AI Email Follow Up

AI Email Follow Up — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

  • SQLBuddy

    SQLBuddy

    SQL Buddy is an open-source web-based application primarily coded in PHP, that allows users to control both MySQL and SQLite database through a web browser. The project was well regarded for its easy installation process and the friendly user interface it offered. The application was further praised for its cross-platform compatibility, meaning users could manage their databases on various operating systems, including Linux, Windows, and macOS. The development of SQL Buddy has stopped, with version 1.3.3 being the final release on January 18, 2011. No further releases are expected.

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  • Sasha Luccioni

    Sasha Luccioni

    Alexandra Sasha Luccioni (née Vorobyova; born 1990) is a computer scientist specializing in the intersection of artificial intelligence (AI) and climate change. Her work focuses on quantifying the environmental impact of AI technologies and promoting sustainable practices in machine learning development. == Early life and education == Alexandra Sasha Vorobyova was born in the Ukrainian Soviet Socialist Republic in 1990. When she was four years old, her family relocated to Ontario, Canada. Her interest in science is influenced by her family's history; her mother, grandmother, and great-grandmother all pursued careers in scientific fields. Luccioni earned a B.A. in language science from University of Paris III: Sorbonne Nouvelle in 2010. Subsequently, she completed a M.S. in cognitive science, with a minor in natural language processing, at École normale supérieure in Paris in 2012. Luccioni obtained her PhD in cognitive computing from Université du Québec à Montréal (UQAM) in 2018. == Career == Luccioni began her professional career at Nuance Communications in 2017, where she focused on natural language processing (NLP) and machine learning (ML) techniques to enhance conversational agents. She then joined Morgan Stanley’s AI/ML Center of Excellence in 2018, working on explainable artificial intelligence (AI) and decision-making systems. In 2019, she became a postdoctoral researcher at Université de Montréal and Mila, collaborating with computer scientist Yoshua Bengio on a project titled This Climate Does Not Exist. This initiative used generative adversarial networks to visualize the effects of climate change. During this time, she also contributed to integrating fairness and accountability into machine learning education at Mila. Luccioni briefly worked with the United Nations Global Pulse in 2021, developing tools to monitor COVID-19 misinformation. Later that year, she joined Hugging Face as a research scientist. Her role includes quantifying the carbon footprint of AI systems, co-chairing the carbon working group in the Big Science project, and advancing responsible machine learning practices. She helped create "CodeCarbon," an open-source software tool that estimates the carbon emissions produced during the training and operation of machine learning models. In addition to her research, she has developed tools to measure the environmental impact of AI models, communicated findings through media engagements, and presented at international conferences, including a TED Talk. In 2024, she was listed on BBC 100 Women and Time 100 AI.

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  • Top 10 AI Headshot Generators Compared (2026)

    Top 10 AI Headshot Generators Compared (2026)

    Trying to pick the best AI headshot generator? An AI headshot generator is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI headshot generator slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  • How to Choose an AI Voice Assistant

    How to Choose an AI Voice Assistant

    Curious about the best AI voice assistant? An AI voice assistant 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 voice assistant slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  • Tesla Dojo

    Tesla Dojo

    Tesla Dojo is a series of supercomputers designed and built by Tesla for computer vision video processing and recognition. It was used for training Tesla's machine learning models to improve its Full Self-Driving (FSD) advanced driver-assistance system. It went into production in July 2023. Dojo's goal was to efficiently process millions of terabytes of video data captured from real-life driving situations from Tesla's 4+ million cars. This goal led to a considerably different architecture than conventional supercomputer designs. In August 2025, Bloomberg News reported that the Dojo project had been disbanded, though it was restarted in January 2026. == History == Tesla operates several massively parallel computing clusters for developing its Autopilot advanced driver assistance system. Its primary unnamed cluster using 5,760 Nvidia A100 graphics processing units (GPUs) was touted by Andrej Karpathy in 2021 at the fourth International Joint Conference on Computer Vision and Pattern Recognition (CCVPR 2021) to be "roughly the number five supercomputer in the world" at approximately 81.6 petaflops, based on scaling the performance of the Nvidia Selene supercomputer, which uses similar components. However, the performance of the primary Tesla GPU cluster has been disputed, as it was not clear if this was measured using single-precision or double-precision floating point numbers (FP32 or FP64). Tesla also operates a second 4,032 GPU cluster for training and a third 1,752 GPU cluster for automatic labeling of objects. The primary unnamed Tesla GPU cluster has been used for processing one million video clips, each ten seconds long, taken from Tesla Autopilot cameras operating in Tesla cars in the real world, running at 36 frames per second. Collectively, these video clips contained six billion object labels, with depth and velocity data; the total size of the data set was 1.5 petabytes. This data set was used for training a neural network intended to help Autopilot computers in Tesla cars understand roads. By August 2022, Tesla had upgraded the primary GPU cluster to 7,360 GPUs. Dojo was first mentioned by Elon Musk in April 2019 during Tesla's "Autonomy Investor Day". In August 2020, Musk stated it was "about a year away" due to power and thermal issues. Dojo was officially announced at Tesla's Artificial Intelligence (AI) Day on August 19, 2021. Tesla revealed details of the D1 chip and its plans for "Project Dojo", a datacenter that would house 3,000 D1 chips; the first "Training Tile" had been completed and delivered the week before. In October 2021, Tesla released a "Dojo Technology" whitepaper describing the Configurable Float8 (CFloat8) and Configurable Float16 (CFloat16) floating point formats and arithmetic operations as an extension of Institute of Electrical and Electronics Engineers (IEEE) standard 754. At the follow-up AI Day in September 2022, Tesla announced it had built several System Trays and one Cabinet. During a test, the company stated that Project Dojo drew 2.3 megawatts (MW) of power before tripping a local San Jose, California power substation. At the time, Tesla was assembling one Training Tile per day. In August 2023, Tesla powered on Dojo for production use as well as a new training cluster configured with 10,000 Nvidia H100 GPUs. In January 2024, Musk described Dojo as "a long shot worth taking because the payoff is potentially very high. But it's not something that is a high probability." In June 2024, Musk explained that ongoing construction work at Gigafactory Texas is for a computing cluster claiming that it is planned to comprise an even mix of "Tesla AI" and Nvidia/other hardware with a total thermal design power of at first 130 MW and eventually exceeding 500 MW. In August 2025, Bloomberg News reported that the Dojo project was disbanded, though Musk announced it would be restarted in January 2026 with a new chip iteration. == Technical architecture == The fundamental unit of the Dojo supercomputer is the D1 chip, designed by a team at Tesla led by ex-AMD CPU designer Ganesh Venkataramanan, including Emil Talpes, Debjit Das Sarma, Douglas Williams, Bill Chang, and Rajiv Kurian. The D1 chip is manufactured by the Taiwan Semiconductor Manufacturing Company (TSMC) using 7 nanometer (nm) semiconductor nodes, has 50 billion transistors and a large die size of 645 mm2 (1.0 square inch). Updating at Artificial Intelligence (AI) Day in 2022, Tesla announced that Dojo would scale by deploying multiple ExaPODs, in which there would be: 10 Cabinets per ExaPOD (1,062,000 cores, 3,000 D1 chips) 2 System Trays per Cabinet (106,200 cores, 300 D1 chips) 6 Training Tiles per System Tray (53,100 cores, along with host interface hardware) 25 D1 chips per Training Tile (8,850 cores) 354 computing cores per D1 chip According to Venkataramanan, Tesla's senior director of Autopilot hardware, Dojo will have more than an exaflop (a million teraflops) of computing power. For comparison, according to Nvidia, in August 2021, the (pre-Dojo) Tesla AI-training center used 720 nodes, each with eight Nvidia A100 Tensor Core GPUs for 5,760 GPUs in total, providing up to 1.8 exaflops of performance. === D1 chip === Each node (computing core) of the D1 processing chip is a general purpose 64-bit CPU with a superscalar core. It supports internal instruction-level parallelism, and includes simultaneous multithreading (SMT). It doesn't support virtual memory and uses limited memory protection mechanisms. Dojo software/applications manage chip resources. The D1 instruction set supports both 64-bit scalar and 64-byte single instruction, multiple data (SIMD) vector instructions. The integer unit mixes reduced instruction set computer (RISC-V) and custom instructions, supporting 8, 16, 32, or 64 bit integers. The custom vector math unit is optimized for machine learning kernels and supports multiple data formats, with a mix of precisions and numerical ranges, many of which are compiler composable. Up to 16 vector formats can be used simultaneously. ==== Node ==== Each D1 node uses a 32-byte fetch window holding up to eight instructions. These instructions are fed to an eight-wide decoder which supports two threads per cycle, followed by a four-wide, four-way SMT scalar scheduler that has two integer units, two address units, and one register file per thread. Vector instructions are passed further down the pipeline to a dedicated vector scheduler with two-way SMT, which feeds either a 64-byte SIMD unit or four 8×8×4 matrix multiplication units. The network on-chip (NOC) router links cores into a two-dimensional mesh network. It can send one packet in and one packet out in all four directions to/from each neighbor node, along with one 64-byte read and one 64-byte write to local SRAM per clock cycle. Hardware native operations transfer data, semaphores and barrier constraints across memories and CPUs. System-wide double data rate 4 (DDR4) synchronous dynamic random-access memory (SDRAM) memory works like bulk storage. ==== Memory ==== Each core has a 1.25 megabytes (MB) of SRAM main memory. Load and store speeds reach 400 gigabytes (GB) per second and 270 GB/sec, respectively. The chip has explicit core-to-core data transfer instructions. Each SRAM has a unique list parser that feeds a pair of decoders and a gather engine that feeds the vector register file, which together can directly transfer information across nodes. ==== Die ==== Twelve nodes (cores) are grouped into a local block. Nodes are arranged in an 18×20 array on a single die, of which 354 cores are available for applications. The die runs at 2 gigahertz (GHz) and totals 440 MB of SRAM (360 cores × 1.25 MB/core). It reaches 376 teraflops using 16-bit brain floating point (BF16) numbers or using configurable 8-bit floating point (CFloat8) numbers, which is a Tesla proposal, and 22 teraflops at FP32. Each die comprises 576 bi-directional serializer/deserializer (SerDes) channels along the perimeter to link to other dies, and moves 8 TB/sec across all four die edges. Each D1 chip has a thermal design power of approximately 400 watts. === Training Tile === The water-cooled Training Tile packages 25 D1 chips into a 5×5 array. Each tile supports 36 TB/sec of aggregate bandwidth via 40 input/output (I/O) chips - half the bandwidth of the chip mesh network. Each tile supports 10 TB/sec of on-tile bandwidth. Each tile has 11 GB of SRAM memory (25 D1 chips × 360 cores/D1 × 1.25 MB/core). Each tile achieves 9 petaflops at BF16/CFloat8 precision (25 D1 chips × 376 TFLOP/D1). Each tile consumes 15 kilowatts; 288 amperes at 52 volts. === System Tray === Six tiles are aggregated into a System Tray, which is integrated with a host interface. Each host interface includes 512 x86 cores, providing a Linux-based user environment. Previously, the Dojo System Tray was known as the Training Matrix, which includes six Training Tiles, 20 Dojo Interface Processor cards across four host servers, and Ethernet-l

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  • Small language model

    Small language model

    Small language models or compact language models are artificial intelligence language models designed for human natural language processing including language and text generation. They are smaller in scale and scope than large language models. A large language model typically contains hundreds of billions of training parameters, with some models exceeding a trillion parameters. This substantial parameter count enables the model to encode vast amounts of information, thereby improving the generalizability and accuracy of its outputs. However, training such models demands enormous computational resources, rendering it infeasible for an individual to do so using a single computer and graphics processing unit. Small language models, on the other hand, use far fewer parameters, typically ranging from a few thousand to a few hundred million. This make them more feasible to train and host in resource-constrained environments such as a single computer or even a mobile device. Most contemporary (2020s) small language models use the same architecture as a large language model, but with a smaller parameter count and sometimes lower arithmetic precision. Parameter count is reduced by a combination of knowledge distillation and pruning. Precision can be reduced by quantization. Work on large language models mostly translate to small language models: pruning and quantization are also widely used to speed up large language models. == Models == Some notable models are: Below 1B parameters: Llama-Prompt-Guard-2-22M (detects prompt injection and jailbreaking, based on DeBERTa-xsmall), SmolLM2-135M, SmolLM2-360M 1–4B parameters: Llama3.2-1B, Qwen2.5-1.5B, DeepSeek-R1-1.5B, SmolLM2-1.7B, SmolVLM-2.25B, Phi-3.5-Mini-3.8B, Phi-4-Mini-3.8B, Gemma3-4B; closed-weights ones include Gemini Nano 4–14B parameters: Mistral 7B, Gemma 9B, Phi-4 14B. Phi-4 14B is marginally "small" at best, but Microsoft does market it as a small model. == Language model with small pre-training dataset == Traditional AI language systems need enormous computers and vast amounts of data. Pre-training matters, even tiny models show significant performance improvements when pre-trained performance increases with larger pre-training datasets. Classification accuracy improves when pre-training and test datasets share similar tokens. Shallow architectures can replicate deep model performance through collaborative learning.

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  • Barbara Di Eugenio

    Barbara Di Eugenio

    Barbara Di Eugenio is an Italian-American computer scientist, the Collegiate Warren S. McCulloch Professor of Computer Science at the University of Illinois Chicago. Her research focuses on natural language processing and its applications to human–computer interaction, educational technology, and artificial intelligence in healthcare. == Education and career == Di Eugenio is originally from Turin. After an undergraduate education in Italy, she completed her Ph.D. in computer and information science in 1993 at the University of Pennsylvania. Her dissertation, Understanding Natural Language Instructions: A Computational Approach to Purpose Clauses, was supervised by Bonnie Webber. She became a faculty member at the University of Illinois Chicago in 1999, and at that time was the only woman faculty member in the Department of Electrical Engineering and Computer Science. == Recognition == In 2022, Di Eugenio received the Zenith Award of the Association for Women in Science. She was named as a Fellow of the Association for Computational Linguistics in 2023, "for outstanding contributions to natural language generation; intelligent tutoring systems; discourse; intercoder agreement; and applying multimodal interactive systems to health".

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  • Top 10 AI Presentation Makers Compared (2026)

    Top 10 AI Presentation Makers Compared (2026)

    Trying to pick the best AI presentation maker? An AI presentation maker is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI presentation maker slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  • ACL Data Collection Initiative

    ACL Data Collection Initiative

    The ACL Data Collection Initiative (ACL/DCI) was a project established in 1989 by the Association for Computational Linguistics (ACL) to create and distribute large text and speech corpora for computational linguistics research. The initiative aimed to address the growing need for substantial text databases that could support research in areas such as natural language processing, speech recognition, and computational linguistics. By 1993, the initiative’s activities had effectively ceased, with its functions and datasets absorbed by the Linguistic Data Consortium (LDC), which was founded in 1992. == Objectives == The ACL/DCI had several key objectives: To acquire a large and diverse text corpus from various sources To transform the collected texts into a common format based on the Standard Generalized Markup Language (SGML) To make the corpus available for scientific research at low cost with minimal restrictions To provide a common database that would allow researchers to replicate or extend published results To reduce duplication of effort among researchers in obtaining and preparing text data These objectives were designed to address the growing demand for very large amounts of text arising from applications in recognition and analysis of text and speech. Its core objective was to "oversee the acquisition and preparation of a large text corpus to be made available for scientific research at cost and without royalties". == History == By the late 1980s, researchers in computational linguistics and speech recognition faced a significant problem: the lack of large-scale, accessible text corpora for developing statistical models and testing algorithms. Existing generally available text databases were too small to meet the needs of developing applications in text and speech recognition. The initiative was formed to meet this need by collecting, standardizing, and distributing large quantities of text data with minimal restrictions for scientific research. As stated by Liberman (1990), "research workers have been severely hampered by the lack of appropriate materials, and specially by the lack of a large enough body of text on which published results can be replicated or extended by others." The ACL/DCI committee was established in February 1989. The committee included members from academic and industrial research laboratories in the United States and Europe. The initiative was chaired by Mark Liberman from the University of Pennsylvania (formerly of AT&T Bell Laboratories). Other committee members included representatives from organizations such as Bellcore, IBM T.J. Watson Research Center, Cambridge University, Virginia Polytechnic Institute & State University, Northeastern University, University of Pennsylvania, SRI International, MCC, Xerox PARC, ISSCO, and University of Pisa. The project operated initially without dedicated funding, relying on volunteer efforts from committee members and their affiliated institutions. Key supporters included AT&T Bell Labs, Bellcore, IBM, Xerox, and the University of Pennsylvania, which allowed the use of their computing facilities for ACL/DCI-related work. Previously running on volunteer effort pro bono, in 1991, it obtained funding from General Electric and the National Science Foundation (IRI-9113530). == Data == As of 1990, the ACL/DCI had collected hundreds of millions of words of diverse text. The collection included: Wall Street Journal articles (25 to 50 million words); Canadian Hansard (parliamentary records) in parallel English and French versions: cleaned-up English Hansard donated by the IBM alignment models group (100 million words), and original Bilingual Hansard (from a different time period) obtained directly (200 million words). Collins English Dictionary (1979 edition), both as fulltext (3 million words) and as various "database" versions, constructed using "typographers' tape" donated by Collins, which were computer tapes containing the structured digital data used to typeset and print the 1979 edition of the dictionary; Emails from ARPANET newsletters for the ACM Special Interest Group on Information Retrieval Forum (IRLIST) and AIList Digest issues distributed over the ARPANET (AILIST) (5 million words), both collected by Edward A. Fox at VIPSU; Articles on networking (2 million words); U.S. Department of Agriculture Extension Service Fact Sheets (>1 million words); 200,000 scientific abstracts of about 1,500 words each from the Department of Energy (25 million words); Archives of the Challenger Investigation Commission, including transcripts of depositions and hearings (2.5 million words); Books from the Library of America, including works by Mark Twain, Eugene O'Neill, Ralph Waldo Emerson, Herman Melville, W.E.B. DuBois, Willa Cather, and Benjamin Franklin (130 books, 20 million words); Public domain books like the King James Bible, Tristram Shandy, The Federalist Papers; Several million words of transcribed radiologists' reports, donated by Francis Ganong at Kurzweil Applied Intelligence Inc (about 5 million words); The Child Language Data Exchange corpus of child language acquisition transcripts; U.S. Department of Justice Justice Retrieval and Inquiry System (JURIS) materials; The Swiss Civil Code in parallel German, French and Italian; Economic reports from the Union Bank of Switzerland, in parallel English, German, French and Italian; About 12K words of administrative policy manuals and 14K words of administrative memos, contributed by Geoff Pullum of U.C.S.C.; Material from various ACM journals and the ACL journal Computational Linguistics; The CSLI publications series: 50-100 reports (8K words each) and 5-10 books (80K words each). The initiative started with North American English text but expanded to include Canadian French and planned to include Japanese, Chinese, and other Asian languages. At least 5 million words from the collection were tagged under the Penn Treebank project, and those tags were distributed by DCI as well. After DCI was absorbed by the LDC, the datasets were curated under LDC. == Format == The ACL/DCI corpus was coded in a standard form based on SGML (Standard Generalized Markup Language, ISO 8879), consistent with the recommendations of the Text Encoding Initiative (TEI), of which the DCI was an affiliated project. The TEI was a joint project of the ACL, the Association for Computers and the Humanities, and the Association for Literary and Linguistic Computing, aiming to provide a common interchange format for literary and linguistic data. The initiative planned to add annotations reflecting consensually approved linguistic features like part of speech and various aspects of syntactic and semantic structure over time. == Examples == As an example of the use of ACL/DCI, consider the Wall Street Journal (WSJ) corpus for speech recognition research. The WSJ corpus was used as the basis for the DARPA Spoken Language System (SLS) community's Continuous Speech Recognition (CSR) Corpus. The WSJ corpus became a standard benchmark for evaluating speech recognition systems and has been used in numerous research papers. The WSJ CSR Corpus provided DARPA with its first general-purpose English, large vocabulary, natural language, high perplexity corpus containing speech (400 hours) and text (47 million words) during 1987–89. The text corpus was 313 MB in size. The text was preprocessed to remove ambiguity in the word sequence that a reader might choose, ensuring that the unread text used to train language models was representative of the spoken test material. The preprocessing included converting numbers into orthographics, expanding abbreviations, resolving apostrophes and quotation marks, and marking punctuation. As another example, the Yarowsky algorithm used bitext data from DCI to train a simple word-sense disambiguation model that was competitive with advanced models trained on smaller datasets. == Distribution == Materials from the ACL/DCI collection were distributed to research groups on a non-commercial basis. By 1990, about 25 research groups and individual researchers had received tapes containing various portions of the collected material. To obtain the data, researchers had to sign an agreement not to redistribute the data or make direct commercial use of it. However, commercial application of "analytical materials" derived from the text, such as statistical tables or grammar rules, was explicitly permitted. The initiative first distributed data via 12-inch reels of 9-track tape, then via CD-ROMs. Each such tape could contain 30 million words compressed via the Lempel-Ziv algorithms. The first CD-ROM distribution was in 1991, funded by Dragon Systems Inc. It contained Collins English Dictionary, WSJ, scientific abstracts provided by the U.S. Department of Energy, and the Penn Treebank.

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  • AI Virtual Assistants: Free vs Paid (2026)

    AI Virtual Assistants: Free vs Paid (2026)

    Trying to pick the best AI virtual assistant? An AI virtual assistant is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI virtual assistant slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

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  • Is an AI Text-to-video Tool Worth It in 2026?

    Is an AI Text-to-video Tool Worth It in 2026?

    In search of the best AI text-to-video tool? An AI text-to-video tool is software that uses machine learning to help you get more done — it turns a rough idea into a polished result in seconds. When choosing one, weigh output quality, pricing, export formats, and how well it fits the tools you already use. Whether you are a beginner or a pro, the right AI text-to-video tool 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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  • Markov chain

    Markov chain

    In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov. Markov chains have many applications as statistical models of real-world processes. They provide the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in areas including Bayesian statistics, biology, chemistry, economics, finance, information theory, physics, signal processing, and speech processing. The adjectives Markovian and Markov are used to describe something that is related to a Markov process. == Principles == === Definition === A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as "memorylessness"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history. In other words, conditional on the present state of the system, its future and past states are independent. A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it is also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space). === Types of Markov chains === The system's state space and time parameter index need to be specified. The following table gives an overview of the different instances of Markov processes for different levels of state space generality for both discrete and continuous time: Note that there is no definitive agreement in the literature on the use of some of the terms that signify special cases of Markov processes. Usually the term "Markov chain" is reserved for a process with a discrete set of times, that is, a discrete-time Markov chain (DTMC), but a few authors use the term "Markov process" to refer to a continuous-time Markov chain (CTMC) without explicit mention. In addition, there are other extensions of Markov processes that are referred to as such but do not necessarily fall within any of these four categories (see Markov model). Moreover, the time index need not necessarily be real-valued; like with the state space, there are conceivable processes that move through index sets with other mathematical constructs. Notice that the general state space continuous-time Markov chain is general to such a degree that it has no designated term. While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis. Besides time-index and state-space parameters, there are many other variations, extensions and generalizations (see Variations). For simplicity, most of this article concentrates on the discrete-time, discrete state-space case, unless mentioned otherwise. === Transitions === The changes of state of the system are called transitions. The probabilities associated with various state changes are called transition probabilities. The process is characterized by a state space, a transition matrix describing the probabilities of particular transitions, and an initial state (or initial distribution) across the state space. By convention, we assume all possible states and transitions have been included in the definition of the process, so there is always a next state, and the process does not terminate. A discrete-time random process involves a system which is in a certain state at each step, with the state changing randomly between steps. The steps are often thought of as moments in time, but they can equally well refer to physical distance or any other discrete measurement. Formally, the steps are the integers or natural numbers, and the random process is a mapping of these to states. The Markov property states that the conditional probability distribution for the system at the next step (and in fact at all future steps) depends only on the current state of the system, and not additionally on the state of the system at previous steps. Since the system changes randomly, it is generally impossible to predict with certainty the state of a Markov chain at a given point in the future. However, the statistical properties of the system's future can be predicted. In many applications, it is these statistical properties that are important. == History == Andrey Markov studied Markov processes in the early 20th century, publishing his first paper on the topic in 1906. Markov processes in continuous time were discovered long before his work in the early 20th century in the form of the Poisson process. Markov was interested in studying an extension of independent random sequences, motivated by a disagreement with Pavel Nekrasov who claimed independence was necessary for the weak law of large numbers to hold. In his first paper on Markov chains, published in 1906, Markov showed that under certain conditions the average outcomes of the Markov chain would converge to a fixed vector of values, so proving a weak law of large numbers without the independence assumption, which had been commonly regarded as a requirement for such mathematical laws to hold. Markov later used Markov chains to study the distribution of vowels in Eugene Onegin, written by Alexander Pushkin, and proved a central limit theorem for such chains. In 1912 Henri Poincaré studied Markov chains on finite groups with an aim to study card shuffling. Other early uses of Markov chains include a diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branching process, introduced by Francis Galton and Henry William Watson in 1873, preceding the work of Markov. After the work of Galton and Watson, it was later revealed that their branching process had been independently discovered and studied around three decades earlier by Irénée-Jules Bienaymé. Starting in 1928, Maurice Fréchet became interested in Markov chains, eventually resulting in him publishing in 1938 a detailed study on Markov chains. Andrey Kolmogorov developed in a 1931 paper a large part of the early theory of continuous-time Markov processes. Kolmogorov was partly inspired by Louis Bachelier's 1900 work on fluctuations in the stock market as well as Norbert Wiener's work on Einstein's model of Brownian movement. He introduced and studied a particular set of Markov processes known as diffusion processes, where he derived a set of differential equations describing the processes. Independent of Kolmogorov's work, Sydney Chapman derived in a 1928 paper an equation, now called the Chapman–Kolmogorov equation, in a less mathematically rigorous way than Kolmogorov, while studying Brownian movement. The differential equations are now called the Kolmogorov equations or the Kolmogorov–Chapman equations. Other mathematicians who contributed significantly to the foundations of Markov processes include William Feller, starting in 1930s, and then later Eugene Dynkin, starting in the 1950s. == Examples == Mark V. Shaney is a third-order Markov chain program, and a Markov text generator. It ingests the sample text (the Tao Te Ching, or the posts of a Usenet group) and creates a massive list of every sequence of three successive words (triplet) which occurs in the text. It then chooses two words at random, and looks for a word which follows those two in one of the triplets in its massive list. If there is more than one, it picks at random (identical triplets count separately, so a sequence which occurs twice is twice as likely to be picked as one which only occurs once). It then adds that word to the generated text. Then, in the same way, it picks a triplet that starts with the second and third words in the generated text, and that gives a fourth word. It adds the fourth word, then repeats with the third and fourth words, and so on. Random walks based on integers and the gambler's ruin problem are ex

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  • Floyd–Steinberg dithering

    Floyd–Steinberg dithering

    Floyd–Steinberg dithering is an image dithering algorithm first published in 1976 by Robert W. Floyd and Louis Steinberg. It is commonly used by image manipulation software, for example, when converting an image from a Truecolor 24-bit PNG format into a GIF format, which is restricted to a maximum of 256 colors. == Implementation == The algorithm achieves dithering using error diffusion, meaning it pushes (adds) the residual quantization error of a pixel onto its neighboring pixels, to be quantized after. It spreads the debt out according to the distribution (shown as a map of the neighboring pixels): [ ∗ 7 16 … … 3 16 5 16 1 16 … ] {\displaystyle {\begin{bmatrix}&&&{\frac {\displaystyle 7}{\displaystyle 16}}&\ldots \\\ldots &{\frac {\displaystyle 3}{\displaystyle 16}}&{\frac {\displaystyle 5}{\displaystyle 16}}&{\frac {\displaystyle 1}{\displaystyle 16}}&\ldots \\\end{bmatrix}}} The pixel indicated with a star () indicates the pixel currently being scanned, and the blank pixels are the previously scanned pixels. The specific values (7/16, 3/16, 5/16, 1/16) were originally found by trial-and-error, "guided by the desire to have a region of desired density 0.5 come out as a checkerboard pattern". The algorithm scans the image from left to right, top to bottom, quantizing pixel values one by one. Each time, the quantization error is transferred to the neighboring pixels, while not affecting the pixels that already have been quantized. Hence, if a number of pixels have been rounded downwards, it becomes more likely that the next pixel is rounded upwards, such that on average, the quantization error is close to zero. The diffusion coefficients have the property that if the original pixel values are exactly halfway in between the nearest available colors, the dithered result is a checkerboard pattern. For example, 50% grey data could be dithered as a black-and-white checkerboard pattern. For optimal dithering, the counting of quantization errors should be in sufficient accuracy to prevent rounding errors from affecting the result. For correct results, all values should be linearized first, rather than operating directly on sRGB values as is common for images stored on computers. In some implementations, the horizontal direction of scan alternates between lines; this is called "serpentine scanning" or boustrophedon transform dithering. The algorithm described above is in the following pseudocode. This works for any approximately linear encoding of pixel values, such as 8-bit integers, 16-bit integers or real numbers in the range [0, 1]. for each y from top to bottom do for each x from left to right do oldpixel := pixels[x][y] newpixel := find_closest_palette_color(oldpixel) pixels[x][y] := newpixel quant_error := oldpixel - newpixel pixels[x + 1][y ] := pixels[x + 1][y ] + quant_error × 7 / 16 pixels[x - 1][y + 1] := pixels[x - 1][y + 1] + quant_error × 3 / 16 pixels[x ][y + 1] := pixels[x ][y + 1] + quant_error × 5 / 16 pixels[x + 1][y + 1] := pixels[x + 1][y + 1] + quant_error × 1 / 16 When converting grayscale pixel values from a high to a low bit depth (e.g. 8-bit grayscale to 1-bit black-and-white), find_closest_palette_color() may perform just a simple rounding, for example: find_closest_palette_color(oldpixel) = round(oldpixel / 255) The pseudocode can result in pixel values exceeding the valid values (such as greater than 255 in 8-bit grayscale images). Such values should ideally be handled by the find_closest_palette_color() function, rather than clipping the intermediate values, since a subsequent error may bring the value back into range. However, if fixed-width integers are used, wrapping of intermediate values would cause inversion of black and white, and so should be avoided. The find_closest_palette_color() implementation is nontrivial for a palette that is not evenly distributed, however small inaccuracies in selecting the correct palette color have minimal visual impact due to error being propagated to future pixels. A nearest neighbor search in 3D is frequently used.

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  • The Best Free AI Photo Editor for Beginners

    The Best Free AI Photo Editor for Beginners

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

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  • EuroMatrixPlus

    EuroMatrixPlus

    The EuroMatrixPlus is a project that ran from March 2009 to February 2012. EuroMatrixPlus succeeded a project called EuroMatrix (September 2006 to February 2009) and continued in further development and improvement of machine translation (MT) systems for languages of the European Union (EU). == Project objectives == EuroMatrixPlus focused on achieving several goals: To continue advance of MT technology (create MT systems for all official EU languages and provide other MT researchers with existing data and infrastructure). To continually expand and investigate different MT approaches and techniques; to stay open to novel combinations of methods of MT. To bring MT to the users. Users post-edit output of statistical models and the system learns from the feedback and improves itself. Two groups of users were aimed at: Professional translators and translation agencies Users who voluntarily translate texts into their native language To contribute to MT research in Europe. To produce sample application for automatic translation of news and web pages and make that application freely accessible. == Outcome == EuroMatrixPlus contributed to MT field in several ways. It continued in development of an open source statistical MT engine Moses. The project worked on research in hybrid approaches to MT (combination of rule-based and statistical techniques). Several “MT Marathons” and annual evaluation campaigns were organized by the project. The project also resulted in releasing of 196 scientific publications. The results of the work were arranged into ten work packages: WP1: Rich Tree-Based Statistical Translation WP2: Hybrid Machine Translation WP3: Advanced Learning Methods for MT WP4: Open Source Tools and Data WP5: "WikiTrans" Translation Environments WP6: Integrated Localisation Workflow WP7: Evaluation Campaign WP8: Project Management and Dissemination WP9: Integrating Slovak Language Resources WP10: HPSG-based Statistical Translation === Software and data === Here is a list of software and data that were released by the project: Appraise – an open source tool for manual evaluation of MT output BURGER – Bulgarian Resource BulTreeBank – Treebank of Bulgarian CSLM toolkit – free tool for training continuous space language models (CSLM) to large tasks Caitra – tool for post-editing MT results Europarl – European Parliament parallel corpus IRSTLM toolkit – tool for training language models Joshua – an open-source statistical machine translation decoder for hierarchical and syntax-based MT MT Server Land – an open-source architecture for MT Moses – statistical MT MultiUN Corpora – parallel corpus extracted from the United Nations Website PCEDT 2.0 – Prague Czech-English Dependency Treebank PEDT 2.0 – English part of the Prague Czech-English Dependency Treebank Slovak corpora – English-Slovak and Czech-Slovak as well as a Slovak-English and a Slovak-Czech parallel corpus Slovak treebank – A dependency treebank TermEx – RBMT-Suited Statistical Terminology Extraction Tool Treex, TectoMT == Funding == The EuroMatrixPlus project was sponsored by EU Information Society Technology program. Total cost of the project was 5 942 121 €, from which the European Union contributed 4 266 896 €. == Project members == To ensure advance in MT, several organizations that are experts in various disciplines (linguistics, computer science, mathematics, translation) were brought together to cooperate on EuroMatrixPlus. The consortium consisted of academic as well as commercial partners. Academic partners were the University of Edinburgh (United Kingdom), DFKI – German Research Centre for Artificial Intelligence (Germany), Charles University (Czech Republic), Johns Hopkins University (United States), University of Le Mans (France), Fondazione Bruno Kessler (Italy), Dublin City University (Ireland). Two institutions joined about one year into the project. These were the L'udovít Štúr Institute of Linguistics (Slovak Republic) and IICT – Institute of Information and Communication Technologies at the Bulgarian Academy of Sciences (Bulgaria). Commercial partners included Lucy Software and Services GmbH (Germany) and CEET s.r.o. (Czech Republic). Coordination of the project was in hands of DFKI with its Language Technology Lab in Saarbrücken. The principal investigator and scientific coordinator was Hans Uszkoreit, a professor of Computational Linguistics at Saarland University.

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