AI Coding Claude

AI Coding Claude — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

Curve (tonality)

In image editing, a curve is a remapping of image tonality, specified as a function from input level to output level, used as a way to emphasize colours or other elements in a picture. Curves can usually be applied to all channels together in an image, or to each channel individually. Applying a curve to all channels typically changes the brightness in part of the spectrum. Light parts of a picture can be easily made lighter and dark parts darker to increase contrast. Applying a curve to individual channels can be used to stress a colour. This is particularly efficient in the Lab colour space due to the separation of luminance and chromaticity, but it can also be used in RGB, CMYK or whatever other colour models the software supports.
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Vinberg's algorithm

In mathematics, Vinberg's algorithm is an algorithm, introduced by Ernest Borisovich Vinberg, for finding a fundamental domain of a hyperbolic reflection group. Conway (1983) used Vinberg's algorithm to describe the automorphism group of the 26-dimensional even unimodular Lorentzian lattice II25,1 in terms of the Leech lattice. == Description of the algorithm == Let Γ < I s o m ( H n ) {\displaystyle \Gamma <\mathrm {Isom} (\mathbb {H} ^{n})} be a hyperbolic reflection group. Choose any point v 0 ∈ H n {\displaystyle v_{0}\in \mathbb {H} ^{n}} ; we shall call it the basic (or initial) point. The fundamental domain P 0 {\displaystyle P_{0}} of its stabilizer Γ v 0 {\displaystyle \Gamma _{v_{0}}} is a polyhedral cone in H n {\displaystyle \mathbb {H} ^{n}} . Let H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} be the faces of this cone, and let a 1 , . . . , a m {\displaystyle a_{1},...,a_{m}} be outer normal vectors to it. Consider the half-spaces H k − = { x ∈ R n , 1 | ( x , a k ) ≤ 0 } . {\displaystyle H_{k}^{-}=\{x\in \mathbb {R} ^{n,1}|(x,a_{k})\leq 0\}.} There exists a unique fundamental polyhedron P {\displaystyle P} of Γ {\displaystyle \Gamma } contained in P 0 {\displaystyle P_{0}} and containing the point v 0 {\displaystyle v_{0}} . Its faces containing v 0 {\displaystyle v_{0}} are formed by faces H 1 , . . . , H m {\displaystyle H_{1},...,H_{m}} of the cone P 0 {\displaystyle P_{0}} . The other faces H m + 1 , . . . {\displaystyle H_{m+1},...} and the corresponding outward normals a m + 1 , . . . {\displaystyle a_{m+1},...} are constructed by induction. Namely, for H j {\displaystyle H_{j}} we take a mirror such that the root a j {\displaystyle a_{j}} orthogonal to it satisfies the conditions (1) ( v 0 , a j ) < 0 {\displaystyle (v_{0},a_{j})<0} ; (2) ( a i , a j ) ≤ 0 {\displaystyle (a_{i},a_{j})\leq 0} for all i < j {\displaystyle i Read more →
Best arm identification

Best arm identification (BAI) is a sequential one-player game where the player has to find the best action (arm) among a list of actions (arms) by collecting information in the most efficient way. It is a multi-armed bandit game as a player only gets information about an arm by playing it. The most common objective in multi-armed bandit games is to minimize the regret (i.e., play the best action as much as possible), but in BAI, the goal is to find the best arm as efficiently as possible. This problem naturally arises in scenarios such as adaptive clinical trials where the number of patients is limited and the quantification of the confidence in a treatment is important. It also arises in hyperparameter optimization where the goal is to find the optimal choice of hyperparameters for an algorithm with the smallest possible number of experiments, as it can be costly in terms of time, energy, or money. == Stochastic multi-armed bandit == The stochastic multi-armed bandit (MAB) is a sequential game with one player and K {\displaystyle K} actions (arms). Each arm has an unknown probability distribution associated with it. At each turn, the player has to choose one action and receive an observation from the probability distribution associated with the arm. The more you play an arm, the more you get information on its probability distribution. === Best arm identification === In BAI the goal is to find the arm that has the probability distribution with the highest mean. BAI may be either fixed confidence or fixed horizon. In a fixed-confidence game, a confidence level δ {\displaystyle \delta } is fixed at the beginning of the game and the goal is to find the best arm with this confidence level in as few turns as possible. In a fixed horizon game, the number of turns T {\displaystyle T} is fixed, and the goal is to find the best arm with the highest possible confidence in T {\displaystyle T} turns. === Math formalisation === We have one player and K {\displaystyle K} actions (arms). Behind each arm k ∈ { 1 , … , K } {\displaystyle k\in \{1,\ldots ,K\}} lies an unknown distribution ν k {\displaystyle \nu _{k}} with mean μ k {\displaystyle \mu _{k}} . Each distribution ν k {\displaystyle \nu _{k}} belongs to a known family D {\displaystyle {\mathcal {D}}} (such as the set of Gaussian distributions or Bernoulli distributions). At each time step t {\displaystyle t} , the player selects an arm a t {\displaystyle a_{t}} and observes an independent sample X t ∼ ν a t {\displaystyle X_{t}\sim \nu _{a_{t}}} from the corresponding distribution. We will note μ ∗ := max μ a {\displaystyle \mu ^{}:=\max \mu _{a}} the highest mean. An arm a {\displaystyle a} that satisfies μ a = μ ∗ {\displaystyle \mu _{a}=\mu ^{}} is called an optimal arm; otherwise it is called suboptimal arm. In best arm identification (BAI) the objective is to identify an optimal arm. Two main settings for BAI appear in the literature: Fixed confidence: In this setting, one typically assumes that there exists a unique optimal arm. A confidence level δ ∈ ( 0 , 1 ) {\displaystyle \delta \in (0,1)} is specified at the beginning. The algorithm must stop at some finite stopping time τ δ < + ∞ {\displaystyle \tau _{\delta }<+\infty } and return an arm a ^ τ δ {\displaystyle {\hat {a}}_{\tau _{\delta }}} such that the probability of error is bounded: P ( a ^ τ δ ≠ a ∗ ) ≤ δ {\displaystyle \mathbb {P} ({\hat {a}}_{\tau _{\delta }}\neq a^{})\leq \delta } . The objective is to minimize the expected sample complexity E [ τ δ ] {\displaystyle \mathbb {E} [\tau _{\delta }]} . Such a setting appears, for example, when a constraint on the confidence is required (for example, if we require a confidence level of 95%, so δ = 1 − 0.95 = 0.05 {\displaystyle \delta =1-0.95=0.05} ). Fixed horizon: In this setting, the number of samples T {\displaystyle T} is fixed in advance. The goal is to design an algorithm that minimizes the probability of misidentifying the optimal arm: P ( a ^ T ≠ a ∗ ) {\displaystyle \mathbb {P} ({\hat {a}}_{T}\neq a^{})} . This setting appears when the number of experiments is limited (for drug tests, the number of patients can be fixed in advance). === Example of simple modelling === In the case where we have K {\displaystyle K} treatments and we want to be sure with a confidence level of 95% which treatment is the best to heal a specific disease. Each treatment heals or does not heal the disease with a probability μ k {\displaystyle \mu _{k}} , which means that each distribution is a Bernoulli distribution, so D {\displaystyle {\mathcal {D}}} is the set of Bernoulli distributions. We can use a BAI algorithm to minimize E [ τ 0.05 ] {\displaystyle \mathbb {E} [\tau _{0.05}]} , the number of patients required to find the best treatment with probability 95%. == Applications == Best arm identification naturally arises in several practical domains: Adaptive clinical trials: The objective is to identify the most effective treatment based on sequentially collected patient data. Each treatment can be modeled as having an underlying distribution of outcomes. The goal is to identify the treatment with the highest expected outcome with high confidence (fixed confidence setting δ {\displaystyle \delta } ) while minimizing the number of drug test patients (minimise E [ τ δ ] {\displaystyle \mathbb {E} [\tau _{\delta }]} ), as it costs to pay patients for this and we would like to use as little as possible less effective drugs. Hyperparameter tuning: Selecting the best configuration for machine learning models efficiently by treating each hyperparameter setting as an arm. The goal is to find the best hyperparameter with as few experiments possible as experiments are costly in time and in energy == Fixed confidence level == In the fixed-confidence setting, the goal is to design an algorithm that identifies the best arm with a prescribed confidence level δ {\displaystyle \delta } while minimizing the expected number of samples. Any such algorithm requires two key components: Stopping rule: A decision criterion that determines when to stop sampling. Formally, this defines a stopping time τ δ {\displaystyle \tau _{\delta }} and returns an arm a ^ τ δ {\displaystyle {\hat {a}}_{\tau _{\delta }}} such that P ( a ^ τ δ ≠ a ⋆ ) ≤ δ {\displaystyle \mathbb {P} ({\hat {a}}_{\tau _{\delta }}\neq a^{\star })\leq \delta } and P ( τ δ < + ∞ ) = 1 {\displaystyle \mathbb {P} (\tau _{\delta }<+\infty )=1} . Sampling rule: A policy π {\displaystyle \pi } that, at each round t {\displaystyle t} , selects the next arm to sample a t {\displaystyle a_{t}} based on all previous observations ( a s , X s ) s < t {\displaystyle (a_{s},X_{s})_{s Read more →
Nike+iPod

The Nike+iPod Sport Kit is an activity tracker device, developed by Nike, Inc., which measures and records the distance and pace of a walk or run. The Nike+iPod consists of a small transmitter device attached to or embedded in a shoe, which communicates with either the Nike+ Sportband, or a receiver plugged into an iPod Nano. It can also work directly with a 2nd Generation iPod Touch (or higher), iPhone 3GS, iPhone 4, iPhone 4S, iPhone 5, The Nike+iPod was announced on May 23, 2006. On September 7, 2010, Nike released the Nike+ Running App (originally called Nike+ GPS) on the App Store, which used a tracking engine powered by MotionX that does not require the separate shoe sensor or pedometer. This application works using the accelerometer and GPS of the iPhone and the accelerometer of the iPod Touch, which does not have a GPS chip. Nike+Running is compatible with the iPhone 6 and iPhone 6 Plus down to iPhone 3GS and iPod touch. On June 21, 2012, Nike released Nike+ Running App for Android. The current app is compatible with all Android phones running 4.0.3 and up. == Overview == The sensor and iPod kit were revealed on May 20, 2006. The kit stores information such as the elapsed time of the workout, the distance traveled, pace, and calories burned by the individual. Nike+ was a collaboration between Nike and Apple; the platform consisted of an iPod, a wireless chip, Nike shoes that accepted the wireless chip, an iTunes membership, and a Nike+ online community. iPods using Nike iPod require a sensor and remote. The next upgraded product was the Sportband kit, which was announced in April 2008. The kit allows users to store run information without the iPod Nano. The Sportband consists of two parts: a rubber holding strap which is worn around the wrist, and a receiver which resembles a USB key-disk. The receiver displays information comparable to that of the iPod kit on the built-in display. After a run, the receiver can be plugged straight into a USB port and the software will upload the run information automatically to the Nike+ website. As of August 2008 "Nike+iPod for the Gym" launched, allowing users to record their cardio workouts directly to their iPods. No Sport kit or shoe sensor is required; all that is needed is a compatible iPod (1st–6th generation iPod Nano or 2nd/3rd gen iPod Touch) and an enabled piece of cardio equipment. As of March 2009, the seven largest commercial equipment providers were shipping enabled equipment (Life Fitness, Technogym, Precor USA, Star Trac, Cybex International, Matrix Fitness and Free Motion). The models of compatible cardio equipment include treadmills, stationary bicycles, stair climbers, ellipticals, and others such as Precor's Adaptive Motion Trainer. Once the user syncs an iPod with iTunes, the cardio workouts are automatically stored at Nikeplus.com, where each workout is visualized and tracked based on the number of calories burned. The calories are converted to "CardioMiles", at a ratio of 100:1, allowing cardio users to take full advantage of all the tools and features of Nikeplus.com, and allow them to engage in challenges with other runners, walkers and cardio users, using a common currency. With the release of the second-generation iPod Touch in 2008, Apple Inc. included a built-in ability to receive Nike+ signals, which allowed the iPod to connect directly to the wireless sensor thus eliminating the need for an external receiver to be connected. Apple also added this capability to the iPhone 3GS (released 2009), iPhone 4 (2010), and third-generation iPod Touch (2009). Those devices use their Broadcom Bluetooth chipset to receive the signals. On June 7, 2010, Polar and Nike introduced the Polar WearLink+ that works with Nike+. This new product works with the Nike+ SportBand and the fifth generation iPod nano in conjunction with the Nike+ iPod Sport Kit. Polar WearLink+ that works with Nike+ communicates directly with the fifth generation iPod nano and Nike+ SportBand using a proprietary digital protocol but it is dual-mode so it is also compatible with most Polar training computers (all those using 5 kHz analog transmission technology). Nike+ had 18 million global users as of April 2013. One year later, Nike updated the number of global users to 28 million. In iOS 6.1.2 (and possibly higher), a hole in the compatibility for the app has allowed jailbroken iPad users to use the native Nike + iPod iPhone and iPod app by moving the app bundle and setting permissions for the app. On April 30, 2018, Nike retired services for legacy Nike wearable devices, such as the Nike+ FuelBand and the Nike+ SportWatch GPS, and previous versions of apps, including Nike Run Club and Nike Training Club version 4.X and lower. Likewise, Nike no longer supported the Nike+ Connect software that transferred data to a NikePlus Profile or the Nike+ Fuel/FuelBand and Nike+ Move apps. == Sports kit equipment == The kit consists of two pieces: a piezoelectric sensor with a Nordic Semiconductor nRF2402 transmitter that is mounted under the inner sole of the shoe and a receiver that connects to the iPod. They communicate using a 2.4 GHz wireless radio and use Nordic Semiconductor's "ShockBurst" network protocol. The wireless data is encrypted in transit, but some uniquely identifying data is sent in the plain. The wireless protocol was reverse engineered and documented by Dmitry Grinberg in 2011. Nike recommends that the shoe be a Nike+ model with a special pocket in which to place the device. Nike has released the sensor for individual sale meaning that consumers no longer have to purchase the whole set (the iPod receiver and sensor). As the sensor battery cannot be replaced, a new one must be purchased every time the battery runs out. Aftermarket solutions are available to users who do not want to use shoes with built-in or hand-made pockets for the foot sensor, such as shoe pouches and containment devices designed to affix the sensor against the shoe laces. No matter how the sensor is integrated with the user's shoes, care must be taken that it is firmly fixed in place and will not jerk around while in use, which would degrade the accuracy. == Sports kit usage == The Sports Kit can be used to track running, which it refers to as "workouts". New workouts are started by plugging the receiving unit into the iPod, then navigating through the iPod menu system. The user chooses a goal for the workout, which might be to cover a specific distance, or burn a number of calories, or work out for a specified time. A workout can also be started without a goal, which is called a "Basic Workout". When the workout goal has been set, the receiver seeks the sensor, possibly asking the user to "walk around to activate [the] sensor". The user then must press the center button on the iPod to begin the workout. Audio feedback is provided in the user's choice of generic male or female voice by the iPod over the course of the workout, depending on the type of workout chosen. For goal-oriented workouts, the feedback will correspond to significant milestones toward the goal. In a distance workout, for example, the audio feedback will inform the user as each mile or kilometer has been completed, as well as the half-way point of the workout, and a countdown of four 100-meter increments at the end of the workout. The iPod's control wheel functions change slightly during a workout. The Pause button now not only pauses the music but also the workout. Similarly, the Menu button is used to access the controls to end the workout. The Forward and Back buttons are unchanged, performing audio track skip and reverse functions. The Center button has two functions: audio feedback about the current distance, time, and pace are provided when the button is tapped once, while if the button is held down the iPod skips to the "PowerSong" - an audio track chosen by the user, generally intended for motivation. In addition to the in-workout audio feedback, there are pre-recorded congratulations provided by Lance Armstrong, Tiger Woods, Joan Benoit Samuelson, and Paula Radcliffe whenever a user achieves a personal best (such as fastest mile, fastest 5K, fastest 10K, longest run yet) or reaches certain long-term milestones (such as 250 miles, 500 kilometers). This "celebrity feedback" is heard after the usual end-of-run statistics. While the Sports Kit can be used immediately after purchase, it will report more accurate results if it is calibrated before the first usage and then regularly afterwards. For calibration, the user finds a fixed known distance of at least 0.25 mile or 400 meters and then sets the Nike+ to calibration mode for the walk or run over that distance. When the walk or run is complete, the device calibrates itself and future workout reporting will reflect statistics closer to that individual user's workout style. Consumer Reports magazine tested the device and found it accurate as long as you keep an even pace. In workouts with varied pa
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MovieRide FX

MovieRide FX is a patented automated special visual effects video compositing engine used in the MovieRide FX mobile application for Android (requires Android 2.3 or later) and iOS (compatible with iPhone 4 and up, iPad, and iPod Touch (new generation), requires iOS 7 or later). MovieRide FX allows the user to personalize a "Hollywood-style" movie clip by inserting themself into the clip as the "actor". == Features == The MovieRide FX app uses the relevant mobile device's camera to record a video of the user and insert it into a pre-packaged "Hollywood style" movie clip. The "actor" is extracted from their recorded video clip through various known effects such as masking, keying, and motion tracking. The "actor" is then inserted into one of the pre-packaged movie clips created by the MovieRide FX visual effects artists. This is done through an automated process requiring little or no artistic or technical skill from the user. The custom movie clips pre-packaged with MovieRide FX offer the user a variety of movie scenarios. Additional clips based on popular television and movie themes are continually being developed and are available on a freemium basis. == Sharing == Once the user's footage has automatically been composited into a movie clip and rendered as an .mp4 file, it can be shared via social media, such as Facebook, YouTube, and Twitter, and by e-mail. == History == === 2012 === MovieRide FX was created by Grant Waterston and Johann Mynhardt, who started development in 2012. === 2013 === The beta version was released on Google Play in July 2013. In August 2013 MovieRide FX was a New Media Award winner in the "New Media" category of the Accolade International Awards in Los Angeles. In October 2013 MovieRide FX was awarded exhibitor space in the ‘start-up village’ at the Apps-World Expo in London. === 2014 === MovieRide FX reached the 100 000 – 500 000 downloads category on the Google Play Store in June 2014. The official Android version was launched in July 2014. iOS version released in August 2014. MovieRide FX was selected as one of the "Top 150" startups at the Pioneer Festival in Vienna in September 2014. In November 2014 MovieRide FX was shortlisted for the Appster Awards in the "Best Entertainment App" and "Most Innovative App" categories and was awarded exhibitor space at the ‘start-up village’ at the Apps-World Expo in London. Patent applications were filed in South Africa, the EU and USA in April 2014. === 2015 === In September 2015 MovieRide FX was shortlisted for "Best Software innovation" at The Technology Expo Awards in London. === 2016 === In April 2016 MovieRide FX was nominated for a National Science and Technology Forum (NSTF) award for 'Research leading to Innovation by a corporate organization' In August 2016 Movie Ride FX won two Gold Awards at the 2016 Mobile Marketing Awards (MMA Smarties SA). These two Gold awards were for the 'Innovation' and 'Best in Show’ categories. In December 2016 FlicJam Inc. was formed in the US to access the larger global market. EU patent application was published in March 2016. === 2017 === South African patent was granted in February 2017. === 2018 === US patent was granted in March 2018.
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Golden record (informatics)

In informatics, a golden record is the valid version of a data element (record) in a single source of truth system. It may refer to a database, specific table or data field, or any unit of information used. A golden copy is a consolidated data set, and is supposed to provide a single source of truth and a "well-defined version of all the data entities in an organizational ecosystem". Other names sometimes used include master source or master version. The term has been used in conjunction with data quality, master data management, and similar topics. (Different technical solutions exist, see master data management). == Master data == In master data management (MDM), the golden copy refers to the master data (master version) of the reference data which works as an authoritative source for the "truth" for all applications in a given IT landscape.
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Timeline of algorithms

The following timeline of algorithms outlines the development of algorithms (mainly "mathematical recipes") since their inception. == Antiquity == Before – writing about "recipes" (on cooking, rituals, agriculture and other themes) c. 1700–2000 BC – Egyptians develop earliest known algorithms for multiplying two numbers c. 1600 BC – Babylonians develop earliest known algorithms for factorization and finding square roots c. 300 BC – Euclid's algorithm c. 200 BC – the Sieve of Eratosthenes 263 AD – Gaussian elimination described by Liu Hui == Medieval Period == 628 – Chakravala method described by Brahmagupta c. 820 – Al-Khawarizmi described algorithms for solving linear equations and quadratic equations in his Algebra; the word algorithm comes from his name 825 – Al-Khawarizmi described the algorism, algorithms for using the Hindu–Arabic numeral system, in his treatise On the Calculation with Hindu Numerals, which was translated into Latin as Algoritmi de numero Indorum, where "Algoritmi", the translator's rendition of the author's name gave rise to the word algorithm (Latin algorithmus) with a meaning "calculation method" c. 850 – cryptanalysis and frequency analysis algorithms developed by Al-Kindi (Alkindus) in A Manuscript on Deciphering Cryptographic Messages, which contains algorithms on breaking encryptions and ciphers c. 1025 – Ibn al-Haytham (Alhazen), was the first mathematician to derive the formula for the sum of the fourth powers, and in turn, he develops an algorithm for determining the general formula for the sum of any integral powers c. 1400 – Ahmad al-Qalqashandi gives a list of ciphers in his Subh al-a'sha which include both substitution and transposition, and for the first time, a cipher with multiple substitutions for each plaintext letter; he also gives an exposition on and worked example of cryptanalysis, including the use of tables of letter frequencies and sets of letters which can not occur together in one word == Before 1940 == 1540 – Lodovico Ferrari discovered a method to find the roots of a quartic polynomial 1545 – Gerolamo Cardano published Cardano's method for finding the roots of a cubic polynomial 1614 – John Napier develops method for performing calculations using logarithms 1671 – Newton–Raphson method developed by Isaac Newton 1690 – Newton–Raphson method independently developed by Joseph Raphson 1706 – John Machin develops a quickly converging inverse-tangent series for π and computes π to 100 decimal places 1768 – Leonhard Euler publishes his method for numerical integration of ordinary differential equations in problem 85 of Institutiones calculi integralis 1789 – Jurij Vega improves Machin's formula and computes π to 140 decimal places, 1805 – FFT-like algorithm known by Carl Friedrich Gauss 1842 – Ada Lovelace writes the first algorithm for a computing engine 1903 – A fast Fourier transform algorithm presented by Carle David Tolmé Runge 1918 - Soundex 1926 – Borůvka's algorithm 1926 – Primary decomposition algorithm presented by Grete Hermann 1927 – Hartree–Fock method developed for simulating a quantum many-body system in a stationary state. 1934 – Delaunay triangulation developed by Boris Delaunay 1936 – Turing machine, an abstract machine developed by Alan Turing, with others developed the modern notion of algorithm. == 1940s == 1942 – A fast Fourier transform algorithm developed by G.C. Danielson and Cornelius Lanczos 1945 – Merge sort developed by John von Neumann 1947 – Simplex algorithm developed by George Dantzig == 1950s == 1950 – Hamming codes developed by Richard Hamming 1952 – Huffman coding developed by David A. Huffman 1953 – Simulated annealing introduced by Nicholas Metropolis 1954 – Radix sort computer algorithm developed by Harold H. Seward 1964 – Box–Muller transform for fast generation of normally distributed numbers published by George Edward Pelham Box and Mervin Edgar Muller. Independently pre-discovered by Raymond E. A. C. Paley and Norbert Wiener in 1934. 1956 – Kruskal's algorithm developed by Joseph Kruskal 1956 – Ford–Fulkerson algorithm developed and published by R. Ford Jr. and D. R. Fulkerson 1957 – Prim's algorithm developed by Robert Prim 1957 – Bellman–Ford algorithm developed by Richard E. Bellman and L. R. Ford, Jr. 1959 – Dijkstra's algorithm developed by Edsger Dijkstra 1959 – Shell sort developed by Donald L. Shell 1959 – De Casteljau's algorithm developed by Paul de Casteljau 1959 – QR factorization algorithm developed independently by John G.F. Francis and Vera Kublanovskaya 1959 – Rabin–Scott powerset construction for converting NFA into DFA published by Michael O. Rabin and Dana Scott == 1960s == 1960 – Karatsuba multiplication 1961 – CRC (Cyclic redundancy check) invented by W. Wesley Peterson 1962 – AVL trees 1962 – Quicksort developed by C. A. R. Hoare 1962 – Bresenham's line algorithm developed by Jack E. Bresenham 1962 – Gale–Shapley 'stable-marriage' algorithm developed by David Gale and Lloyd Shapley 1964 – Heapsort developed by J. W. J. Williams 1964 – multigrid methods first proposed by R. P. Fedorenko 1965 – Cooley–Tukey algorithm rediscovered by James Cooley and John Tukey 1965 – Levenshtein distance developed by Vladimir Levenshtein 1965 – Cocke–Younger–Kasami (CYK) algorithm independently developed by Tadao Kasami 1965 – Buchberger's algorithm for computing Gröbner bases developed by Bruno Buchberger 1965 – LR parsers invented by Donald Knuth 1966 – Dantzig algorithm for shortest path in a graph with negative edges 1967 – Viterbi algorithm proposed by Andrew Viterbi 1967 – Cocke–Younger–Kasami (CYK) algorithm independently developed by Daniel H. Younger 1968 – A graph search algorithm described by Peter Hart, Nils Nilsson, and Bertram Raphael 1968 – Risch algorithm for indefinite integration developed by Robert Henry Risch 1969 – Strassen algorithm for matrix multiplication developed by Volker Strassen == 1970s == 1970 – Dinic's algorithm for computing maximum flow in a flow network by Yefim (Chaim) A. Dinitz 1970 – Knuth–Bendix completion algorithm developed by Donald Knuth and Peter B. Bendix 1970 – BFGS method of the quasi-Newton class 1970 – Needleman–Wunsch algorithm published by Saul B. Needleman and Christian D. Wunsch 1972 – Edmonds–Karp algorithm published by Jack Edmonds and Richard Karp, essentially identical to Dinic's algorithm from 1970 1972 – Graham scan developed by Ronald Graham 1972 – Red–black trees and B-trees discovered 1973 – RSA encryption algorithm discovered by Clifford Cocks 1973 – Jarvis march algorithm developed by R. A. Jarvis 1973 – Hopcroft–Karp algorithm developed by John Hopcroft and Richard Karp 1974 – Pollard's p − 1 algorithm developed by John Pollard 1974 – Quadtree developed by Raphael Finkel and J.L. Bentley 1975 – Genetic algorithms popularized by John Holland 1975 – Pollard's rho algorithm developed by John Pollard 1975 – Aho–Corasick string matching algorithm developed by Alfred V. Aho and Margaret J. Corasick 1975 – Cylindrical algebraic decomposition developed by George E. Collins 1976 – Salamin–Brent algorithm independently discovered by Eugene Salamin and Richard Brent 1976 – Knuth–Morris–Pratt algorithm developed by Donald Knuth and Vaughan Pratt and independently by J. H. Morris 1977 – Boyer–Moore string-search algorithm for searching the occurrence of a string into another string. 1977 – RSA encryption algorithm rediscovered by Ron Rivest, Adi Shamir, and Len Adleman 1977 – LZ77 algorithm developed by Abraham Lempel and Jacob Ziv 1977 – multigrid methods developed independently by Achi Brandt and Wolfgang Hackbusch 1978 – LZ78 algorithm developed from LZ77 by Abraham Lempel and Jacob Ziv 1978 – Bruun's algorithm proposed for powers of two by Georg Bruun 1979 – Khachiyan's ellipsoid method developed by Leonid Khachiyan 1979 – ID3 decision tree algorithm developed by Ross Quinlan == 1980s == 1980 – Brent's Algorithm for cycle detection Richard P. Brendt 1981 – Quadratic sieve developed by Carl Pomerance 1981 – Smith–Waterman algorithm developed by Temple F. Smith and Michael S. Waterman 1983 – Simulated annealing developed by S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi 1983 – Classification and regression tree (CART) algorithm developed by Leo Breiman, et al. 1984 – LZW algorithm developed from LZ78 by Terry Welch 1984 – Karmarkar's interior-point algorithm developed by Narendra Karmarkar 1984 – ACORN PRNG discovered by Roy Wikramaratna and used privately 1985 – Simulated annealing independently developed by V. Cerny 1985 – Car–Parrinello molecular dynamics developed by Roberto Car and Michele Parrinello 1985 – Splay trees discovered by Sleator and Tarjan 1986 – Blum Blum Shub proposed by L. Blum, M. Blum, and M. Shub 1986 – Push relabel maximum flow algorithm by Andrew Goldberg and Robert Tarjan 1986 – Barnes–Hut tree method developed by Josh Barnes and Piet Hut for fast approximate simulation of n-body problems 1987 – Fast multipole method developed by Leslie Greengard and Vladimir
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Virtual facility

A Virtual Facility (VF) is a highly realistic digital representation of a data center, used to model all relevant aspects of a physical data center with a high degree of precision. The term "virtual" in Virtual Facility refers to its use of virtual reality, rather than the abstraction of computer resources as seen in platform virtualization. The VF mirrors the characteristics of a physical facility over time and allows for detailed analysis and modeling. == VF Model features == A standard VF model includes: Three-dimensional physical facility layout Network connectivity of facility equipment Full inventory of facility equipment, including electronics and electrical systems such as power distribution units (PDUs) and uninterruptible power supplies (UPSs) Full air conditioning system (ACUs) and controls within the room The term Virtual Facility was introduced to address the emerging environmental problems facing modern Mission Critical Facilities (MCFs). This concept combines virtual reality (VR), computer simulation, and expert systems applied to the domain of facilities. The VF type of computer simulation allows for detailed analysis and prototyping of airflow in the data center using computational fluid dynamics (CFD) techniques. This enables the visualization and numerical analysis of airflow and temperatures within the facility, helping to predict real-world outcomes. == VF applications == The VF model can be used to assist with the following: Greenfield design Asset management Troubleshooting existing data centers Making existing data centers more resilient Making existing data centers more energy efficient Cost prediction Staff training Capacity planning Load growth management Many organizations use VF models to virtually assess scenarios before committing resources to physical changes. This allows for better decision-making regarding the addition or modification of equipment, helping to avoid logistical or thermal problems.
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Co-occurrence matrix

A co-occurrence matrix or co-occurrence distribution (also referred to as : gray-level co-occurrence matrices GLCMs) is a matrix that is defined over an image to be the distribution of co-occurring pixel values (grayscale values, or colors) at a given offset. It is used as an approach to texture analysis with various applications especially in medical image analysis. == Method == Given a grey-level image I {\displaystyle I} , co-occurrence matrix computes how often pairs of pixels with a specific value and offset occur in the image. The offset, ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} , is a position operator that can be applied to any pixel in the image (ignoring edge effects): for instance, ( 1 , 2 ) {\displaystyle (1,2)} could indicate "one down, two right". An image with p {\displaystyle p} different pixel values will produce a p × p {\displaystyle p\times p} co-occurrence matrix, for the given offset. The ( i , j ) th {\displaystyle (i,j)^{\text{th}}} value of the co-occurrence matrix gives the number of times in the image that the i th {\displaystyle i^{\text{th}}} and j th {\displaystyle j^{\text{th}}} pixel values occur in the relation given by the offset. For an image with p {\displaystyle p} different pixel values, the p × p {\displaystyle p\times p} co-occurrence matrix C is defined over an n × m {\displaystyle n\times m} image I {\displaystyle I} , parameterized by an offset ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} , as: C Δ x , Δ y ( i , j ) = ∑ x = 1 n ∑ y = 1 m { 1 , if I ( x , y ) = i and I ( x + Δ x , y + Δ y ) = j 0 , otherwise {\displaystyle C_{\Delta x,\Delta y}(i,j)=\sum _{x=1}^{n}\sum _{y=1}^{m}{\begin{cases}1,&{\text{if }}I(x,y)=i{\text{ and }}I(x+\Delta x,y+\Delta y)=j\\0,&{\text{otherwise}}\end{cases}}} where: i {\displaystyle i} and j {\displaystyle j} are the pixel values; x {\displaystyle x} and y {\displaystyle y} are the spatial positions in the image I; the offsets ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} define the spatial relation for which this matrix is calculated; and I ( x , y ) {\displaystyle I(x,y)} indicates the pixel value at pixel ( x , y ) {\displaystyle (x,y)} . The 'value' of the image originally referred to the grayscale value of the specified pixel, but could be anything, from a binary on/off value to 32-bit color and beyond. (Note that 32-bit color will yield a 232 × 232 co-occurrence matrix!) Co-occurrence matrices can also be parameterized in terms of a distance, d {\displaystyle d} , and an angle, θ {\displaystyle \theta } , instead of an offset ( Δ x , Δ y ) {\displaystyle (\Delta x,\Delta y)} . Any matrix or pair of matrices can be used to generate a co-occurrence matrix, though their most common application has been in measuring texture in images, so the typical definition, as above, assumes that the matrix is an image. It is also possible to define the matrix across two different images. Such a matrix can then be used for color mapping. == Aliases == Co-occurrence matrices are also referred to as: GLCMs (gray-level co-occurrence matrices) GLCHs (gray-level co-occurrence histograms) spatial dependence matrices == Application to image analysis == Whether considering the intensity or grayscale values of the image or various dimensions of color, the co-occurrence matrix can measure the texture of the image. Because co-occurrence matrices are typically large and sparse, various metrics of the matrix are often taken to get a more useful set of features. Features generated using this technique are usually called Haralick features, after Robert Haralick. Texture analysis is often concerned with detecting aspects of an image that are rotationally invariant. To approximate this, the co-occurrence matrices corresponding to the same relation, but rotated at various regular angles (e.g. 0, 45, 90, and 135 degrees), are often calculated and summed. Texture measures like the co-occurrence matrix, wavelet transforms, and model fitting have found application in medical image analysis in particular. == Other applications == Co-occurrence matrices are also used for words processing in natural language processing (NLP).
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Upper ontology

In information science, an upper ontology (also known as a top-level ontology, upper model, or foundation ontology) is an ontology (in the sense used in information science) that consists of very general terms (such as "object", "property", "relation") that are common across all domains. An important function of an upper ontology is to support broad semantic interoperability among a large number of domain-specific ontologies by providing a common starting point for the formulation of definitions. Terms in the domain ontology are ranked under the terms in the upper ontology, e.g., the upper ontology classes are superclasses or supersets of all the classes in the domain ontologies. A number of upper ontologies have been proposed, each with its own proponents. Library classification systems predate upper ontology systems. Though library classifications organize and categorize knowledge using general concepts that are the same across all knowledge domains, neither system is a replacement for the other. == Development == Any standard foundational ontology is likely to be contested among different groups, each with its own idea of "what exists". One factor exacerbating the failure to arrive at a common approach has been the lack of open-source applications that would permit the testing of different ontologies in the same computational environment. The differences have thus been debated largely on theoretical grounds, or are merely the result of personal preferences. Foundational ontologies can however be compared on the basis of adoption for the purposes of supporting interoperability across domain ontologies. No particular upper ontology has yet gained widespread acceptance as a de facto standard. Different organizations have attempted to define standards for specific domains. The 'Process Specification Language' (PSL) created by the National Institute of Standards and Technology (NIST) is one example. Another important factor leading to the absence of wide adoption of any existing upper ontology is the complexity. Some upper ontologies—Cyc is often cited as an example in this regard—are very large, ranging up to thousands of elements (classes, relations), with complex interactions among them and with a complexity similar to that of a human natural language, and the learning process can be even longer than for a natural language because of the unfamiliar format and logical rules. The motivation to overcome this learning barrier is largely absent because of the paucity of publicly accessible examples of use. As a result, those building domain ontologies for local applications tend to create the simplest possible domain-specific ontology, not related to any upper ontology. Such domain ontologies may function adequately for the local purpose, but they are very time-consuming to relate accurately to other domain ontologies. To solve this problem, some genuinely top level ontologies have been developed, which are deliberately designed to have minimal overlap with any domain ontologies. Examples are Basic Formal Ontology and the DOLCE (see below). === Arguments for the infeasibility of an upper ontology === Historically, many attempts in many societies have been made to impose or define a single set of concepts as more primal, basic, foundational, authoritative, true or rational than all others. A common objection to such attempts points out that humans lack the sort of transcendent perspective — or God's eye view — that would be required to achieve this goal. Humans are bound by language or culture, and so lack the sort of objective perspective from which to observe the whole terrain of concepts and derive any one standard. Thomasson, under the headline "1.5 Skepticism about Category Systems", wrote: "category systems, at least as traditionally presented, seem to presuppose that there is a unique true answer to the question of what categories of entity there are – indeed the discovery of this answer is the goal of most such inquiries into ontological categories. [...] But actual category systems offered vary so much that even a short survey of past category systems like that above can undermine the belief that such a unique, true and complete system of categories may be found. Given such a diversity of answers to the question of what the ontological categories are, by what criteria could we possibly choose among them to determine which is uniquely correct?" Another objection is the problem of formulating definitions. Top level ontologies are designed to maximize support for interoperability across a large number of terms. Such ontologies must therefore consist of terms expressing very general concepts, but such concepts are so basic to our understanding that there is no way in which they can be defined, since the very process of definition implies that a less basic (and less well understood) concept is defined in terms of concepts that are more basic and so (ideally) more well understood. Very general concepts can often only be elucidated, for example by means of examples, or paraphrase. There is no self-evident way of dividing the world up into concepts, and certainly no non-controversial one There is no neutral ground that can serve as a means of translating between specialized (or "lower" or "application-specific") ontologies Human language itself is already an arbitrary approximation of just one among many possible conceptual maps. To draw any necessary correlation between English words and any number of intellectual concepts, that we might like to represent in our ontologies, is just asking for trouble. (WordNet, for instance, is successful and useful, precisely because it does not pretend to be a general-purpose upper ontology; rather, it is a tool for semantic / syntactic / linguistic disambiguation, which is richly embedded in the particulars and peculiarities of the English language.) Any hierarchical or topological representation of concepts must begin from some ontological, epistemological, linguistic, cultural, and ultimately pragmatic perspective. Such pragmatism does not allow for the exclusion of politics between persons or groups, indeed it requires they be considered as perhaps more basic primitives than any that are represented. Those who doubt the feasibility of general purpose ontologies are more inclined to ask "what specific purpose do we have in mind for this conceptual map of entities and what practical difference will this ontology make?" This pragmatic philosophical position surrenders all hope of devising the encoded ontology version of "The world is everything that is the case." (Wittgenstein, Tractatus Logico-Philosophicus). Finally, there are objections similar to those against artificial intelligence. Technically, the complex concept acquisition and the social / linguistic interactions of human beings suggest any axiomatic foundation of "most basic" concepts must be cognitive biological or otherwise difficult to characterize since we don't have axioms for such systems. Ethically, any general-purpose ontology could quickly become an actual tyranny by recruiting adherents into a political program designed to propagate it and its funding means, and possibly defend it by violence. Historically, inconsistent and irrational belief systems have proven capable of commanding obedience to the detriment or harm of persons both inside and outside a society that accepts them. How much more harmful would a consistent rational one be, were it to contain even one or two basic assumptions incompatible with human life? === Arguments for the feasibility of an upper ontology === Many of those who doubt the possibility of developing wide agreement on a common upper ontology fall into one of two traps: they assert that there is no possibility of universal agreement on any conceptual scheme; but they argue that a practical common ontology does not need to have universal agreement, it only needs a large enough user community (as is the case for human languages) to make it profitable for developers to use it as a means to general interoperability, and for third-party developer to develop utilities to make it easier to use; and they point out that developers of data schemes find different representations congenial for their local purposes; but they do not demonstrate that these different representations are in fact logically inconsistent. In fact, different representations of assertions about the real world (though not philosophical models), if they accurately reflect the world, must be logically consistent, even if they focus on different aspects of the same physical object or phenomenon. If any two assertions about the real world are logically inconsistent, one or both must be wrong, and that is a topic for experimental investigation, not for ontological representation. In practice, representations of the real world are created as and known to be approximations to the basic reality, and their use is circumscribed by the limits of e
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Savepoint

A savepoint is a way of implementing subtransactions (also known as nested transactions) within a relational database management system by indicating a point within a transaction that can be "rolled back to" without affecting any work done in the transaction before the savepoint was created. Multiple savepoints can exist within a single transaction. Savepoints are useful for implementing complex error recovery in database applications. If an error occurs in the midst of a multiple-statement transaction, the application may be able to recover from the error (by rolling back to a savepoint) without needing to abort the entire transaction. A savepoint can be declared by issuing a SAVEPOINT name statement. All changes made after a savepoint has been declared can be undone by issuing a ROLLBACK TO SAVEPOINT name command. Issuing RELEASE SAVEPOINT name will cause the named savepoint to be discarded, but will not otherwise affect anything. Issuing the commands ROLLBACK or COMMIT will also discard any savepoints created since the start of the main transaction. Savepoints are defined in the SQL standard and are supported by all established SQL relational databases, including PostgreSQL, Oracle Database, Microsoft SQL Server, MySQL, IBM Db2, SQLite (since 3.6.8), Firebird, H2 Database Engine, and Informix (since version 11.50xC3).
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Information seeking

Information seeking is the process or activity of attempting to obtain information in both human and technological contexts. Information seeking is related to, but different from, information retrieval (IR). == Compared to information retrieval == Traditionally, IR tools have been designed for IR professionals to enable them to effectively and efficiently retrieve information from a source. It is assumed that the information exists in the source and that a well-formed query will retrieve it (and nothing else). It has been argued that laypersons' information seeking on the internet is very different from information retrieval as performed within the IR discourse. Yet, internet search engines are built on IR principles. Since the late 1990s a body of research on how casual users interact with internet search engines has been forming, but the topic is far from fully understood. IR can be said to be technology-oriented, focusing on algorithms and issues such as precision and recall. Information seeking may be understood as a more human-oriented and open-ended process than information retrieval. In information seeking, one does not know whether there exists an answer to one's query, so the process of seeking may provide the learning required to satisfy one's information need. == In different contexts == Much library and information science (LIS) research has focused on the information-seeking practices of practitioners within various fields of professional work. Studies have been carried out into the information-seeking behaviors of librarians, academics, medical professionals, engineers, lawyers and mini-publics(among others). Much of this research has drawn on the work done by Leckie, Pettigrew (now Fisher) and Sylvain, who in 1996 conducted an extensive review of the LIS literature (as well as the literature of other academic fields) on professionals' information seeking. The authors proposed an analytic model of professionals' information seeking behaviour, intended to be generalizable across the professions, thus providing a platform for future research in the area. The model was intended to "prompt new insights... and give rise to more refined and applicable theories of information seeking" (1996, p. 188). The model has been adapted by Wilkinson (2001) who proposes a model of the information seeking of lawyers. Recent studies in this topic address the concept of information-gathering that "provides a broader perspective that adheres better to professionals' work-related reality and desired skills." (Solomon & Bronstein, 2021). == Theories of information-seeking behavior == A variety of theories of information behavior – e.g. Zipf's Principle of Least Effort, Brenda Dervin's Sense Making, Elfreda Chatman's Life in the Round – seek to understand the processes that surround information seeking. In addition, many theories from other disciplines have been applied in investigating an aspect or whole process of information seeking behavior. A review of the literature on information seeking behavior shows that information seeking has generally been accepted as dynamic and non-linear (Foster, 2005; Kuhlthau 2006). People experience the information search process as an interplay of thoughts, feelings and actions (Kuhlthau, 2006). Donald O. Case (2007) also wrote a good book that is a review of the literature. Information seeking has been found to be linked to a variety of interpersonal communication behaviors beyond question-asking, to include strategies such as candidate answers. Robinson's (2010) research suggests that when seeking information at work, people rely on both other people and information repositories (e.g., documents and databases), and spend similar amounts of time consulting each (7.8% and 6.4% of work time, respectively; 14.2% in total). However, the distribution of time among the constituent information seeking stages differs depending on the source. When consulting other people, people spend less time locating the information source and information within that source, similar time understanding the information, and more time problem solving and decision making, than when consulting information repositories. Furthermore, the research found that people spend substantially more time receiving information passively (i.e., information that they have not requested) than actively (i.e., information that they have requested), and this pattern is also reflected when they provide others with information. == Wilson's nested model of conceptual areas == The concepts of information seeking, information retrieval, and information behaviour are objects of investigation of information science. Within this scientific discipline a variety of studies has been undertaken analyzing the interaction of an individual with information sources in case of a specific information need, task, and context. The research models developed in these studies vary in their level of scope. Wilson (1999) therefore developed a nested model of conceptual areas, which visualizes the interrelation of the here mentioned central concepts. Wilson defines models of information behavior to be "statements, often in the form of diagrams, that attempt to describe an information-seeking activity, the causes and consequences of that activity, or the relationships among stages in information-seeking behaviour" (1999: 250).
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Cowrie (honeypot)

Cowrie is a medium interaction SSH and Telnet honeypot designed to log brute force attacks and shell interaction performed by an attacker. Cowrie also functions as an SSH and telnet proxy to observe attacker behavior to another system. Cowrie was developed from Kippo. == Reception == Cowrie has been referenced in published papers. The Book "Hands-On Ethical Hacking and Network Defense" includes Cowrie in a list of 5 commercial honeypots. === Prior uses === Discussing a honeypot effort called the Project Heisenberg Cloud by Rapid7, Bob Rudis, the company's chief data scientist, told eWEEK, "There are custom Rapid7-developed low- and medium-interaction honeypots used within the framework, along with open-source ones, such as Cowrie." Doug Rickert has experimented with the open-source Cowrie SSH honeypot and wrote about it on Medium. Putting up a simple honeypot isn't difficult, and there are many open-source products besides Cowrie, including the original Honeyd to MongoDB and NoSQL honeypots, to ones that emulate web servers. Some appear to be SCADA or other more advanced applications. === Best practices === Researchers at the SysAdmin, Audit, Network and Security (SANS) institute urged administrators and security researchers to run the latest version of Cowrie on a honeypot to monitor shifts in the type of passwords being scanned for and pattern of attacks on IoT devices. === Discussion and further resources === Attack Detection and Forensics Using Honeypot in an IoT Environment calls Cowrie a "medium interaction honeypot" and describes results from using it for 40 days to capture "all communicated sessions in log files." The book Advances on Data Science also devotes chapter two to "Cowrie Honeypot Dataset and Logging." ICCWS 2018 13th International Conference on Cyber Warfare and Security describes using Cowrie. On the Move to Meaningful Internet Systems: OTM 2019 Conferences includes details of using Cowrie. Splunk, a security tool that can receive information from honeypots, outlines how to set up a honeypot using the open-source Cowrie package.
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Berlekamp–Rabin algorithm

In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. == History == The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in F p {\displaystyle \mathbb {F} _{p}} . In 2000 Peralta's method was generalized for cubic equations. == Statement of problem == Let p {\displaystyle p} be an odd prime number. Consider the polynomial f ( x ) = a 0 + a 1 x + ⋯ + a n x n {\textstyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}} over the field F p ≃ Z / p Z {\displaystyle \mathbb {F} _{p}\simeq \mathbb {Z} /p\mathbb {Z} } of remainders modulo p {\displaystyle p} . The algorithm should find all λ {\displaystyle \lambda } in F p {\displaystyle \mathbb {F} _{p}} such that f ( λ ) = 0 {\textstyle f(\lambda )=0} in F p {\displaystyle \mathbb {F} _{p}} . == Algorithm == === Randomization === Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial f z ( x ) = f ( x − z ) = ( x − λ 1 − z ) ( x − λ 2 − z ) ⋯ ( x − λ n − z ) {\textstyle f_{z}(x)=f(x-z)=(x-\lambda _{1}-z)(x-\lambda _{2}-z)\cdots (x-\lambda _{n}-z)} where z {\displaystyle z} is some element of F p {\displaystyle \mathbb {F} _{p}} . If one can represent this polynomial as the product f z ( x ) = p 0 ( x ) p 1 ( x ) {\displaystyle f_{z}(x)=p_{0}(x)p_{1}(x)} then in terms of the initial polynomial it means that f ( x ) = p 0 ( x + z ) p 1 ( x + z ) {\displaystyle f(x)=p_{0}(x+z)p_{1}(x+z)} , which provides needed factorization of f ( x ) {\displaystyle f(x)} . === Classification of === F p {\displaystyle \mathbb {F} _{p}} elements Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly one of following properties holds: The monomial is equal to x {\displaystyle x} if λ = 0 {\displaystyle \lambda =0} , The monomial divides g 0 ( x ) = ( x ( p − 1 ) / 2 − 1 ) {\textstyle g_{0}(x)=(x^{(p-1)/2}-1)} if λ {\displaystyle \lambda } is quadratic residue modulo p {\displaystyle p} , The monomial divides g 1 ( x ) = ( x ( p − 1 ) / 2 + 1 ) {\textstyle g_{1}(x)=(x^{(p-1)/2}+1)} if λ {\displaystyle \lambda } is quadratic non-residual modulo p {\displaystyle p} . Thus if f z ( x ) {\displaystyle f_{z}(x)} is not divisible by x {\displaystyle x} , which may be checked separately, then f z ( x ) {\displaystyle f_{z}(x)} is equal to the product of greatest common divisors gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} and gcd ( f z ( x ) ; g 1 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{1}(x))} . === Berlekamp's method === The property above leads to the following algorithm: Explicitly calculate coefficients of f z ( x ) = f ( x − z ) {\displaystyle f_{z}(x)=f(x-z)} , Calculate remainders of x , x 2 , x 2 2 , x 2 3 , x 2 4 , … , x 2 ⌊ log 2 ⁡ p ⌋ {\textstyle x,x^{2},x^{2^{2}},x^{2^{3}},x^{2^{4}},\ldots ,x^{2^{\lfloor \log _{2}p\rfloor }}} modulo f z ( x ) {\displaystyle f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of x ( p − 1 ) / 2 {\textstyle x^{(p-1)/2}} modulo f z ( x ) {\textstyle f_{z}(x)} , If x ( p − 1 ) / 2 ≢ ± 1 ( mod f z ( x ) ) {\textstyle x^{(p-1)/2}\not \equiv \pm 1{\pmod {f_{z}(x)}}} then gcd {\displaystyle \gcd } mentioned below provide a non-trivial factorization of f z ( x ) {\displaystyle f_{z}(x)} , Otherwise all roots of f z ( x ) {\displaystyle f_{z}(x)} are either residues or non-residues simultaneously and one has to choose another z {\displaystyle z} . If f ( x ) {\displaystyle f(x)} is divisible by some non-linear primitive polynomial g ( x ) {\displaystyle g(x)} over F p {\displaystyle \mathbb {F} _{p}} then when calculating gcd {\displaystyle \gcd } with g 0 ( x ) {\displaystyle g_{0}(x)} and g 1 ( x ) {\displaystyle g_{1}(x)} one will obtain a non-trivial factorization of f z ( x ) / g z ( x ) {\displaystyle f_{z}(x)/g_{z}(x)} , thus algorithm allows to find all roots of arbitrary polynomials over F p {\displaystyle \mathbb {F} _{p}} . === Modular square root === Consider equation x 2 ≡ a ( mod p ) {\textstyle x^{2}\equiv a{\pmod {p}}} having elements β {\displaystyle \beta } and − β {\displaystyle -\beta } as its roots. Solution of this equation is equivalent to factorization of polynomial f ( x ) = x 2 − a = ( x − β ) ( x + β ) {\textstyle f(x)=x^{2}-a=(x-\beta )(x+\beta )} over F p {\displaystyle \mathbb {F} _{p}} . In this particular case problem it is sufficient to calculate only gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} . For this polynomial exactly one of the following properties will hold: GCD is equal to 1 {\displaystyle 1} which means that z + β {\displaystyle z+\beta } and z − β {\displaystyle z-\beta } are both quadratic non-residues, GCD is equal to f z ( x ) {\displaystyle f_{z}(x)} which means that both numbers are quadratic residues, GCD is equal to ( x − t ) {\displaystyle (x-t)} which means that exactly one of these numbers is quadratic residue. In the third case GCD is equal to either ( x − z − β ) {\displaystyle (x-z-\beta )} or ( x − z + β ) {\displaystyle (x-z+\beta )} . It allows to write the solution as β = ( t − z ) ( mod p ) {\textstyle \beta =(t-z){\pmod {p}}} . === Example === Assume we need to solve the equation x 2 ≡ 5 ( mod 11 ) {\textstyle x^{2}\equiv 5{\pmod {11}}} . For this we need to factorize f ( x ) = x 2 − 5 = ( x − β ) ( x + β ) {\displaystyle f(x)=x^{2}-5=(x-\beta )(x+\beta )} . Consider some possible values of z {\displaystyle z} : Let z = 3 {\displaystyle z=3} . Then f z ( x ) = ( x − 3 ) 2 − 5 = x 2 − 6 x + 4 {\displaystyle f_{z}(x)=(x-3)^{2}-5=x^{2}-6x+4} , thus gcd ( x 2 − 6 x + 4 ; x 5 − 1 ) = 1 {\displaystyle \gcd(x^{2}-6x+4;x^{5}-1)=1} . Both numbers 3 ± β {\displaystyle 3\pm \beta } are quadratic non-residues, so we need to take some other z {\displaystyle z} . Let z = 2 {\displaystyle z=2} . Then f z ( x ) = ( x − 2 ) 2 − 5 = x 2 − 4 x − 1 {\displaystyle f_{z}(x)=(x-2)^{2}-5=x^{2}-4x-1} , thus gcd ( x 2 − 4 x − 1 ; x 5 − 1 ) ≡ x − 9 ( mod 11 ) {\textstyle \gcd(x^{2}-4x-1;x^{5}-1)\equiv x-9{\pmod {11}}} . From this follows x − 9 = x − 2 − β {\textstyle x-9=x-2-\beta } , so β ≡ 7 ( mod 11 ) {\displaystyle \beta \equiv 7{\pmod {11}}} and − β ≡ − 7 ≡ 4 ( mod 11 ) {\textstyle -\beta \equiv -7\equiv 4{\pmod {11}}} . A manual check shows that, indeed, 7 2 ≡ 49 ≡ 5 ( mod 11 ) {\textstyle 7^{2}\equiv 49\equiv 5{\pmod {11}}} and 4 2 ≡ 16 ≡ 5 ( mod 11 ) {\textstyle 4^{2}\equiv 16\equiv 5{\pmod {11}}} . == Correctness proof == The algorithm finds factorization of f z ( x ) {\displaystyle f_{z}(x)} in all cases except for ones when all numbers z + λ 1 , z + λ 2 , … , z + λ n {\displaystyle z+\lambda _{1},z+\lambda _{2},\ldots ,z+\lambda _{n}} are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are all residues or non-residues simultaneously (that is, when z = 0 {\displaystyle z=0} would fail) may be estimated as 2 − k {\displaystyle 2^{-k}} where k {\displaystyle k} is the number of distinct values in λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . In this way even for the worst case of k = 1 {\displaystyle k=1} and f ( x ) = ( x − λ ) n {\displaystyle f(x)=(x-\lambda )^{n}} , the probability of error may be estimated as 1 / 2 {\displaystyle 1/2} and for modular square root case error probability is at most 1 / 4 {\displaystyle 1/4} . == Complexity == Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − z ) {\displaystyle f(x-z)} in O ( n 2 ) {\displaystyle O(n^{2})} time. Polynomial multiplication a
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International Philosophical Bibliography

The International Philosophical Bibliography (IPB), also known in French as Répertoire bibliographique de la philosophie (RBP), is a bibliographic database covering publications on the history of philosophy and continental philosophy. The database comprises records of publications in over 30 languages. Annually, about 12,000 records are added. The indexes include, among other elements, over 84,000 names of authors, editors, translators, reviewers, and collaborators, as well as more than 3,000 commentaries on philosophical works, making it the world's most complete index in Philosophy. Since 1934, the IPB has been developed by the Higher Institute of Philosophy at the University of Louvain (UCLouvain), first in Leuven and since 1978 in Louvain-la-Neuve. The online version was launched by Peeters Publishers in 1997 and continues to be updated quarterly.
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