Retained mode in computer graphics is a major pattern of API design in graphics libraries, in which the graphics library, instead of the client, retains the scene (complete object model of the rendering primitives) to be rendered and the client calls into the graphics library do not directly cause actual rendering, but make use of extensive indirection to resources, managed – thus retained – by the graphics library. It does not preclude the use of double-buffering. Immediate mode is an alternative approach. Historically, retained mode has been the dominant style in GUI libraries; however, both can coexist in the same library and are not necessarily exclusionary in practice. == Overview == In retained mode the client calls do not directly cause actual rendering, but instead update an abstract internal model (typically a list of objects) which is maintained within the library's data space. This allows the library to optimize when actual rendering takes place along with the processing of related objects. Some techniques to optimize rendering include: managing double buffering treatment of hidden surfaces by backface culling/occlusion culling (Z-buffering) only transferring data that has changed from one frame to the next from the application to the library Example of coexistence with immediate mode in the same library is OpenGL. OpenGL has immediate mode functions that can use previously defined server side objects (textures, vertex buffers and index buffers, shaders, etc.) without resending unchanged data. Examples of retained mode rendering systems include Windows Presentation Foundation, SceneKit on macOS, and PHIGS.
Learning to rank
Learning to rank (LTR) or machine-learned ranking (MLR) is the application of machine learning, often supervised, semi-supervised or reinforcement learning, in the construction of ranking models for information retrieval and recommender systems. Training data may, for example, consist of lists of items with some partial order specified between items in each list. This order is typically induced by giving a numerical or ordinal score or a binary judgment (e.g. "relevant" or "not relevant") for each item. The goal of constructing the ranking model is to rank new, unseen lists in a similar way to rankings in the training data. == Applications == === In information retrieval === Ranking is a central part of many information retrieval problems, such as document retrieval, collaborative filtering, sentiment analysis, and online advertising. A possible architecture of a machine-learned search engine is shown in the accompanying figure. Training data consists of queries and documents matching them together with the relevance degree of each match. It may be prepared manually by human assessors (or raters, as Google calls them), who check results for some queries and determine relevance of each result. It is not feasible to check the relevance of all documents, and so typically a technique called pooling is used — only the top few documents, retrieved by some existing ranking models are checked. This technique may introduce selection bias. Alternatively, training data may be derived automatically by analyzing clickthrough logs (i.e. search results which got clicks from users), query chains, or such search engines' features as Google's (since-replaced) SearchWiki. Clickthrough logs can be biased by the tendency of users to click on the top search results on the assumption that they are already well-ranked. Training data is used by a learning algorithm to produce a ranking model which computes the relevance of documents for actual queries. Typically, users expect a search query to complete in a short time (such as a few hundred milliseconds for web search), which makes it impossible to evaluate a complex ranking model on each document in the corpus, and so a two-phase scheme is used. First, a small number of potentially relevant documents are identified using simpler retrieval models which permit fast query evaluation, such as the vector space model, Boolean model, weighted AND, or BM25. This phase is called top- k {\displaystyle k} document retrieval and many heuristics were proposed in the literature to accelerate it, such as using a document's static quality score and tiered indexes. In the second phase, a more accurate but computationally expensive machine-learned model is used to re-rank these documents. === In other areas === Learning to rank algorithms have been applied in areas other than information retrieval: In machine translation for ranking a set of hypothesized translations; In computational biology for ranking candidate 3-D structures in protein structure prediction problems; In recommender systems for identifying a ranked list of related news articles to recommend to a user after he or she has read a current news article. == Feature vectors == For the convenience of MLR algorithms, query-document pairs are usually represented by numerical vectors, which are called feature vectors. Such an approach is sometimes called bag of features and is analogous to the bag of words model and vector space model used in information retrieval for representation of documents. Components of such vectors are called features, factors or ranking signals. They may be divided into three groups (features from document retrieval are shown as examples): Query-independent or static features — those features, which depend only on the document, but not on the query. For example, PageRank or document's length. Such features can be precomputed in off-line mode during indexing. They may be used to compute document's static quality score (or static rank), which is often used to speed up search query evaluation. Query-dependent or dynamic features — those features, which depend both on the contents of the document and the query, such as TF-IDF score or other non-machine-learned ranking functions. Query-level features or query features, which depend only on the query. For example, the number of words in a query. Some examples of features, which were used in the well-known LETOR dataset: TF, TF-IDF, BM25, and language modeling scores of document's zones (title, body, anchors text, URL) for a given query; Lengths and IDF sums of document's zones; Document's PageRank, HITS ranks and their variants. Selecting and designing good features is an important area in machine learning, which is called feature engineering. == Evaluation measures == There are several measures (metrics) which are commonly used to judge how well an algorithm is doing on training data and to compare the performance of different MLR algorithms. Often a learning-to-rank problem is reformulated as an optimization problem with respect to one of these metrics. Examples of ranking quality measures: Mean average precision (MAP); DCG and NDCG; Precision@n, NDCG@n, where "@n" denotes that the metrics are evaluated only on top n documents; Mean reciprocal rank; Kendall's tau; Spearman's rho. DCG and its normalized variant NDCG are usually preferred in academic research when multiple levels of relevance are used. Other metrics such as MAP, MRR and precision, are defined only for binary judgments. Recently, there have been proposed several new evaluation metrics which claim to model user's satisfaction with search results better than the DCG metric: Expected reciprocal rank (ERR); Yandex's pfound. Both of these metrics are based on the assumption that the user is more likely to stop looking at search results after examining a more relevant document, than after a less relevant document. == Approaches == Learning to Rank approaches are often categorized using one of three approaches: pointwise (where individual documents are ranked), pairwise (where pairs of documents are ranked into a relative order), and listwise (where an entire list of documents are ordered). Tie-Yan Liu of Microsoft Research Asia has analyzed existing algorithms for learning to rank problems in his book Learning to Rank for Information Retrieval. He categorized them into three groups by their input spaces, output spaces, hypothesis spaces (the core function of the model) and loss functions: the pointwise, pairwise, and listwise approach. In practice, listwise approaches often outperform pairwise approaches and pointwise approaches. This statement was further supported by a large scale experiment on the performance of different learning-to-rank methods on a large collection of benchmark data sets. In this section, without further notice, x {\displaystyle x} denotes an object to be evaluated, for example, a document or an image, f ( x ) {\displaystyle f(x)} denotes a single-value hypothesis, h ( ⋅ ) {\displaystyle h(\cdot )} denotes a bi-variate or multi-variate function and L ( ⋅ ) {\displaystyle L(\cdot )} denotes the loss function. === Pointwise approach === In this case, it is assumed that each query-document pair in the training data has a numerical or ordinal score. Then the learning-to-rank problem can be approximated by a regression problem — given a single query-document pair, predict its score. Formally speaking, the pointwise approach aims at learning a function f ( x ) {\displaystyle f(x)} predicting the real-value or ordinal score of a document x {\displaystyle x} using the loss function L ( f ; x j , y j ) {\displaystyle L(f;x_{j},y_{j})} . A number of existing supervised machine learning algorithms can be readily used for this purpose. Ordinal regression and classification algorithms can also be used in pointwise approach when they are used to predict the score of a single query-document pair, and it takes a small, finite number of values. === Pairwise approach === In this case, the learning-to-rank problem is approximated by a classification problem — learning a binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} that can tell which document is better in a given pair of documents. The classifier shall take two documents as its input and the goal is to minimize a loss function L ( h ; x u , x v , y u , v ) {\displaystyle L(h;x_{u},x_{v},y_{u,v})} . The loss function typically reflects the number and magnitude of inversions in the induced ranking. In many cases, the binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} is implemented with a scoring function f ( x ) {\displaystyle f(x)} . As an example, RankNet adapts a probability model and defines h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} as the estimated probability of the document x u {\displaystyle x_{u}} has higher quality than x v {\displaystyle x_{v}} : P u , v ( f ) = CDF ( f ( x u ) − f ( x v ) ) , {\displaystyle P_{u,v}(f)={\text{CDF}
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
Ubiquitous computing
Ubiquitous computing (or "ubicomp") is a concept in software engineering, hardware engineering and computer science where computing is made to appear seamlessly anytime and everywhere. In contrast to desktop computing, ubiquitous computing implies use on any device, in any location, and in any format. A user interacts with the computer, which can exist in many different forms, including laptop computers, tablets, smart phones and terminals in everyday objects such as a refrigerator or a pair of glasses. The underlying technologies to support ubiquitous computing include the Internet, advanced middleware, kernels, operating systems, mobile codes, sensors, microprocessors, new I/Os and user interfaces, computer networks, mobile protocols, global navigational systems, and new materials. This paradigm is also described as pervasive computing, ambient intelligence, or "everyware". Each term emphasizes slightly different aspects. When primarily concerning the objects involved, it is also known as physical computing, the Internet of Things, haptic computing, and "things that think". Rather than propose a single definition for ubiquitous computing and for these related terms, a taxonomy of properties for ubiquitous computing has been proposed, from which different kinds or flavors of ubiquitous systems and applications can be described. Ubiquitous computing themes include: distributed computing, mobile computing, location computing, mobile networking, sensor networks, human–computer interaction, context-aware smart home technologies, and artificial intelligence. == Core concepts == Ubiquitous computing is the concept of using small internet connected and inexpensive computers to help with everyday functions in an automated fashion. Mark Weiser proposed three basic forms for ubiquitous computing devices: Tabs: a wearable device that is approximately a centimeter in size Pads: a hand-held device that is approximately a decimeter in size Boards: an interactive larger display device that is approximately a meter in size Ubiquitous computing devices proposed by Mark Weiser are all based around flat devices of different sizes with a visual display. These conceptual device categories were later implemented at Xerox PARC in experimental systems including the PARCTab, PARCPad, and LiveBoard, which served as early prototypes of handheld, tablet-style, and large interactive display computing environments. Expanding beyond those concepts there is a large array of other ubiquitous computing devices that could exist. == History == Mark Weiser coined the phrase "ubiquitous computing" around 1988, during his tenure as Chief Technologist of the Xerox Palo Alto Research Center (PARC). Both alone and with PARC Director and Chief Scientist John Seely Brown, Weiser wrote some of the earliest papers on the subject, largely defining it and sketching out its major concerns. == Recognizing the effects of extending processing power == Recognizing that the extension of processing power into everyday scenarios would necessitate understandings of social, cultural and psychological phenomena beyond its proper ambit, Weiser was influenced by many fields outside computer science, including "philosophy, phenomenology, anthropology, psychology, post-Modernism, sociology of science and feminist criticism". He was explicit about "the humanistic origins of the 'invisible ideal in post-modernist thought'", referencing as well the ironically dystopian Philip K. Dick novel Ubik. Andy Hopper from Cambridge University UK proposed and demonstrated the concept of "Teleporting" – where applications follow the user wherever he/she moves. Roy Want (now at Google), while at Olivetti Research Ltd, designed the first "Active Badge System", which is an advanced location computing system where personal mobility is merged with computing. Later at Xerox PARC, he designed and built the "PARCTab" or simply "Tab", widely recognized as the world's first Context-Aware computer, which has great similarity to the modern smartphone. Bill Schilit (now at Google) also did some earlier work in this topic, and participated in the early Mobile Computing workshop held in Santa Cruz in 1996. Ken Sakamura of the University of Tokyo, Japan leads the Ubiquitous Networking Laboratory (UNL), Tokyo as well as the T-Engine Forum. The joint goal of Sakamura's Ubiquitous Networking specification and the T-Engine forum, is to enable any everyday device to broadcast and receive information. MIT has also contributed significant research in this field, notably Things That Think consortium (directed by Hiroshi Ishii, Joseph A. Paradiso and Rosalind Picard) at the Media Lab and the CSAIL effort known as Project Oxygen. Other major contributors include University of Washington (Shwetak Patel, Anind Dey and James Landay), Dartmouth College's HealthX Lab (directed by Andrew Campbell), Georgia Tech's College of Computing (Gregory Abowd and Thad Starner), Cornell Tech's People Aware Computing Lab (directed by Tanzeem Choudhury), NYU's Interactive Telecommunications Program, UC Irvine's Department of Informatics, Microsoft Research, Intel Research and Equator, Ajou University UCRi & CUS. == Examples == One of the earliest ubiquitous systems was artist Natalie Jeremijenko's "Live Wire", also known as "Dangling String", installed at Xerox PARC during Mark Weiser's time there. This was a piece of string attached to a stepper motor and controlled by a LAN connection; network activity caused the string to twitch, yielding a peripherally noticeable indication of traffic. Weiser called this an example of calm technology. A present manifestation of this trend is the widespread diffusion of mobile phones. Many mobile phones support high speed data transmission, video services, and other services with powerful computational ability. Although these mobile devices are not necessarily manifestations of ubiquitous computing, there are examples, such as Japan's Yaoyorozu ("Eight Million Gods") Project in which mobile devices, coupled with radio frequency identification tags demonstrate that ubiquitous computing is already present in some form. Ambient Devices has produced an "orb", a "dashboard", and a "weather beacon": these decorative devices receive data from a wireless network and report current events, such as stock prices and the weather, like the Nabaztag, which was invented by Rafi Haladjian and Olivier Mével, and manufactured by the company Violet. The Australian futurist Mark Pesce has produced a highly configurable 52-LED LAMP enabled lamp which uses Wi-Fi named MooresCloud after Gordon Moore. The Unified Computer Intelligence Corporation launched a device called Ubi – The Ubiquitous Computer designed to allow voice interaction with the home and provide constant access to information. Ubiquitous computing research has focused on building an environment in which computers allow humans to focus attention on select aspects of the environment and operate in supervisory and policy-making roles. Ubiquitous computing emphasizes the creation of a human computer interface that can interpret and support a user's intentions. For example, MIT's Project Oxygen seeks to create a system in which computation is as pervasive as air: In the future, computation will be human centered. It will be freely available everywhere, like batteries and power sockets, or oxygen in the air we breathe...We will not need to carry our own devices around with us. Instead, configurable generic devices, either handheld or embedded in the environment, will bring computation to us, whenever we need it and wherever we might be. As we interact with these "anonymous" devices, they will adopt our information personalities. They will respect our desires for privacy and security. We won't have to type, click, or learn new computer jargon. Instead, we'll communicate naturally, using speech and gestures that describe our intent... This is a fundamental transition that does not seek to escape the physical world and "enter some metallic, gigabyte-infested cyberspace" but rather brings computers and communications to us, making them "synonymous with the useful tasks they perform". Network robots link ubiquitous networks with robots, contributing to the creation of new lifestyles and solutions to address a variety of social problems including the aging of population and nursing care. The "Continuity" set of features, introduced by Apple in OS X Yosemite, can be seen as an example of ubiquitous computing. == Issues == Privacy is easily the most often-cited criticism of ubiquitous computing (ubicomp), and may be the greatest barrier to its long-term success. == Research centres == This is a list of notable institutions who claim to have a focus on Ubiquitous computing sorted by country: Canada Topological Media Lab, Concordia University, Canada Finland Community Imaging Group, University of Oulu, Finland Germany Telecooperation Office (TECO), Karlsruhe Institute of Technology, Ger
Tuple
In mathematics, a tuple is a finite sequence (or ordered list) of numbers. More generally, it is a sequence of mathematical objects, called the elements of the tuple. An n-tuple is a tuple of n elements, where n is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences". Tuples are usually written by listing the elements within parentheses "( )" and separated by commas; for example, (2, 7, 4, 1, 7) denotes a 5-tuple. Other types of brackets are sometimes used, although they may have a different meaning. An n-tuple can be formally defined as the image of a function that has the set of the first n natural numbers as its domain (1, 2, ..., n). Tuples may be also defined from ordered pairs by a recurrence starting from an ordered pair; indeed, an n-tuple can be identified with the ordered pair of its (n − 1) first elements and its nth element, for example, ( ( ( 1 , 2 ) , 3 ) , 4 ) = ( 1 , 2 , 3 , 4 ) {\displaystyle \left(\left(\left(1,2\right),3\right),4\right)=\left(1,2,3,4\right)} . In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types, tightly associated with algebraic data types, pattern matching, and destructuring assignment. Many programming languages offer an alternative to tuples, known as record types, featuring unordered elements accessed by label. A few programming languages combine ordered tuple product types and unordered record types into a single construct, as in C structs and Haskell records. Relational databases may formally identify their rows (records) as tuples. Tuples also occur in relational algebra; when programming the semantic web with the Resource Description Framework (RDF); in linguistics; and in philosophy. == Etymology == The term originated as an abstraction of the sequence: single, couple/double, triple, quadruple, quintuple, sextuple, septuple, octuple, ..., n‑tuple, ..., where the prefixes are taken from the Latin names of the numerals. The unique 0-tuple is called the null tuple or empty tuple. A 1‑tuple is called a single (or singleton), a 2‑tuple is called an ordered pair or couple, and a 3‑tuple is called a triple (or triplet). The number n can be any nonnegative integer. For example, a complex number can be represented as a 2‑tuple of reals, a quaternion can be represented as a 4‑tuple, an octonion can be represented as an 8‑tuple, and a sedenion can be represented as a 16‑tuple. Although these uses treat ‑tuple as the suffix, the original suffix was ‑ple as in "triple" (three-fold) or "decuple" (ten‑fold). This originates from medieval Latin plus (meaning "more") related to Greek ‑πλοῦς, which replaced the classical and late antique ‑plex (meaning "folded"), as in "duplex". == Properties == The general rule for the identity of two n-tuples is ( a 1 , a 2 , … , a n ) = ( b 1 , b 2 , … , b n ) {\displaystyle (a_{1},a_{2},\ldots ,a_{n})=(b_{1},b_{2},\ldots ,b_{n})} if and only if a 1 = b 1 , a 2 = b 2 , … , a n = b n {\displaystyle a_{1}=b_{1},{\text{ }}a_{2}=b_{2},{\text{ }}\ldots ,{\text{ }}a_{n}=b_{n}} . Thus a tuple has properties that distinguish it from a set: A tuple may contain multiple instances of the same element, so tuple ( 1 , 2 , 2 , 3 ) ≠ ( 1 , 2 , 3 ) {\displaystyle (1,2,2,3)\neq (1,2,3)} ; but set { 1 , 2 , 2 , 3 } = { 1 , 2 , 3 } {\displaystyle \{1,2,2,3\}=\{1,2,3\}} . Tuple elements are ordered: tuple ( 1 , 2 , 3 ) ≠ ( 3 , 2 , 1 ) {\displaystyle (1,2,3)\neq (3,2,1)} , but set { 1 , 2 , 3 } = { 3 , 2 , 1 } {\displaystyle \{1,2,3\}=\{3,2,1\}} . A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements. == Definitions == There are several definitions of tuples that give them the properties described in the previous section. === Tuples as functions === The 0 {\displaystyle 0} -tuple may be identified as the empty function. For n ≥ 1 , {\displaystyle n\geq 1,} the n {\displaystyle n} -tuple ( a 1 , … , a n ) {\displaystyle \left(a_{1},\ldots ,a_{n}\right)} may be identified with the surjective function F : { 1 , … , n } → { a 1 , … , a n } {\displaystyle F~:~\left\{1,\ldots ,n\right\}~\to ~\left\{a_{1},\ldots ,a_{n}\right\}} with domain domain F = { 1 , … , n } = { i ∈ N : 1 ≤ i ≤ n } {\displaystyle \operatorname {domain} F=\left\{1,\ldots ,n\right\}=\left\{i\in \mathbb {N} :1\leq i\leq n\right\}} and with codomain codomain F = { a 1 , … , a n } , {\displaystyle \operatorname {codomain} F=\left\{a_{1},\ldots ,a_{n}\right\},} that is defined at i ∈ domain F = { 1 , … , n } {\displaystyle i\in \operatorname {domain} F=\left\{1,\ldots ,n\right\}} by F ( i ) := a i . {\displaystyle F(i):=a_{i}.} That is, F {\displaystyle F} is the function defined by 1 ↦ a 1 ⋮ n ↦ a n {\displaystyle {\begin{alignedat}{3}1\;&\mapsto &&\;a_{1}\\\;&\;\;\vdots &&\;\\n\;&\mapsto &&\;a_{n}\\\end{alignedat}}} in which case the equality ( a 1 , a 2 , … , a n ) = ( F ( 1 ) , F ( 2 ) , … , F ( n ) ) {\displaystyle \left(a_{1},a_{2},\dots ,a_{n}\right)=\left(F(1),F(2),\dots ,F(n)\right)} necessarily holds. Tuples as sets of ordered pairs Functions are commonly identified with their graphs, which is a certain set of ordered pairs. Indeed, many authors use graphs as the definition of a function. Using this definition of "function", the above function F {\displaystyle F} can be defined as: F := { ( 1 , a 1 ) , … , ( n , a n ) } . {\displaystyle F~:=~\left\{\left(1,a_{1}\right),\ldots ,\left(n,a_{n}\right)\right\}.} === Tuples as nested ordered pairs === Another way of modeling tuples in set theory is as nested ordered pairs. This approach assumes that the notion of ordered pair has already been defined. The 0-tuple (i.e. the empty tuple) is represented by the empty set ∅ {\displaystyle \emptyset } . An n-tuple, with n > 0, can be defined as an ordered pair of its first entry and an (n − 1)-tuple (which contains the remaining entries when n > 1): ( a 1 , a 2 , a 3 , … , a n ) = ( a 1 , ( a 2 , a 3 , … , a n ) ) {\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},a_{3},\ldots ,a_{n}))} This definition can be applied recursively to the (n − 1)-tuple: ( a 1 , a 2 , a 3 , … , a n ) = ( a 1 , ( a 2 , ( a 3 , ( … , ( a n , ∅ ) … ) ) ) ) {\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=(a_{1},(a_{2},(a_{3},(\ldots ,(a_{n},\emptyset )\ldots ))))} Thus, for example: ( 1 , 2 , 3 ) = ( 1 , ( 2 , ( 3 , ∅ ) ) ) ( 1 , 2 , 3 , 4 ) = ( 1 , ( 2 , ( 3 , ( 4 , ∅ ) ) ) ) {\displaystyle {\begin{aligned}(1,2,3)&=(1,(2,(3,\emptyset )))\\(1,2,3,4)&=(1,(2,(3,(4,\emptyset ))))\\\end{aligned}}} A variant of this definition starts "peeling off" elements from the other end: The 0-tuple is the empty set ∅ {\displaystyle \emptyset } . For n > 0: ( a 1 , a 2 , a 3 , … , a n ) = ( ( a 1 , a 2 , a 3 , … , a n − 1 ) , a n ) {\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((a_{1},a_{2},a_{3},\ldots ,a_{n-1}),a_{n})} This definition can be applied recursively: ( a 1 , a 2 , a 3 , … , a n ) = ( ( … ( ( ( ∅ , a 1 ) , a 2 ) , a 3 ) , … ) , a n ) {\displaystyle (a_{1},a_{2},a_{3},\ldots ,a_{n})=((\ldots (((\emptyset ,a_{1}),a_{2}),a_{3}),\ldots ),a_{n})} Thus, for example: ( 1 , 2 , 3 ) = ( ( ( ∅ , 1 ) , 2 ) , 3 ) ( 1 , 2 , 3 , 4 ) = ( ( ( ( ∅ , 1 ) , 2 ) , 3 ) , 4 ) {\displaystyle {\begin{aligned}(1,2,3)&=(((\emptyset ,1),2),3)\\(1,2,3,4)&=((((\emptyset ,1),2),3),4)\\\end{aligned}}} === Tuples as nested sets === Using Kuratowski's representation for an ordered pair, the second definition above can be reformulated in terms of pure set theory: The 0-tuple (i.e. the empty tuple) is represented by the empty set ∅ {\displaystyle \emptyset } ; Let x {\displaystyle x} be an n-tuple ( a 1 , a 2 , … , a n ) {\displaystyle (a_{1},a_{2},\ldots ,a_{n})} , and let x → b ≡ ( a 1 , a 2 , … , a n , b ) {\displaystyle x\rightarrow b\equiv (a_{1},a_{2},\ldots ,a_{n},b)} . Then, x → b ≡ { { x } , { x , b } } {\displaystyle x\rightarrow b\equiv \{\{x\},\{x,b\}\}} . (The right arrow, → {\displaystyle \rightarrow } , could be read as "adjoined with".) In this formulation: ( ) = ∅ ( 1 ) = ( ) → 1 = { { ( ) } , { ( ) , 1 } } = { { ∅ } , { ∅ , 1 } } ( 1 , 2 ) = ( 1 ) → 2 = { { ( 1 ) } , { ( 1 ) , 2 } } = { { { { ∅ } , { ∅ , 1 } } } , { { { ∅ } , { ∅ , 1 } } , 2 } } ( 1 , 2 , 3 ) = ( 1 , 2 ) → 3 = { { ( 1 , 2 ) } , { ( 1 , 2 ) , 3 } } = { { { { { { ∅ } , { ∅ , 1 } } } , { { { ∅ } , { ∅ , 1 } } , 2 } } } , { { { { { ∅ } , { ∅ , 1 } } } , { { { ∅ } , { ∅ , 1 } } , 2 } } , 3 } } {\displaystyle {\begin{array}{lclcl}()&&&=&\emptyset \\&&&&\\(1)&=&()\rightarrow 1&=&\{\{()\},\{(),1\}\}\\&&&=&\{\{\emptyset \},\{\emptyset ,1\}\}\\&&&&\\(1,2)&=&(1)\rightarrow 2&=&\{\{(1)\},\{(1),2\}\}\\&&&=&\{\{\{\{\emptyset \},\{\emptyset ,1\}\}\},\\&&&&\{\{\{\emptyset \},\{\emptyset ,1\}\},2\}\}\\&&&&\\(1,2,3)&=&(1,2)\rightarrow 3&=&\{\{(1,2)\},\{(1,2),3\}\}\\&&&=&\{\{\{\{\{\{\empty
Packed pixel
In packed pixel or chunky framebuffer organization, the bits defining each pixel are clustered and stored consecutively. For example, if there are 16 bits per pixel, each pixel is represented in two consecutive (contiguous) 8-bit bytes in the framebuffer. If there are 4 bits per pixel, each framebuffer byte defines two pixels, one in each nibble. The latter example is as opposed to storing a single 4-bit pixel in a byte, leaving 4 bits of the byte unused. If a pixel has more than one channel, the channels are interleaved when using packed pixel organization. Packed pixel displays were common on early microcomputer system that shared a single main memory for both the central processing unit (CPU) and display driver. In such systems, memory was normally accessed a byte at a time, so by packing the pixels, the display system could read out several pixels worth of data in a single read operation. Packed pixel is one of two major ways to organize graphics data in memory, the other being planar organization, where each pixel is made of individual bits stored in their own plane. For a 4-bit color value, memory would be organized as four screen-sized planes of one bit each and a single pixel's value built up by selecting the appropriate bit from each plane. Planar organization has the advantage that the data can be accessed in parallel, and is used when memory bandwidth is an issue.
Seismological Facility for the Advancement of Geoscience
The U.S. National Science Foundation's Seismological Facility for the Advancement of Geoscience (NSF SAGE) is a distributed, multi-user national facility that provides support for state of-the-art seismic research. It is operated by EarthScope Consortium. Its previous operator was the Incorporated Research Institutions for Seismology (IRIS), until its merger with UNAVCO to become EarthScope Consortium. NSF SAGE is one of the two premier geophysical facilities in support of geoscience and geoscience education of the National Science Foundation. The other premiere geophysical facility is NSF GAGE, the Geodetic Facility for the Advancement of Geoscience. The services of the facility include support for the Global Seismographic Network (GSN), Data Services, and instrument support via the EarthScope Primary Instrument Center (EPIC), including magnetotelluric (MT) geophysical research. == Global Seismographic Network (GSN) == NSF SAGE manages 40 stations of the 152-station Global Seismographic Network (GSN) for basic global seismicity and Earth structure research. The GSN also enables earthquake hazard mission-related data operations such as: Earthquake location and characterization Tsunami warning Nuclear explosion monitoring == Data Services == SAGE Data Services (DS) is the largest facility for the archiving, curation, and distribution of seismological and other geophysical data in the world. == EarthScope Primary Instrument Center (EPIC) == The EPIC facility maintains the largest open access, shared-use pool of portable seismic sensors in the world. It is located on the campus of New Mexico Tech. == MT == NSF SAGE provides instruments for magnetotelluric (MT) or electromagnetic geophysical research for the recording of our planet's ambient electric and magnetic fields, which allow for the characterization of the conductivity of the area consisting of the shallow crust to upper mantle. This helps with analysis of results obtained from seismic imaging methodologies. The NSF SAGE facility is: Developing open source MT data formatting and processing software. Providing access to proprietary software products.