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.
Workplace impact of artificial intelligence
The impact of artificial intelligence on workers includes both applications to improve worker safety and health, and potential hazards that must be controlled. One potential application is using AI to eliminate hazards by removing humans from hazardous situations that involve risk of stress, overwork, or musculoskeletal injuries. Predictive analytics may also be used to identify conditions that may lead to hazards such as fatigue, repetitive strain injuries, or toxic substance exposure, leading to earlier interventions. Another is to streamline workplace safety and health workflows through automating repetitive tasks, enhancing safety training programs through virtual reality, or detecting and reporting near misses. When used in the workplace, AI also presents the possibility of new hazards. These may arise from machine learning techniques leading to unpredictable behavior and inscrutability in their decision-making, or from cybersecurity and information privacy issues. Many hazards of AI are psychosocial due to its potential to cause changes in work organization. These include increased monitoring leading to micromanagement, algorithms unintentionally or intentionally mimicking undesirable human biases, and assigning blame for machine errors to the human operator instead. AI may also lead to physical hazards in the form of human–robot collisions, and ergonomic risks of control interfaces and human–machine interactions. Hazard controls include cybersecurity and information privacy measures, communication and transparency with workers about data usage, and limitations on collaborative robots. From a workplace safety and health perspective, only "weak" or "narrow" AI that is tailored to a specific task is relevant, as there are many examples that are currently in use or expected to come into use in the near future. Certain digital technologies are predicted to result in job losses. Starting in the 2020s, the adoption of modern robotics has led to net employment growth. However, many businesses anticipate that automation, or employing robots would result in job losses in the future. This is especially true for companies in Central and Eastern Europe. Other digital technologies, such as platforms or big data, are projected to have a more neutral impact on employment. A large number of tech workers have been laid off starting in 2023; many such job cuts have been attributed to artificial intelligence. == Health and safety applications == In order for any potential AI health and safety application to be adopted, it requires acceptance by both managers and workers. For example, worker acceptance may be diminished by concerns about information privacy, or from a lack of trust and acceptance of the new technology, which may arise from inadequate transparency or training. Alternatively, managers may emphasize increases in economic productivity rather than gains in worker safety and health when implementing AI-based systems. === Eliminating hazardous tasks === AI may increase the scope of work tasks where a worker can be removed from a situation that carries risk. In a sense, while traditional automation can replace the functions of a worker's body with a robot, AI effectively replaces the functions of their brain with a computer. Hazards that can be avoided include stress, overwork, musculoskeletal injuries, and boredom. This can expand the range of affected job sectors into white-collar and service sector jobs such as in medicine, finance, and information technology. === Analytics to reduce risk === Machine learning is used for people analytics to make predictions about worker behavior to assist management decision-making, such as hiring and performance assessment. These could also be used to improve worker health. The analytics may be based on inputs such as online activities, monitoring of communications, location tracking, and voice analysis and body language analysis of filmed interviews. For example, sentiment analysis may be used to spot fatigue to prevent overwork. Decision support systems have a similar ability to be used to, for example, prevent industrial disasters or make disaster response more efficient. For manual material handling workers, predictive analytics and artificial intelligence may be used to reduce musculoskeletal injury. Traditional guidelines are based on statistical averages and are geared towards anthropometrically typical humans. The analysis of large amounts of data from wearable sensors may allow real-time, personalized calculation of ergonomic risk and fatigue management, as well as better analysis of the risk associated with specific job roles. Wearable sensors may also enable earlier intervention against exposure to toxic substances than is possible with area or breathing zone testing on a periodic basis. Furthermore, the large data sets generated could improve workplace health surveillance, risk assessment, and research. === Streamlining safety and health workflows === AI has also been used to attempt to make the workplace safety and health workflow more efficient. One example is coding of workers' compensation claims, which are submitted in a prose narrative form and must manually be assigned standardized codes. AI is being investigated to perform this task faster, more cheaply, and with fewer errors. == Hazards == There are several broad aspects of AI that may give rise to specific hazards. The risks depend on implementation rather than the mere presence of AI. Systems using sub-symbolic AI such as machine learning may behave unpredictably and are more prone to inscrutability in their decision-making. This is especially true if a situation is encountered that was not part of the AI's training dataset, and is exacerbated in environments that are less structured. Undesired behavior may also arise from flaws in the system's perception (arising either from within the software or from sensor degradation), knowledge representation and reasoning, or from software bugs. They may arise from improper training, such as a user applying the same algorithm to two problems that do not have the same requirements. Machine learning applied during the design phase may have different implications than that applied at runtime. Systems using symbolic AI are less prone to unpredictable behavior. The use of AI also increases cybersecurity risks relative to platforms that do not use AI, and information privacy concerns about collected data may pose a hazard to workers. === Psychosocial === Psychosocial hazards are those that arise from the way work is designed, organized, and managed, or its economic and social contexts, rather than arising from a physical substance or object. They cause not only psychiatric and psychological outcomes such as occupational burnout, anxiety disorders, and depression, but they can also cause physical injury or illness such as cardiovascular disease or musculoskeletal injury. Many hazards of AI are psychosocial in nature due to its potential to cause changes in work organization, in terms of increasing complexity and interaction between different organizational factors. However, psychosocial risks are often overlooked by designers of advanced manufacturing systems. Einola and Khoreva explore how different organizational groups perceive and interact with AI technologies. Their research shows that successful AI integration depends on human ownership and contextual understanding. They caution against blind technological optimism and stress the importance of tailoring AI use to specific workplace ecosystems. This perspective reinforces the need for inclusive design and transparent implementation strategies. ==== Changes in work practices ==== Over-reliance on AI tools may lead to deskilling of some professions. When AI becomes a substitute for traditional peer collaboration and mentorship, there is a risk of diminishing opportunities for interpersonal skill development and team-based learning. Increased monitoring may lead to micromanagement and thus to stress and anxiety. A perception of surveillance may also lead to stress. Controls for these include consultation with worker groups, extensive testing, and attention to introduced bias. Wearable sensors, activity trackers, and augmented reality may also lead to stress from micromanagement, both for assembly line workers and gig workers. Gig workers also lack the legal protections and rights of formal workers. Newell & Marabelli argue that AI alters power dynamics and employee autonomy, requiring a more nuanced understanding of its social and organizational implications. There is also the risk of people being forced to work at a robot's pace, or to monitor robot performance at nonstandard hours. A 2025 preprint paper based on users' interactions with the AI chatbot Microsoft Copilot identified forty jobs that the author's claimed had high overlaps with the capabilities of AI. Some media outlets used this paper to report on jobs becoming obsolete. Cri
AlphaTensor
AlphaTensor is an artificial intelligence system developed by DeepMind for discovering efficient matrix multiplication algorithms using reinforcement learning. Introduced in 2022, the system was based on AlphaZero and formulated the search for matrix multiplication algorithms as a single-player game called TensorGame. AlphaTensor was designed to search for new ways to multiply matrices with fewer scalar multiplication operations. Matrix multiplication is a fundamental operation in linear algebra, numerical analysis, scientific computing, computer graphics, and machine learning. The system discovered thousands of matrix multiplication algorithms, including algorithms that rediscovered known human-designed methods and others that improved on previously known results for particular matrix sizes and mathematical settings. == Background == Matrix multiplication is one of the basic operations in numerical computing. The standard algorithm for multiplying two square matrices has cubic time complexity, while faster algorithms such as the Strassen algorithm reduce the number of multiplication operations by using more complex algebraic decompositions. Finding optimal matrix multiplication algorithms can be difficult because it involves searching through a large space of possible tensor decompositions. AlphaTensor approached this problem by representing algorithm discovery as TensorGame, in which each move corresponds to an operation that reduces a tensor representing matrix multiplication. The goal of the game is to find a low-rank decomposition of the matrix multiplication tensor, corresponding to an efficient multiplication algorithm. == Development == AlphaTensor was developed by DeepMind and described in a paper published in Nature in October 2022. The system built on the reinforcement-learning approach used in AlphaZero, which had previously been applied to games such as Go, chess, and shogi. Unlike those games, TensorGame involved a very large search space, requiring changes to the AlphaZero-style search method and neural network architecture. DeepMind released source code and discovered algorithms associated with the publication through a public GitHub repository. == Results == AlphaTensor discovered matrix multiplication algorithms over both standard arithmetic and finite fields. One widely reported result was a method for multiplying 4 × 4 matrices over the field with two elements using 47 multiplication operations, improving on the 49 operations required by applying Strassen's algorithm recursively in that setting. The system also found algorithms optimized for particular computer hardware, including algorithms designed for graphics processing units and Tensor Processing Units. DeepMind stated that some of the hardware-specific algorithms improved practical execution time compared with commonly used algorithms on the tested hardware. == Significance == AlphaTensor was described as an example of using machine learning not only to apply existing algorithms, but to assist in discovering new ones. The work was connected to broader research in algorithm discovery, automated machine learning, program synthesis, and computational complexity theory, especially the open problem of determining the optimal complexity of matrix multiplication. AlphaTensor later became part of a broader group of Google DeepMind systems for algorithm and mathematical discovery, alongside systems such as AlphaDev and AlphaEvolve.
Information flow
In discourse-based grammatical theory, information flow is any tracking of referential information by speakers. Information may be new, i.e., just introduced into the conversation; given, i.e., already active in the speakers' consciousness; or old, i.e., no longer active. The various types of activation, and how these are defined, are model-dependent. Information flow affects grammatical structures such as: Word order (topic, focus, and afterthought constructions). Active, passive, or middle voice. Choice of deixis, such as articles; "medial" deictics such as Spanish ese and Japanese sore are generally determined by the familiarity of a referent rather than by physical distance. Overtness of information, such as whether an argument of a verb is indicated by a lexical noun phrase, a pronoun, or not mentioned at all. Clefting: Splitting a single clause into two clauses, each with its own verb, e.g. ‘The chicken turtles tasted like chicken.’ becomes ‘It was the chicken turtle | that tasted like chicken.’ In this case, clefting is used to shift the focus of the sentence to the subject, the chicken turtle. Front focus: Placing at the start (front) of a sentence information that would normally occur later in the sentence, to give it extra prominence. For example, in pop culture, Yoda's speech often utilizes such syntactic construction, such as when he says 'much to learn you still have' to Luke Skywalker. End focus (or end weight): Given or familiar information followed by new information. This gives prominence to the final part of the sentences and can enable suspense to build, e.g. ‘Through the door came a gigantic wolf’.(Umer Prince)
Zassenhaus algorithm
In mathematics, the Zassenhaus algorithm is a method to calculate a basis for the intersection and sum of two subspaces of a vector space. It is named after Hans Zassenhaus, but no publication of this algorithm by him is known. It is used in computer algebra systems. == Algorithm == === Input === Let V be a vector space and U, W two finite-dimensional subspaces of V with the following spanning sets: U = ⟨ u 1 , … , u n ⟩ {\displaystyle U=\langle u_{1},\ldots ,u_{n}\rangle } and W = ⟨ w 1 , … , w k ⟩ . {\displaystyle W=\langle w_{1},\ldots ,w_{k}\rangle .} Finally, let B 1 , … , B m {\displaystyle B_{1},\ldots ,B_{m}} be linearly independent vectors so that u i {\displaystyle u_{i}} and w i {\displaystyle w_{i}} can be written as u i = ∑ j = 1 m a i , j B j {\displaystyle u_{i}=\sum _{j=1}^{m}a_{i,j}B_{j}} and w i = ∑ j = 1 m b i , j B j . {\displaystyle w_{i}=\sum _{j=1}^{m}b_{i,j}B_{j}.} === Output === The algorithm computes the base of the sum U + W {\displaystyle U+W} and a base of the intersection U ∩ W {\displaystyle U\cap W} . === Algorithm === The algorithm creates the following block matrix of size ( ( n + k ) × ( 2 m ) ) {\displaystyle ((n+k)\times (2m))} : ( a 1 , 1 a 1 , 2 ⋯ a 1 , m a 1 , 1 a 1 , 2 ⋯ a 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a n , 1 a n , 2 ⋯ a n , m a n , 1 a n , 2 ⋯ a n , m b 1 , 1 b 1 , 2 ⋯ b 1 , m 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ b k , 1 b k , 2 ⋯ b k , m 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,m}&a_{1,1}&a_{1,2}&\cdots &a_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,m}&a_{n,1}&a_{n,2}&\cdots &a_{n,m}\\b_{1,1}&b_{1,2}&\cdots &b_{1,m}&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\b_{k,1}&b_{k,2}&\cdots &b_{k,m}&0&0&\cdots &0\end{pmatrix}}} Using elementary row operations, this matrix is transformed to the row echelon form. Then, it has the following shape: ( c 1 , 1 c 1 , 2 ⋯ c 1 , m ∙ ∙ ⋯ ∙ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ c q , 1 c q , 2 ⋯ c q , m ∙ ∙ ⋯ ∙ 0 0 ⋯ 0 d 1 , 1 d 1 , 2 ⋯ d 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 d ℓ , 1 d ℓ , 2 ⋯ d ℓ , m 0 0 ⋯ 0 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}c_{1,1}&c_{1,2}&\cdots &c_{1,m}&\bullet &\bullet &\cdots &\bullet \\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\c_{q,1}&c_{q,2}&\cdots &c_{q,m}&\bullet &\bullet &\cdots &\bullet \\0&0&\cdots &0&d_{1,1}&d_{1,2}&\cdots &d_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&d_{\ell ,1}&d_{\ell ,2}&\cdots &d_{\ell ,m}\\0&0&\cdots &0&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&0&0&\cdots &0\end{pmatrix}}} Here, ∙ {\displaystyle \bullet } stands for arbitrary numbers, and the vectors ( c p , 1 , c p , 2 , … , c p , m ) {\displaystyle (c_{p,1},c_{p,2},\ldots ,c_{p,m})} for every p ∈ { 1 , … , q } {\displaystyle p\in \{1,\ldots ,q\}} and ( d p , 1 , … , d p , m ) {\displaystyle (d_{p,1},\ldots ,d_{p,m})} for every p ∈ { 1 , … , ℓ } {\displaystyle p\in \{1,\ldots ,\ell \}} are nonzero. Then ( y 1 , … , y q ) {\displaystyle (y_{1},\ldots ,y_{q})} with y i := ∑ j = 1 m c i , j B j {\displaystyle y_{i}:=\sum _{j=1}^{m}c_{i,j}B_{j}} is a basis of U + W {\displaystyle U+W} and ( z 1 , … , z ℓ ) {\displaystyle (z_{1},\ldots ,z_{\ell })} with z i := ∑ j = 1 m d i , j B j {\displaystyle z_{i}:=\sum _{j=1}^{m}d_{i,j}B_{j}} is a basis of U ∩ W {\displaystyle U\cap W} . === Proof of correctness === First, we define π 1 : V × V → V , ( a , b ) ↦ a {\displaystyle \pi _{1}:V\times V\to V,(a,b)\mapsto a} to be the projection to the first component. Let H := { ( u , u ) ∣ u ∈ U } + { ( w , 0 ) ∣ w ∈ W } ⊆ V × V . {\displaystyle H:=\{(u,u)\mid u\in U\}+\{(w,0)\mid w\in W\}\subseteq V\times V.} Then π 1 ( H ) = U + W {\displaystyle \pi _{1}(H)=U+W} and H ∩ ( 0 × V ) = 0 × ( U ∩ W ) {\displaystyle H\cap (0\times V)=0\times (U\cap W)} . Also, H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} is the kernel of π 1 | H {\displaystyle {\pi _{1}|}_{H}} , the projection restricted to H. Therefore, dim ( H ) = dim ( U + W ) + dim ( U ∩ W ) {\displaystyle \dim(H)=\dim(U+W)+\dim(U\cap W)} . The Zassenhaus algorithm calculates a basis of H. In the first m columns of this matrix, there is a basis y i {\displaystyle y_{i}} of U + W {\displaystyle U+W} . The rows of the form ( 0 , z i ) {\displaystyle (0,z_{i})} (with z i ≠ 0 {\displaystyle z_{i}\neq 0} ) are obviously in H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} . Because the matrix is in row echelon form, they are also linearly independent. All rows which are different from zero ( ( y i , ∙ ) {\displaystyle (y_{i},\bullet )} and ( 0 , z i ) {\displaystyle (0,z_{i})} ) are a basis of H, so there are dim ( U ∩ W ) {\displaystyle \dim(U\cap W)} such z i {\displaystyle z_{i}} s. Therefore, the z i {\displaystyle z_{i}} s form a basis of U ∩ W {\displaystyle U\cap W} . == Example == Consider the two subspaces U = ⟨ ( 1 − 1 0 1 ) , ( 0 0 1 − 1 ) ⟩ {\displaystyle U=\left\langle \left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right\rangle } and W = ⟨ ( 5 0 − 3 3 ) , ( 0 5 − 3 − 2 ) ⟩ {\displaystyle W=\left\langle \left({\begin{array}{r}5\\0\\-3\\3\end{array}}\right),\left({\begin{array}{r}0\\5\\-3\\-2\end{array}}\right)\right\rangle } of the vector space R 4 {\displaystyle \mathbb {R} ^{4}} . Using the standard basis, we create the following matrix of dimension ( 2 + 2 ) × ( 2 ⋅ 4 ) {\displaystyle (2+2)\times (2\cdot 4)} : ( 1 − 1 0 1 1 − 1 0 1 0 0 1 − 1 0 0 1 − 1 5 0 − 3 3 0 0 0 0 0 5 − 3 − 2 0 0 0 0 ) . {\displaystyle \left({\begin{array}{rrrrrrrr}1&-1&0&1&&1&-1&0&1\\0&0&1&-1&&0&0&1&-1\\\\5&0&-3&3&&0&0&0&0\\0&5&-3&-2&&0&0&0&0\end{array}}\right).} Using elementary row operations, we transform this matrix into the following matrix: ( 1 0 0 0 ∙ ∙ ∙ ∙ 0 1 0 − 1 ∙ ∙ ∙ ∙ 0 0 1 − 1 ∙ ∙ ∙ ∙ 0 0 0 0 1 − 1 0 1 ) {\displaystyle \left({\begin{array}{rrrrrrrrr}1&0&0&0&&\bullet &\bullet &\bullet &\bullet \\0&1&0&-1&&\bullet &\bullet &\bullet &\bullet \\0&0&1&-1&&\bullet &\bullet &\bullet &\bullet \\\\0&0&0&0&&1&-1&0&1\end{array}}\right)} (Some entries have been replaced by " ∙ {\displaystyle \bullet } " because they are irrelevant to the result.) Therefore ( ( 1 0 0 0 ) , ( 0 1 0 − 1 ) , ( 0 0 1 − 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\0\\0\\0\end{array}}\right),\left({\begin{array}{r}0\\1\\0\\-1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right)} is a basis of U + W {\displaystyle U+W} , and ( ( 1 − 1 0 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right)\right)} is a basis of U ∩ W {\displaystyle U\cap W} .
Invoicera
Invoicera is an online invoicing software. The software was created by a company with the same name that was founded in 2006, had 125 employees, and is based in India. It allows users to monitor, dispatch, and accept invoices in one web service. After signing up for the service, users are assigned a personal subdomain to set up their invoice configuration. It allows users to add clients' data to the service through uploading a Microsoft Excel file. Invoicera is compatible with businesses of varying sizes, including freelancers, small businesses, and large businesses. It is compatible with Basecamp, a project-management tool, so Invoicera can upload data from Basecamp. The software interfaces with more than 25 payment gateways. It supports subscriptions and repeated invoices and allows clients to schedule late fees when payments have not been made on time. Invoicera uses freemium model, letting users dispatch an unrestricted number of invoices to at most three customers. Chelsea Krause wrote in a 2019 review for Merchant Maverick, "Unfortunately, the software isn't as developed as it could be. Time tracking and reporting are limited and there are no live bank feeds — which is surprising for a company so focused on automation (especially since even many of the worst invoicing options out there still offer live bank feeds)." She further criticized Invoicera for having bad customer service and the software for not having recent changes. Brian Turner wrote in TechRadar that Invoicera had fewer templates compared to the other services he reviewed but "the ones offered are fully customizable". Rob Clymo wrote in TechRadar that "Invoicera lets you automate your invoicing and billing needs without too much in the way of hassle" and that although it "isn't a complete accounts solution ... it's a powerful supplement".
External memory algorithm
In computing, external memory algorithms or out-of-core algorithms are algorithms that are designed to process data that are too large to fit into a computer's main memory at once. Such algorithms must be optimized to efficiently fetch and access data stored in slow bulk memory (auxiliary memory) such as hard drives or tape drives, or when memory is on a computer network. External memory algorithms are analyzed in the external memory model. == Model == External memory algorithms are analyzed in an idealized model of computation called the external memory model (or I/O model, or disk access model). The external memory model is an abstract machine similar to the RAM machine model, but with a cache in addition to main memory. The model captures the fact that read and write operations are much faster in a cache than in main memory, and that reading long contiguous blocks is faster than reading randomly using a disk read-and-write head. The running time of an algorithm in the external memory model is defined by the number of reads and writes to memory required. The model was introduced by Alok Aggarwal and Jeffrey Vitter in 1988. The external memory model is related to the cache-oblivious model, but algorithms in the external memory model may know both the block size and the cache size. For this reason, the model is sometimes referred to as the cache-aware model. The model consists of a processor with an internal memory or cache of size M, connected to an unbounded external memory. Both the internal and external memory are divided into blocks of size B. One input/output or memory transfer operation consists of moving a block of B contiguous elements from external to internal memory, and the running time of an algorithm is determined by the number of these input/output operations. == Algorithms == Algorithms in the external memory model take advantage of the fact that retrieving one object from external memory retrieves an entire block of size B. This property is sometimes referred to as locality. Searching for an element among N objects is possible in the external memory model using a B-tree with branching factor B. Using a B-tree, searching, insertion, and deletion can be achieved in O ( log B N ) {\displaystyle O(\log _{B}N)} time (in Big O notation). Information theoretically, this is the minimum running time possible for these operations, so using a B-tree is asymptotically optimal. External sorting is sorting in an external memory setting. External sorting can be done via distribution sort, which is similar to quicksort, or via a M B {\displaystyle {\tfrac {M}{B}}} -way merge sort. Both variants achieve the asymptotically optimal runtime of O ( N B log M B N B ) {\displaystyle O\left({\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)} to sort N objects. This bound also applies to the fast Fourier transform in the external memory model. The permutation problem is to rearrange N elements into a specific permutation. This can either be done either by sorting, which requires the above sorting runtime, or inserting each element in order and ignoring the benefit of locality. Thus, permutation can be done in O ( min ( N , N B log M B N B ) ) {\displaystyle O\left(\min \left(N,{\frac {N}{B}}\log _{\frac {M}{B}}{\frac {N}{B}}\right)\right)} time. == Applications == The external memory model captures the memory hierarchy, which is not modeled in other common models used in analyzing data structures, such as the random-access machine, and is useful for proving lower bounds for data structures. The model is also useful for analyzing algorithms that work on datasets too big to fit in internal memory. A typical example is geographic information systems, especially digital elevation models, where the full data set easily exceeds several gigabytes or even terabytes of data. This methodology extends beyond general purpose CPUs and also includes GPU computing as well as classical digital signal processing. In general-purpose computing on graphics processing units (GPGPU), powerful graphics cards (GPUs) with little memory (compared with the more familiar system memory, which is most often referred to simply as RAM) are utilized with relatively slow CPU-to-GPU memory transfer (when compared with computation bandwidth). == History == An early use of the term "out-of-core" as an adjective is in 1962 in reference to devices that are other than the core memory of an IBM 360. An early use of the term "out-of-core" with respect to algorithms appears in 1971.