more →
  • Pattern theory

    Pattern theory

    Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language. Broad in its mathematical coverage, Pattern Theory spans algebra and statistics, as well as local topological and global entropic properties. In addition to the new algebraic vocabulary, its statistical approach is novel in its aim to: Identify the hidden variables of a data set using real world data rather than artificial stimuli, which was previously commonplace. Formulate prior distributions for hidden variables and models for the observed variables that form the vertices of a Gibbs-like graph. Study the randomness and variability of these graphs. Create the basic classes of stochastic models applied by listing the deformations of the patterns. Synthesize (sample) from the models, not just analyze signals with them. The Brown University Pattern Theory Group was formed in 1972 by Ulf Grenander. Many mathematicians are currently working in this group, noteworthy among them being the Fields Medalist David Mumford. Mumford regards Grenander as his "guru" in Pattern Theory.

    Read more →