AI App Just Like Chatgpt

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  List of Fortran software and tools

  This is a list of Fortran software and tools, including IDEs, compilers, libraries, debugging tools, numerical and scientific computing tools, and related projects. == Fortran compilers == Absoft Pro Fortran — Absoft Pro Fortran is discontinued and ran on Linux and macOS AOCC — from AMD Classic Flang — part of the LLVM Project LLVM Flang — part of the LLVM Project Fortran 77 — Fortran 77 was developed by Digital Equipment Corporation, it is discontinued. G95 – portable open-source Fortran 95 compiler GCC (GNU Fortran) PGI compilers – NVIDIA developed compilers after acquiring The Portland Group IBM XL Fortran — IBM XL Fortran is current and runs on Linux (Power/AIX) and integrates with Eclipse Intel Fortran Compiler – part of Intel OneAPI HPC toolkit LFortran — LFortran is current, cross-platform, and has IDE support. MinGW – cross compiler and forked into Mingw-w64 nAG Fortran Compiler - from nAG Open64 — Open64 is an open-source compiler that has been terminated and ran on Linux Open Watcom — Open Watcom is current, runs on MS-DOS and OS/2, and has IDE support. Oracle Fortran — Oracle Fortran is discontinued, ran on Linux and Solaris. ROSE — source-to-source compiler framework developed at Lawrence Livermore National Laboratory Silverfrost FTN95 — FTN95 from Silverfrost is current, runs on Windows, and has IDE support. == Integrated development environments (IDEs) and editors == Code::Blocks — supports Fortran with plugins Eclipse IDE — with Fortran support via Photran Emacs — extensible text editor with built-in Fortran modes and support for modern tooling via language servers Geany — lightweight cross-platform IDE based on GTK IntelliJ IDEA — cross-platform IDE by JetBrains with Fortran pluggin KDevelop — KDE-based IDE NetBeans — Apache software foundation IDE with Fortran configuration OpenWatcom — IDE and compiler suite for C, C++, and Fortran Simply Fortran — standalone Fortran IDE for Windows, Linux, and macOS Vim — modal text editor with native Fortran syntax support and extensive plugin-based development features Visual Studio — with Intel Fortran integration Visual Studio Code — supports Fortran via extensions == Mathematical libraries == == Scientific libraries == ABINIT — software suite to calculate optical, mechanical, vibrational, and other observable properties of materials Cantera — chemical kinetics, thermodynamics, and transport tool suite CERN Program Library — collection of Fortran libraries for physics applications from CERN CP2K — quantum chemistry and solid-state physics software package for atomistic simulations Dalton — molecular electronic structure program FFTPACK — subroutines for the fast Fourier transform Kinetic PreProcessor – open-source software tool used in atmospheric chemistry MESA — Modules for Experiments in Stellar Astrophysics Nek5000 — MPI parallel higher-order spectral element CFD solver NWChem — open-source high-performance computational chemistry software Octopus — real-space Time-Dependent Density Functional Theory code MODTRAN – model atmospheric propagation of electromagnetic radiation MOLCAS — quantum chemistry software package for multiconfigurational electronic structure calculations NOVAS – software library for astrometry-related numerical computations Physics Analysis Workstation – data analysis and graphical presentation in high-energy physics Quantum ESPRESSO — integrated suite for electronic-structure calculations and materials modeling SIESTA — first-principles materials simulation code using density functional theory Tinker — software tools for molecular design == Debugging and performance tools == GDB — GNU Debugger with Fortran support Valgrind — memory debugging and profiling tool VTune Profiler — performance analysis tool Allinea Forge — debugger and profiler for HPC applications == Build and package management == Autotools — build system supporting Fortran projects CMake — cross-platform build system supporting Fortran Make — build automation tool Spack — package manager for HPC software including Fortran libraries == Machine learning and AI libraries == Athena Fiats (Functional Inference And Training for Surrogates) FNN (Fortran Neural Network) FortNN Fortran-TF-lib (Fortran interface to TensorFlow) FTorch (Fortran interface to PyTorch) MlFortran RoseNNa == Parallel and high-performance computing tools == MPI Fortran bindings — standard interface for distributed-memory parallelism OpenMP — shared-memory parallel programming support through compiler directives Coarray Fortran — parallel programming model introduced in Fortran 2008 ScaLAPACK — parallel linear algebra package built on top of LAPACK == Testing frameworks == FUnit — open-source unit testing framework developed at NASA’s Langley Research Center, for Fortran 90, 95, and 2003. pFUnit — unit testing framework for Fortran, modeled after JUnit == Documentation and code analysis tools == FORD — automatic documentation generator for modern Fortran projects SQuORE — software quality and management platform with code analysis support Understand — static analysis and code comprehension tool for large Fortran projects

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  Artificial intelligence in India

  The artificial intelligence (AI) market in India is projected to reach $8 billion by 2025, growing at 40% CAGR from 2020 to 2025. This growth is part of the broader AI boom, a global period of rapid technological advancements with India being pioneer starting in the early 2010s with NLP based Chatbots from Haptik, Corover.ai, Niki.ai and then gaining prominence in the early 2020s based on reinforcement learning, marked by breakthroughs such as generative AI models from Krutrim, Sarvam, CoRover, OpenAI and Alphafold by Google DeepMind. In India, the development of AI has been similarly transformative, with applications in healthcare, finance, and education, bolstered by government initiatives like NITI Aayog's 2018 National Strategy for Artificial Intelligence. Institutions such as the Indian Statistical Institute and the Indian Institute of Science published breakthrough AI research papers and patents. India's transformation to AI is primarily being driven by startups and government initiatives & policies like Digital India. By fostering technological trust through digital public infrastructure, India is tackling socioeconomic issues by taking a bottom-up approach to AI. NASSCOM and Boston Consulting Group estimate that by 2027, India's AI services might be valued at $17 billion. According to 2025 Technology and Innovation Report, by UN Trade and Development, India ranks 10th globally for private sector investments in AI. According to Mary Meeker, India has emerged as a key market for AI platforms, accounting for the largest share of ChatGPT's mobile app users and having the third-largest user base for DeepSeek in 2025. While AI presents significant opportunities for economic growth and social development in India, challenges such as data privacy concerns, skill shortages, and ethical considerations need to be addressed for responsible AI deployment. The growth of AI in India has also led to an increase in the number of cyberattacks that use AI to target organizations. == History == === Early days (1960s-1980s) === The TIFRAC (Tata Institute of Fundamental Research Automatic Calculator) was designed and developed by a team led by Rangaswamy Narasimhan between 1954 and 1960. He worked on pattern recognition from 1961 to 1964 at the University of Illinois Urbana-Champaign's Digital Computer Laboratory. In order to conduct research on database technology, computer networking, computer graphics, and systems software, he and M. G. K. Menon founded the National Centre for Software Development and Computing Techniques. In 1965, he established the Computer Society of India and supervised the initial research work on AI at Tata Institute of Fundamental Research. Jagdish Lal launched the first computer science program in 1976 at Motilal Nehru Regional Engineering College. H. K. Kesavan from the University of Waterloo and Vaidyeswaran Rajaraman from the University of Wisconsin–Madison joined the IIT Kanpur Electrical Engineering Department in 1963–1964 as Assistant Professor and Head of Department, respectively. H.N. Mahabala, who was employed at Bendix Corporation's Computer Division, joined the department in 1965. He previously worked with Marvin Minsky. The IIT Kanpur Computer Center was led by H. K. Kesavan, with Vaidyeswaran Rajaraman serving as his deputy. Kesavan informally permitted Rajaraman and Mahabala to introduce artificial intelligence into computer science classes. The computer science program was approved by IIT Kanpur in 1971 and split out from the electrical engineering department. In 1973, an IBM System/370 Model 155 was installed at IIT Madras. John McCarthy, head of the Artificial Intelligence Laboratory at Stanford University visited IIT Kanpur in 1971. He donated PDP-1 with a time-sharing operating system. During the 1970s, the balance of payments deficit in India restricted import of computers. The Department of Computer Science and Automation at the Indian Institute of Science established in 1969, played an important role in nurturing the development of data science and artificial intelligence in India. First course on AI was introduced in the 1970s by G. Krishna. B. L. Deekshatulu introduced the first course on pattern recognition in the early 1970s. === Foundation phase === ==== 1980s ==== In the 1980s, the Indian Statistical Institute's Optical Character Recognition Project was one of the country's first attempts at studying artificial intelligence and machine learning. OCR technology has benefited greatly from the work of ISI's Computer Vision and Pattern Recognition Unit, which is headed by Bidyut Baran Chaudhuri. He also contributed in the development of computer vision and digital image processing. As part of the Indian Fifth Generation Computer Systems Research Programme, the Department of Electronics, with support from the United Nations Development Programme, initiated the Knowledge Based Computer Systems Project in 1986, marking the beginning of India's first major AI research program. Prime Minister Rajiv Gandhi requested that the Department of Electronics and IISc to initiate the Parallel Processing Project in 1986–1987. The Center for Development of Advanced Computing eventually joined those efforts. IIT Madras was selected to develop system diagnosis, ISI for image processing, National Centre for Software Technology for natural language processing and TIFR for speech processing. In 1987, the proposal of N. Seshagiri, Director General of the National Informatics Centre for the prototype development of supercomputer was cleared. Negotiations for a Cray supercomputer were underway between the Reagan administration and the Rajiv Gandhi government. US Defense Secretaries Frank Carlucci and Caspar Weinberger visited New Delhi after the US approved the transfer in 1988. The sale of a lower-end XMP-14 supercomputer was permitted in lieu of the Cray XMP-24 supercomputer due to security concerns. The Center for Development of Advanced Computing was formally established in March 1988 by the Ministry of Communications and Information Technology (previously the Ministry of IT) within the Department of Information Technology (formerly the Department of Electronics) in response to a recommendation made to the Prime Minister by the Scientific Advisory Council. The National Initiative in Supercomputing, which produced the PARAM series, was led by Vijay P. Bhatkar. For the first ten years, supercomputing and Indian language computing were the two main focus areas. C-DAC has expanded its operations in order to meet the needs in a number of domains, including network and internet software, real-time systems, artificial intelligence, and NLP. Under the direction of Professor KV Ramakrishnamacharyulu from National Sanskrit University and Professor Rajeev Sangal from the International Institute of Information Technology, Hyderabad, the Akshar Bharati Research Group was established in 1984 with support from IIT Kanpur and the University of Hyderabad for computational processing of Indian languages. They focused on computational linguistics, NLP with ontological database systems, and Indian language/translation theories with linguistic tradition. ==== 1990s ==== From IIT Kanpur, Mohan Tambe joined C-DAC in the 1990s to work on Graphics and Intelligence based Script Technology (GIST), which addressed the challenge of adapting personal computer software based on Latin script to Devanagiri and a number of other Indian language scripts. He was previously working on the Machine Translation for Indian languages Project. Within C-DAC, he established the GIST group. The technology was expanded to encompass NLP, artificial intelligence-based machine-aided language learning and translation, multimedia and multilingual computing solutions, and more. GIST resulted in the creation of G-CLASS (GIST cross language search plug-ins suite), a cross-language search engine. The Applied Artificial Intelligence Group at C-DAC has developed some basic and novel applications in the field of NLP, including machine translation, information extraction/retrieval, automatic summarization, speech recognition, text-to-speech synthesis, intelligent language teaching, and natural language-based document management with Decision Support Systems. These applications are the result of the foundation laid by previous language technology activities. Software firms in the Indian private sector began looking into AI applications, mostly in the area of business process automation. In order to allow machines to read, comprehend, and interpret human languages, the Language Technologies Research Center was founded in October 1999 at the International Institute of Information Technology, Hyderabad. It focused on the advancements in semantic parsing, information extraction, natural language generation, sentiment analysis, and dialogue systems. Some of the early AI research in India was driven by societal needs. For example; Eklavya, a knowledge-based program created by I

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  Hindley–Milner type system

  A Hindley–Milner (HM) type system is a classical type system for the lambda calculus with parametric polymorphism. It is also known as Damas–Milner or Damas–Hindley–Milner. It was first described by J. Roger Hindley and later rediscovered by Robin Milner. Luis Damas contributed a close formal analysis and proof of the method in his PhD thesis. Among HM's more notable properties are its completeness and its ability to infer the most general type of a given program without programmer-supplied type annotations or other hints. Algorithm W is an efficient type inference method in practice and has been successfully applied on large code bases, although it has a high theoretical complexity. HM is preferably used for functional programming languages. It was first implemented as part of the type system of the programming language ML. Since then, HM has been extended in various ways, most notably with type class constraints like those in Haskell. == Introduction == As a type inference method, Hindley–Milner is able to deduce the types of variables, expressions and functions from programs written in an entirely untyped style. Being scope sensitive, it is not limited to deriving the types only from a small portion of source code, but rather from complete programs or modules. Being able to cope with parametric types, too, it is core to the type systems of many functional programming languages. It was first applied in this manner in the ML programming language. The origin is the type inference algorithm for the simply typed lambda calculus that was devised by Haskell Curry and Robert Feys in 1958. In 1969, J. Roger Hindley extended this work and proved that their algorithm always inferred the most general type. In 1978, Robin Milner, independently of Hindley's work, provided an equivalent algorithm, Algorithm W. In 1982, Luis Damas finally proved that Milner's algorithm is complete and extended it to support systems with polymorphic references. === Monomorphism vs. polymorphism === In the simply typed lambda calculus, types T are either atomic type constants or function types of form T → T {\displaystyle T\rightarrow T} . Such types are monomorphic. Typical examples are the types used in arithmetic values: 3 : N u m b e r a d d 3 4 : N u m b e r a d d : N u m b e r → N u m b e r → N u m b e r {\displaystyle {\begin{array}{ll}3&:{\mathtt {Number}}\\{\mathtt {add}}\ 3\ 4&:{\mathtt {Number}}\\{\mathtt {add}}&:{\mathtt {Number}}\rightarrow {\mathtt {Number}}\rightarrow {\mathtt {Number}}\end{array}}} Contrary to this, the untyped lambda calculus is neutral to typing at all, and many of its functions can be meaningfully applied to all type of arguments. The trivial example is the identity function i d ≡ λ x . x {\displaystyle {\mathtt {id}}\equiv \lambda x.x} which simply returns whatever value it is applied to. Less trivial examples include parametric types like lists. While polymorphism in general means that operations accept values of more than one type, the polymorphism used here is parametric. One finds the notation of type schemes in the literature, too, emphasizing the parametric nature of the polymorphism. Additionally, constants may be typed with (quantified) type variables. For example, the following type schemes quantify universally over α {\displaystyle \alpha } , meaning that they are true for all possible α {\displaystyle \alpha } : c o n s : ∀ α . α → L i s t α → L i s t α n i l : ∀ α . L i s t α i d : ∀ α . α → α {\displaystyle {\begin{array}{ll}{\mathtt {cons}}&:\forall \alpha .\alpha \rightarrow {\mathtt {List}}\ \alpha \rightarrow {\mathtt {List}}\ \alpha \\{\mathtt {nil}}&:\forall \alpha .{\mathtt {List}}\ \alpha \\{\mathtt {id}}&:\forall \alpha .\alpha \rightarrow \alpha \end{array}}} Polymorphic types can become monomorphic by consistent substitution of their variables. Examples of monomorphic instances are: i d ′ : S t r i n g → S t r i n g n i l ′ : L i s t N u m b e r {\displaystyle {\begin{array}{ll}{\mathtt {id}}'&:{\mathtt {String}}\rightarrow {\mathtt {String}}\\{\mathtt {nil}}'&:{\mathtt {List}}\ {\mathtt {Number}}\end{array}}} More generally, types are polymorphic when they contain type variables, while types without them are monomorphic. Contrary to the type systems used for example in Pascal (1970) or C (1972), which only support monomorphic types, HM is designed with emphasis on parametric polymorphism. The successors of the languages mentioned, like C++ (1985), focused on different types of polymorphism, namely subtyping in connection with object-oriented programming and overloading. While subtyping is incompatible with HM, a variant of systematic overloading is available in the HM-based type system of Haskell. === Let-polymorphism === When extending the type inference for the simply-typed lambda calculus towards polymorphism, one has to decide whether assigning a polymorphic type not only as type of an expression, but also as the type of a λ-bound variable is admissible. This would allow the generic identity type to be assigned to the variable 'id' in: (λ id . ... (id 3) ... (id "text") ... ) (λ x . x) Allowing this gives rise to the polymorphic lambda calculus; however, type inference in this system is not decidable. Instead, HM distinguishes variables that are immediately bound to an expression from more general λ-bound variables, calling the former let-bound variables, and allows polymorphic types to be assigned only to these. This leads to let-polymorphism where the above example takes the form let id = λ x . x in ... (id 3) ... (id "text") ... which can be typed with a polymorphic type for 'id'. As indicated, the expression syntax is extended to make the let-bound variables explicit, and by restricting the type system to allow only let-bound variable to have polymorphic types, while the parameters in lambda-abstractions must get a monomorphic type, type inference becomes decidable. == Overview == The remainder of this article proceeds as follows: The HM type system is defined. This is done by describing a deduction system that makes precise what expressions have what type, if any. From there, it works towards an implementation of the type inference method. After introducing a syntax-driven variant of the above deductive system, it sketches an efficient implementation (algorithm J), appealing mostly to the reader's metalogical intuition. Because it remains open whether algorithm J indeed realises the initial deduction system, a less efficient implementation (algorithm W), is introduced and its use in a proof is hinted. Finally, further topics related to the algorithm are discussed. The same description of the deduction system is used throughout, even for the two algorithms, to make the various forms in which the HM method is presented directly comparable. == The Hindley–Milner type system == The type system can be formally described by syntax rules that fix a language for the expressions, types, etc. The presentation here of such a syntax is not too formal, in that it is written down not to study the surface grammar, but rather the depth grammar, and leaves some syntactical details open. This form of presentation is usual. Building on this, typing rules are used to define how expressions and types are related. As before, the form used is a bit liberal. === Syntax === The expressions to be typed are exactly those of the lambda calculus extended with a let-expression as shown in the adjacent table. Parentheses can be used to disambiguate an expression. The application is left-binding and binds stronger than abstraction or the let-in construct. Types are syntactically split into two groups, monotypes and polytypes. ==== Monotypes ==== Monotypes always designate a particular type. Monotypes τ {\displaystyle \tau } are syntactically represented as terms. Examples of monotypes include type constants like i n t {\displaystyle {\mathtt {int}}} or s t r i n g {\displaystyle {\mathtt {string}}} , and parametric types like M a p ( S e t s t r i n g ) i n t {\displaystyle {\mathtt {Map\ (Set\ string)\ int}}} . The latter types are examples of applications of type functions, for example, from the set { M a p 2 , S e t 1 , s t r i n g 0 , i n t 0 , → 2 } {\displaystyle \{{\mathtt {Map^{2},\ Set^{1},\ string^{0},\ int^{0}}},\ \rightarrow ^{2}\}} , where the superscript indicates the number of type parameters. The complete set of type functions C {\displaystyle C} is arbitrary in HM, except that it must contain at least → 2 {\displaystyle \rightarrow ^{2}} , the type of functions. It is often written in infix notation for convenience. For example, a function mapping integers to strings has type i n t → s t r i n g {\displaystyle {\mathtt {int}}\rightarrow {\mathtt {string}}} . Again, parentheses can be used to disambiguate a type expression. The application binds stronger than the infix arrow, which is right-binding. Type variables are admitted as monotypes. Monotypes are not to be confused with monomorphic types, which exc

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  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

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  Sensory, Inc.

  Sensory, Inc. is an American company which develops software AI technologies for speech, sound and vision. It is based in Santa Clara, California. Sensory’s technologies have shipped in over three billion products from hundreds of leading consumer electronics manufacturers including AT&T, Hasbro, Huawei, Google, Amazon, Samsung, LG, Mattel, Motorola, Plantronics, GoPro, Sony, Tencent, Garmin, LG, Microsoft, Lenovo, and more. Sensory has over 60 issued patents covering speech recognition in consumer electronics, biometric authentication, sensor/speech combinations, wake word technology, and more. == History == Sensory, Inc. was founded in 1994, originally as Sensory Circuits, by Forrest Mozer, Mike Mozer and Todd Mozer. The three had also co-founded ESS Technology years earlier. In 1999 Sensory acquired Fluent Speech Technologies, which was formed and started by a group of professors out of the Oregon Graduate Institute (formerly OGI, now OHSU). Fluent Speech Technologies developed high performance embedded speech engines, the technology from this acquisition is now the core technology used throughout Sensory's chip and software line. === Company timeline === 1994 – Founded 1995 – Introduces the RSC 164 - first commercially successful speech recognition IC 1998 – Introduces first speaker verification IC 2000 – Acquires Oregon based Fluent-Speech Technologies 2002 – Acquires Texas Instruments line of speech output ICs (the SC series) 2007 – Introduces first Voice User Interface for Bluetooth silicon (CSR BC-5) - BlueGenie 2008 - Sensory and BlueAnt partner on the V1 - Revolutionary new Bluetooth headset with a voice user interface. First wearable to use a voice user interface for control and best-reviewed speech recognition product in history 2009 – Introduced world's smallest text to speech system (TTS) and Truly HandsfreeTM Triggers/ wake words. 2010 – Introduced the NLP-5x – First Natural Language Voice Processor and TrulyHandsfree wake words in SDKs for Android, iOS, Linux, and Windows. NLP5x used the first generation of TrulyHandsfree wake words with low power and enhanced accuracy. 2011 – Sensory partners with Google and Microsoft to enable TrulyHandsfree as a front end to Goog411 and Bing411 2012 – Partnered with Tensilica to offer ultra-low power TrulyHandsfree wake words; introduced Speaker Verification and Speaker Identification for mobile phones and other consumer electronics. 2012 - TrulyHandsfree released into Samsung's Galaxy S2 for "Hey Galaxy" wake word 2013 – TrulyHandsfree wake words migrated to many new platforms and began shipping as MotoVoice in the Google-owned MotoX. Sensory's TrulyHandsfree in mobile takes off with the Galaxy S3 and S4 and Galaxy Note and is licensed into wearables like Google Glass. 2014 – Announced new initiative in Vision; added LG and Motorola as customers; received the 2014 Global Mobile Award for Best Mobile Technology Breakthrough at the GSMA Mobile World Congress in Barcelona, Spain (judges commented, "A big advance for the wearables market, this offers many benefits for consumers, increasing uptake and usage of many mobile apps, driving revenue for operators and content providers.") 2015-2018 - Licensed Google, Amazon, MSFT, Baidu, Huawei, ZTE, and many others with TrulyHandsfree wake words. Sensory develops first wake words for OK Google, Hey Siri, and Hey Cortana. 2019 - Sensory launched two new solutions: SoundID, sound identification, and TrulyNatural, embedded large vocabulary speech recognition. Sensory also acquired Vocalize.ai, an independent testing lab. 2020 - Sensory introduced VoiceHub, which allows the automated generation of wake words. 2021 - Sensory expands VoiceHub with speech recognition and NLU capabilities. The company initiated a new cloud platform, SensoryCloud.ai. 2022-Sensory rolls out SensoryCloud.ai with speech to text, text to speech, face & voice biometrics 2024- Sensory Automotive & TrulyNatural Speech-to-text On-Device launched == Technology and products == Sensory originally developed both hardware (Integrated Circuit - IC or "chip") and software platforms but migrated to software only around 2005 and added cloud and hybrid computing capabilities in 2021. Sensory's RSC-164 IC (Integrated Circuit or "chip") was used on NASA's Mars Polar Lander in the Mars Microphone on the Lander. Speech Synthesis SC-6x chips – acquired some speech synthesis technology from Texas Instruments. Sensory’s embedded AI solutions include the following: TrulyHandsfree (THF) - wake word detection and phrase spotting. TrulyNatural (TNL) - large vocabulary continuous speech recognition with NLU. TrulySecure (TS) - face and voice biometrics. TrulySecureSpeakerVerification (TSSV) - speaker and sound identification. VoiceHub - Online portal for creating custom wake words and speech recognition models with NLU. Sensory Automotive- Sensory Automotive is a full voice and vision suite of AI technologies that operate efficiently in the car without connecting to a network. The cloud initiative, SensoryCloud.ai, is targeting Speech To Text (STT), Text To Speech (TTS), Wake Word verification, face and voice recognition, and sound identification.

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  Transaction data

  Transaction data or transaction information is a category of data describing transactions. Transaction data/information gather variables generally referring to reference data or master data – e.g. dates, times, time zones, currencies. Typical transactions are: Financial transactions about orders, invoices, payments; Work transactions about plans, activity records; Logistic transactions about deliveries, storage records, travel records, etc.. == Management == Recording and storing transactions is called records management. The record of the transaction is stored in a place where the retention can be guaranteed and where data is archived or removed following a retention period. Formats of recorded transactions can be digital data in databases and spreadsheets, or handwritten texts in physical documents like former bankbooks. Transaction processing systems are application software that generate transactions and manage transaction data/information, e.g. SAP and Oracle Financials. == Data warehousing == Transaction data can be summarised in a data warehouse, which helps accessibility and analysis of the data.

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  Ontology-based data integration

  Ontology-based data integration involves the use of one or more ontologies to effectively combine data or information from multiple heterogeneous sources. It is one of the multiple data integration approaches and may be classified as Global-As-View (GAV). The effectiveness of ontology‑based data integration is closely tied to the consistency and expressivity of the ontology used in the integration process. == Background == Data from multiple sources are characterized by multiple types of heterogeneity. The following hierarchy is often used: Syntactic heterogeneity: is a result of differences in representation format of data Schematic or structural heterogeneity: the native model or structure to store data differ in data sources leading to structural heterogeneity. Schematic heterogeneity that particularly appears in structured databases is also an aspect of structural heterogeneity. Semantic heterogeneity: differences in interpretation of the 'meaning' of data are source of semantic heterogeneity System heterogeneity: use of different operating system, hardware platforms lead to system heterogeneity Ontologies, as formal models of representation with explicitly defined concepts and named relationships linking them, are used to address the issue of semantic heterogeneity in data sources. In domains like bioinformatics and biomedicine, the rapid development, adoption and public availability of ontologies [1] Archived 2007-06-16 at the Wayback Machine has made it possible for the data integration community to leverage them for semantic integration of data and information. == The role of ontologies == Ontologies enable the unambiguous identification of entities in heterogeneous information systems and assertion of applicable named relationships that connect these entities together. Specifically, ontologies play the following roles: Content Explication The ontology enables accurate interpretation of data from multiple sources through the explicit definition of terms and relationships in the ontology. Query Model In some systems like SIMS, the query is formulated using the ontology as a global query schema. Verification The ontology verifies the mappings used to integrate data from multiple sources. These mappings may either be user specified or generated by a system. == Approaches using ontologies for data integration == There are three main architectures that are implemented in ontology‑based data integration applications, namely, Single ontology approach A single ontology is used as a global reference model in the system. This is the simplest approach as it can be simulated by other approaches. SIMS is a prominent example of this approach. The Structured Knowledge Source Integration component of Research Cyc is another prominent example of this approach. (Title = Harnessing Cyc to Answer Clinical Researchers' Ad Hoc Queries). The Gellish Taxonomic Dictionary-Ontology follows this approach as well. Multiple ontologies Multiple ontologies, each modeling an individual data source, are used in combination for integration. Though, this approach is more flexible than the single ontology approach, it requires creation of mappings between the multiple ontologies. Ontology mapping is a challenging issue and is focus of large number of research efforts in computer science [2]. The OBSERVER system is an example of this approach. Hybrid approaches The hybrid approach involves the use of multiple ontologies that subscribe to a common, top-level vocabulary. The top-level vocabulary defines the basic terms of the domain. Thus, the hybrid approach makes it easier to use multiple ontologies for integration in presence of the common vocabulary.

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  Run-time algorithm specialization

  In computer science, run-time algorithm specialization is a methodology for creating efficient algorithms for costly computation tasks of certain kinds. The methodology originates in the field of automated theorem proving and, more specifically, in the Vampire theorem prover project. The idea is inspired by the use of partial evaluation in optimising program translation. Many core operations in theorem provers exhibit the following pattern. Suppose that we need to execute some algorithm a l g ( A , B ) {\displaystyle {\mathit {alg}}(A,B)} in a situation where a value of A {\displaystyle A} is fixed for potentially many different values of B {\displaystyle B} . In order to do this efficiently, we can try to find a specialization of a l g {\displaystyle {\mathit {alg}}} for every fixed A {\displaystyle A} , i.e., such an algorithm a l g A {\displaystyle {\mathit {alg}}_{A}} , that executing a l g A ( B ) {\displaystyle {\mathit {alg}}_{A}(B)} is equivalent to executing a l g ( A , B ) {\displaystyle {\mathit {alg}}(A,B)} . The specialized algorithm may be more efficient than the generic one, since it can exploit some particular properties of the fixed value A {\displaystyle A} . Typically, a l g A ( B ) {\displaystyle {\mathit {alg}}_{A}(B)} can avoid some operations that a l g ( A , B ) {\displaystyle {\mathit {alg}}(A,B)} would have to perform, if they are known to be redundant for this particular parameter A {\displaystyle A} . In particular, we can often identify some tests that are true or false for A {\displaystyle A} , unroll loops and recursion, etc. == Difference from partial evaluation == The key difference between run-time specialization and partial evaluation is that the values of A {\displaystyle A} on which a l g {\displaystyle {\mathit {alg}}} is specialised are not known statically, so the specialization takes place at run-time. There is also an important technical difference. Partial evaluation is applied to algorithms explicitly represented as codes in some programming language. At run-time, we do not need any concrete representation of a l g {\displaystyle {\mathit {alg}}} . We only have to imagine a l g {\displaystyle {\mathit {alg}}} when we program the specialization procedure. All we need is a concrete representation of the specialized version a l g A {\displaystyle {\mathit {alg}}_{A}} . This also means that we cannot use any universal methods for specializing algorithms, which is usually the case with partial evaluation. Instead, we have to program a specialization procedure for every particular algorithm a l g {\displaystyle {\mathit {alg}}} . An important advantage of doing so is that we can use some powerful ad hoc tricks exploiting peculiarities of a l g {\displaystyle {\mathit {alg}}} and the representation of A {\displaystyle A} and B {\displaystyle B} , which are beyond the reach of any universal specialization methods. == Specialization with compilation == The specialized algorithm has to be represented in a form that can be interpreted. In many situations, usually when a l g A ( B ) {\displaystyle {\mathit {alg}}_{A}(B)} is to be computed on many values of B {\displaystyle B} in a row, a l g A {\displaystyle {\mathit {alg}}_{A}} can be written as machine code instructions for a special abstract machine, and it is typically said that A {\displaystyle A} is compiled. The code itself can then be additionally optimized by answer-preserving transformations that rely only on the semantics of instructions of the abstract machine. The instructions of the abstract machine can usually be represented as records. One field of such a record, an instruction identifier (or instruction tag), would identify the instruction type, e.g. an integer field may be used, with particular integer values corresponding to particular instructions. Other fields may be used for storing additional parameters of the instruction, e.g. a pointer field may point to another instruction representing a label, if the semantics of the instruction require a jump. All instructions of the code can be stored in a traversable data structure such as an array, linked list, or tree. Interpretation (or execution) proceeds by fetching instructions in some order, identifying their type, and executing the actions associated with said type. In many programming languages, such as C and C++, a simple switch statement may be used to associate actions with different instruction identifiers. Modern compilers usually compile a switch statement with constant (e.g. integer) labels from a narrow range by storing the address of the statement corresponding to a value i {\displaystyle i} in the i {\displaystyle i} -th cell of a special array, as a means of efficient optimisation. This can be exploited by taking values for instruction identifiers from a small interval of values. == Data-and-algorithm specialization == There are situations when many instances of A {\displaystyle A} are intended for long-term storage and the calls of a l g ( A , B ) {\displaystyle {\mathit {alg}}(A,B)} occur with different B {\displaystyle B} in an unpredictable order. For example, we may have to check a l g ( A 1 , B 1 ) {\displaystyle {\mathit {alg}}(A_{1},B_{1})} first, then a l g ( A 2 , B 2 ) {\displaystyle {\mathit {alg}}(A_{2},B_{2})} , then a l g ( A 1 , B 3 ) {\displaystyle {\mathit {alg}}(A_{1},B_{3})} , and so on. In such circumstances, full-scale specialization with compilation may not be suitable due to excessive memory usage. However, we can sometimes find a compact specialized representation A ′ {\displaystyle A^{\prime }} for every A {\displaystyle A} , that can be stored with, or instead of, A {\displaystyle A} . We also define a variant a l g ′ {\displaystyle {\mathit {alg}}^{\prime }} that works on this representation and any call to a l g ( A , B ) {\displaystyle {\mathit {alg}}(A,B)} is replaced by a l g ′ ( A ′ , B ) {\displaystyle {\mathit {alg}}^{\prime }(A^{\prime },B)} , intended to do the same job faster.

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  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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  Subject indexing

  Subject indexing is the act of describing or classifying a document by index terms, keywords, or other symbols in order to indicate what different documents are about, to summarize their contents or to increase findability. In other words, the objective is to identify and describe the subject of documents. Indexes are constructed, separately, on three distinct levels: terms in a document, such as a book; objects in a collection, such as a library; and documents (such as books and articles) within a field of knowledge. Subject indexing is used in information retrieval especially to create bibliographic indexes to retrieve documents on a particular subject. Examples of academic indexing services are Zentralblatt MATH, Chemical Abstracts, and PubMed. The index terms were mostly assigned by experts but author keywords are also common. The process of indexing begins with any analysis of the subject of the document. The indexer must then identify terms that appropriately identify the subject, either by extracting words directly from the document or assigning words from a controlled vocabulary. The terms in the index are then presented in a systematic order. Indexers must decide how many terms to include and how specific the terms should be. Together this gives a depth of indexing. == Subject analysis == The first step in indexing is to decide on the subject matter of the document. In manual indexing, the indexer would consider the subject matter in terms of answer to a set of questions such as "Does the document deal with a specific product, condition or phenomenon?". As the analysis is influenced by the knowledge and experience of the indexer, it follows that two indexers may analyze the content differently and so come up with different index terms. This will impact on the success of retrieval. === Automatic vs. manual subject analysis === Automatic indexing follows set processes of analyzing frequencies of word patterns and comparing results to other documents in order to assign to subject categories. This requires no understanding of the material being indexed. This leads to more uniform indexing but at the expense of the true meaning being interpreted. A computer program will not understand the meaning of statements and may therefore fail to assign some relevant terms or assign incorrectly. Human indexers focus their attention on certain parts of the document such as the title, abstract, summary and conclusions, as analyzing the full text in depth is costly and time-consuming. An automated system takes away the time limit and allows the entire document to be analyzed, but also has the option to be directed to particular parts of the document. == Term selection == The second stage of indexing involves the translation of the subject analysis into a set of index terms. This can involve extracting from the document or assigning from a controlled vocabulary. With the ability to conduct a full text search widely available, many people have come to rely on their own expertise in conducting information searches and full text search has become very popular. Subject indexing and its experts, professional indexers, catalogers, and librarians, remains crucial to information organization and retrieval. These experts understand controlled vocabularies and are able to find information that cannot be located by full text search. The cost of expert analysis to create subject indexing is not easily compared to the cost of hardware, software and labor to manufacture a comparable set of full-text, fully searchable materials. With new web applications that allow every user to annotate documents, social tagging has gained popularity especially in the Web. One application of indexing, the book index, remains relatively unchanged despite the information revolution. === Extraction/Derived indexing === Extraction indexing involves taking words directly from the document. It uses natural language and lends itself well to automated techniques where word frequencies are calculated and those with a frequency over a pre-determined threshold are used as index terms. A stop-list containing common words (such as "the", "and") would be referred to and such stop words would be excluded as index terms. Automated extraction indexing may lead to loss of meaning of terms by indexing single words as opposed to phrases. Although it is possible to extract commonly occurring phrases, it becomes more difficult if key concepts are inconsistently worded in phrases. Automated extraction indexing also has the problem that, even with use of a stop-list to remove common words, some frequent words may not be useful for allowing discrimination between documents. For example, the term glucose is likely to occur frequently in any document related to diabetes. Therefore, use of this term would likely return most or all the documents in the database. Post-coordinated indexing where terms are combined at the time of searching would reduce this effect but the onus would be on the searcher to link appropriate terms as opposed to the information professional. In addition terms that occur infrequently may be highly significant for example a new drug may be mentioned infrequently but the novelty of the subject makes any reference significant. One method for allowing rarer terms to be included and common words to be excluded by automated techniques would be a relative frequency approach where frequency of a word in a document is compared to frequency in the database as a whole. Therefore, a term that occurs more often in a document than might be expected based on the rest of the database could then be used as an index term, and terms that occur equally frequently throughout will be excluded. Another problem with automated extraction is that it does not recognize when a concept is discussed but is not identified in the text by an indexable keyword. Since this process is based on simple string matching and involves no intellectual analysis, the resulting product is more appropriately known as a concordance than an index. === Assignment indexing === An alternative is assignment indexing where index terms are taken from a controlled vocabulary. This has the advantage of controlling for synonyms as the preferred term is indexed and synonyms or related terms direct the user to the preferred term. This means the user can find articles regardless of the specific term used by the author and saves the user from having to know and check all possible synonyms. It also removes any confusion caused by homographs by inclusion of a qualifying term. A third advantage is that it allows the linking of related terms whether they are linked by hierarchy or association, e.g. an index entry for an oral medication may list other oral medications as related terms on the same level of the hierarchy but would also link to broader terms such as treatment. Assignment indexing is used in manual indexing to improve inter-indexer consistency as different indexers will have a controlled set of terms to choose from. Controlled vocabularies do not completely remove inconsistencies as two indexers may still interpret the subject differently. == Index presentation == The final phase of indexing is to present the entries in a systematic order. This may involve linking entries. In a pre-coordinated index the indexer determines the order in which terms are linked in an entry by considering how a user may formulate their search. In a post-coordinated index, the entries are presented singly and the user can link the entries through searches, most commonly carried out by computer software. Post-coordination results in a loss of precision in comparison to pre-coordination. == Depth of indexing == Indexers must make decisions about what entries should be included and how many entries an index should incorporate. The depth of indexing describes the thoroughness of the indexing process with reference to exhaustivity and specificity. === Exhaustivity === An exhaustive index is one which lists all possible index terms. Greater exhaustivity gives a higher recall, or more likelihood of all the relevant articles being retrieved, however, this occurs at the expense of precision. This means that the user may retrieve a larger number of irrelevant documents or documents which only deal with the subject in little depth. In a manual system a greater level of exhaustivity brings with it a greater cost as more man-hours are required. The additional time taken in an automated system would be much less significant. At the other end of the scale, in a selective index only the most important aspects are covered. Recall is reduced in a selective index as if an indexer does not include enough terms, a highly relevant article may be overlooked. Therefore, indexers should strive for a balance and consider what the document may be used. They may also have to consider the implications of time and expense. === Specificity === The specificity describes how closel

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  Shapiro–Senapathy algorithm

  The Shapiro—Senapathy algorithm (S&S) is a computational method for identifying splice sites in eukaryotic genes. The algorithm employs a Position Weight Matrix (PWM) scoring formula to predict donor and acceptor splice sites in any given gene. This methodology has been used to discover splice sites and disease-causing splice site mutations in the human genome, and has become a standard tool in clinical genomics. The S&S algorithm has been cited in thousands of clinical studies, according to Google Scholar. It has also formed the basis of widely used software, including Human Splicing Finder, SROOGLE, and Alamut, which identify splice sites and splice site mutations that cause disease. The algorithm has uncovered splicing mutations in diseases ranging from cancers to inherited disorders, and predicted the deleterious effects of these mutations including exon skipping, intron retention, and cryptic splice site activation. == The algorithm == A splice site defines the boundary between a coding exon and a non-coding intron in eukaryotic genes. The S&S algorithm employs a sliding window, corresponding to the length of the splice site motif, to scan a gene sequence and detect potential splice sites. For each sliding window, the algorithm calculates a score by comparing the nucleotide sequence to a Position Weight Matrix (PWM) derived from known splice sites. This formula generates a percentile score, indicating the likelihood that a given sequence functions as a donor or acceptor splice site. The majority of disease-causing mutations in the human genome are located in splice sites. Clinical genomics studies analyze the splice site scores generated by the S&S algorithm to predict the consequences of splice site mutations including exon skipping and intron retention. The algorithm's sensitivity to single-nucleotide changes allows it to determine mutations that may impact RNA splicing and contribute to disease. In addition to identifying real splice sites, the S&S algorithm has been used to discover cryptic splice sites — alternative splice sites activated by mutations — which may disrupt normal splicing. The algorithm detects mutations that lead to the activation of cryptic splice sites, which may be located proximal to real splice sites or deep within non-coding introns. It has thus been used to determine the causes of numerous diseases that are due to cryptic splicing. == Cancer gene discovery using S&S == The S&S algorithm has been used to identify splice-site mutations in genes associated with several cancers. For example, genes causing commonly occurring cancers including breast cancer, ovarian cancer, colorectal cancer, leukemia, head and neck cancers, prostate cancer, retinoblastoma, squamous cell carcinoma, gastrointestinal cancer, melanoma, liver cancer, Lynch syndrome, skin cancer, and neurofibromatosis have been found. In addition, splicing mutations in genes causing less commonly known cancers including gastric cancer, gangliogliomas, Li-Fraumeni syndrome, Loeys–Dietz syndrome, Osteochondromas (bone tumor), Nevoid basal cell carcinoma syndrome, and Pheochromocytomas have been identified. Specific mutations in different splice sites in various genes causing breast cancer (e.g., BRCA1, PALB2), ovarian cancer (e.g., SLC9A3R1, COL7A1, HSD17B7), colon cancer (e.g., APC, MLH1, DPYD), colorectal cancer (e.g., COL3A1, APC, HLA-A), skin cancer (e.g., COL17A1, XPA, POLH), and Fanconi anemia (e.g., FANC, FANA) have been uncovered. The mutations in the donor and acceptor splice sites in different genes causing a variety of cancers that have been identified by S&S are shown in Table 1. == Discovery of genes causing inherited disorders using S&S == Specific mutations in different splice sites in various genes that cause inherited disorders, including, for example, Type 1 diabetes (e.g., PTPN22, TCF1 (HCF-1A)), hypertension (e.g., LDL, LDLR, LPL), Marfan syndrome (e.g., FBN1, TGFBR2, FBN2), cardiac diseases (e.g., COL1A2, MYBPC3, ACTC1), eye disorders (e.g., EVC, VSX1) have been uncovered. A few example mutations in the donor and acceptor splice sites in different genes causing a variety of inherited disorders identified using S&S are shown in Table 2. == Genes causing immune system disorders == More than 100 immune system disorders affect humans, including inflammatory bowel diseases, multiple sclerosis, systemic lupus erythematosus, bloom syndrome, familial cold autoinflammatory syndrome, and dyskeratosis congenita. The Shapiro–Senapathy algorithm has been used to discover genes and mutations involved in many immune disorder diseases, including Ataxia telangiectasia, B-cell defects, epidermolysis bullosa, and X-linked agammaglobulinemia. Xeroderma pigmentosum, an autosomal recessive disorder is caused by faulty proteins formed due to new preferred splice donor site identified using S&S algorithm and resulted in defective nucleotide excision repair. Type I Bartter syndrome (BS) is caused by mutations in the gene SLC12A1. S&S algorithm helped in disclosing the presence of two novel heterozygous mutations c.724 + 4A > G in intron 5 and c.2095delG in intron 16 leading to complete exon 5 skipping. Mutations in the MYH gene, which is responsible for removing the oxidatively damaged DNA lesion are cancer-susceptible in the individuals. The IVS1+5C plays a causative role in the activation of a cryptic splice donor site and the alternative splicing in intron 1, S&S algorithm shows, guanine (G) at the position of IVS+5 is well conserved (at the frequency of 84%) among primates. This also supported the fact that the G/C SNP in the conserved splice junction of the MYH gene causes the alternative splicing of intron 1 of the β type transcript. Splice site scores were calculated according to S&S to find EBV infection in X-linked lymphoproliferative disease. Identification of Familial tumoral calcinosis (FTC) is an autosomal recessive disorder characterized by ectopic calcifications and elevated serum phosphate levels and it is because of aberrant splicing. == Application of S&S in hospitals for clinical practice and research == The Shapiro–Senapathy (S&S) algorithm has played a significant role in advancing the diagnosis and treatment of human diseases through its application in modern clinical genomics. With the widespread adoption of next-generation sequencing (NGS) technologies, the S&S algorithm is now routinely integrated into clinical practice by geneticists and diagnostic laboratories. It is implemented in various computational tools such as Human Splicing Finder (HSF), Splice Site Finder (SSF), and Alamut Visual, which assist in interpreting the functional impact of genetic variants on RNA splicing. The algorithm is particularly useful in identifying pathogenic splice site mutations in cases where the clinical presentation is unclear or where conventional diagnostic methods have failed to identify a causative gene. Its utility has been demonstrated across diverse patient cohorts, including individuals from different ethnic backgrounds with various cancers and inherited genetic disorders. The following are selected examples illustrating its application in clinical research. === Cancers === === Inherited disorders === == S&S - Algorithm for identifying splice sites, exons and split genes == The Shapiro–Senapathy algorithm (SSA) was developed to identify splice sites in uncharacterized genomic sequences, with early applications in the Human Genome Project. The method introduced a Position Weight Matrix (PWM)-based approach to analyze splicing sequences across eukaryotic organisms, marking the first computational framework to systematically define splice sites using probabilistic scoring. Key innovations of the algorithm included: Exon Detection – Exons were defined as sequences bounded by acceptor and donor splice sites with S&S scores above a threshold, requiring an open reading frame (ORF) for validation. Gene Prediction – The method enabled the identification of complete genes by assembling predicted exons, forming a basis for later gene-finding tools. Mutation Analysis – The algorithm distinguishes deleterious splice-site mutations (which disrupt protein function by lowering S&S scores) from neutral variations. This capability allowed researchers to study disease-linked cryptic splice sites in humans, animals, and plants. SSA's PWM-based framework influenced subsequent computational methods, including machine learning and neural network approaches, for splice-site prediction and alternative splicing research. It remains a foundational tool in genomics and disease studies. == Discovering the mechanisms of aberrant splicing in diseases == The Shapiro–Senapathy algorithm has been used to determine the various aberrant splicing mechanisms in genes due to deleterious mutations in the splice sites, which cause numerous diseases. Deleterious splice site mutations impair the normal splicing of the gene transcripts, and thereby make the encoded protei

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  Documentalist

  A documentalist is a professional, trained in documentation science and specializing in assisting researchers in their search for scientific and technical documentation. With the development of bibliographical databases such as MEDLINE, documentalists were professionals who searched such databases on the behalf of users. When the field of documentation changed its name to information science, the terms information specialist or information professional often replaced the term documentalist.

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  Termcap

  Termcap (terminal capability) is a legacy software library and database used on Unix-like computers that enables programs to use display computer terminals in a terminal-independent manner, which greatly simplifies the process of writing portable text mode applications. It was superseded by the terminfo database used by ncurses, tput, and other programs. A termcap database can describe the capabilities of hundreds of different display terminals. This allows programs to have character-based display output, independent of the type of terminal. On-screen text editors such as vi and Emacs are examples of programs that may use termcap. Other programs are listed in the Termcap category. Access to the termcap database was usually provided by separate libraries, e.g. GNU Termcap. Examples of what the database describes: how many columns wide the display is what string to send to move the cursor to an arbitrary position (including how to encode the row and column numbers) how to scroll the screen up one or several lines how much padding is needed for such a scrolling operation. == History == Bill Joy wrote the first termcap library in 1978 for the Berkeley Unix operating system; it has since been ported to most Unix and Unix-like environments, even OS-9. Joy's design was reportedly influenced by the design of the terminal data store in the earlier Incompatible Timesharing System. == Data model == Termcap databases consist of one or more descriptions of terminals. === Indices === Each description must contain the canonical name of the terminal. It may also contain one or more aliases for the name of the terminal. The canonical name or aliases are the keys by which the library searches the termcap database. === Data values === The description contains one or more capabilities, which have conventional names. The capabilities are typed: boolean, numeric and string. The termcap library has no predetermined type for each capability name. It determines the types of each capability by the syntax: string capabilities have an "=" between the capability name and its value, numeric capabilities have a "#" between the capability name and its value, and boolean capabilities have no associated value (they are always true if specified). Applications which use termcap do expect specific types for the commonly used capabilities, and obtain the values of capabilities from the termcap database using library calls that return successfully only when the database contents matches the assumed type. === Hierarchy === Termcap descriptions can be constructed by including the contents of one description in another, suppressing capabilities from the included description or overriding or adding capabilities. No matter what storage model is used, the termcap library constructs the terminal description from the requested description, including, suppressing or overriding at the time of the request. == Storage model == Termcap data is stored as text, making it simple to modify. The text can be retrieved by the termcap library from files or environment variables. === Environment variables === The TERM environment variable contains the terminal type name. The TERMCAP environment variable may contain a termcap database. It is most often used to store a single termcap description, set by a terminal emulator to provide the terminal's characteristics to the shell and dependent programs. The TERMPATH environment variable is supported by newer termcap implementations and defines a search path for termcap files. === Flat file === The original (and most common) implementation of the termcap library retrieves data from a flat text file. Searching a large termcap file, e.g., 500 kB, can be slow. To aid performance, a utility such as reorder is used to put the most frequently used entries near the beginning of the file. === Hashed database === 4.4BSD based implementations of termcap store the terminal description in a hashed database (e.g., something like Berkeley DB version 1.85). These store two types of records: aliases which point to the canonical entry, and the canonical entry itself. The text of the termcap entry is stored literally. == Limitations and extensions == The original termcap implementation was designed to use little memory: the first name is two characters, to fit in 16 bits capability names are two characters descriptions are limited to 1023 characters. only one termcap entry with its definitions can be included, and must be at the end. Newer implementations of the termcap interface generally do not require the two-character name at the beginning of the entry. Capability names are still two characters in all implementations. The tgetent function used to read the terminal description uses a buffer whose size must be large enough for the data, and is assumed to be 1024 characters. Newer implementations of the termcap interface may relax this constraint by allowing a null pointer in place of the fixed buffer, or by hiding the data which would not fit, e.g., via the ZZ capability in NetBSD termcap. The terminfo library interface also emulates the termcap interface, and does not actually use the fixed-size buffer. The terminfo library's emulation of termcap allows multiple other entries to be included without restricting the position. A few other newer implementations of the termcap library may also provide this ability, though it is not well documented. == Obsolete features == A special capability, the "hz" capability, was defined specifically to support the Hazeltine 1500 terminal, which had the unfortunate characteristic of using the ASCII tilde character ('~') as a control sequence introducer. In order to support that terminal, not only did code that used the database have to know about using the tilde to introduce certain control sequences, but it also had to know to substitute another printable character for any tildes in the displayed text, since a tilde in the text would be interpreted by the terminal as the start of a control sequence, resulting in missing text and screen garbling. Additionally, attribute markers (such as start and end of underlining) themselves took up space on the screen. Comments in the database source code often referred to this as "Hazeltine braindamage". Since the Hazeltine 1500 was a widely used terminal in the late 1970s, it was important for applications to be able to deal with its limitations.

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  Systematic review

  A systematic review is a scholarly synthesis of the evidence on a clearly presented topic using critical methods to identify, define and assess research on the topic. A systematic review extracts and interprets data from published studies on the topic (in the scientific literature), then analyzes, describes, critically appraises and summarizes interpretations into a refined evidence-based conclusion. For example, a systematic review of randomized controlled trials is a way of summarizing and implementing evidence-based medicine. Systematic reviews, sometimes along with meta-analyses, are generally considered the highest level of evidence in medical research. While a systematic review may be applied in the biomedical or health care context, it may also be used where an assessment of a precisely defined subject can advance understanding in a field of research. A systematic review may examine clinical tests, public health interventions, environmental interventions, social interventions, adverse effects, qualitative evidence syntheses, methodological reviews, policy reviews, and economic evaluations. Systematic reviews are closely related to meta-analyses, and often the same instance will combine both (being published with a subtitle of "a systematic review and meta-analysis"). The distinction between the two is that a meta-analysis uses statistical methods to induce a single number from the pooled data set (such as an effect size), whereas the strict definition of a systematic review excludes that step. However, in practice, when one is mentioned, the other may often be involved, as it takes a systematic review to assemble the information that a meta-analysis analyzes, and people sometimes refer to an instance as a systematic review, even if it includes the meta-analytical component. An understanding of systematic reviews and how to implement them in practice is common for professionals in health care, public health, and public policy. Systematic reviews contrast with a type of review often called a narrative review. Systematic reviews and narrative reviews both review the literature (the scientific literature), but the term literature review without further specification refers to a narrative review. == Characteristics == A systematic review can be designed to provide a thorough summary of current literature relevant to a research question. A systematic review uses a rigorous and transparent approach for research synthesis, with the aim of assessing and, where possible, minimizing bias in the findings. While many systematic reviews are based on an explicit quantitative meta-analysis of available data, there are also qualitative reviews and other types of mixed-methods reviews that adhere to standards for gathering, analyzing, and reporting evidence. Systematic reviews of quantitative data or mixed-method reviews sometimes use statistical techniques (meta-analysis) to combine results of eligible studies. Scoring levels are sometimes used to rate the quality of the evidence depending on the methodology used, although this is discouraged by the Cochrane Library. As evidence rating can be subjective, multiple people may be consulted to resolve any scoring differences between how evidence is rated. The EPPI-Centre, Cochrane, and the Joanna Briggs Institute have been influential in developing methods for combining both qualitative and quantitative research in systematic reviews. Several reporting guidelines exist to standardise reporting about how systematic reviews are conducted. Such reporting guidelines are not quality assessment or appraisal tools. The Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement suggests a standardized way to ensure a transparent and complete reporting of systematic reviews, and is now required for this kind of research by more than 170 medical journals worldwide. The latest version of this commonly used statement corresponds to PRISMA 2020 (the respective article was published in 2021). Several specialized PRISMA guideline extensions have been developed to support particular types of studies or aspects of the review process, including PRISMA-P for review protocols and PRISMA-ScR for scoping reviews. A list of PRISMA guideline extensions is hosted by the EQUATOR (Enhancing the QUAlity and Transparency Of health Research) Network. However, the PRISMA guidelines have been found to be limited to intervention research and the guidelines have to be changed in order to fit non-intervention research. As a result, Non-Interventional, Reproducible, and Open (NIRO) Systematic Reviews was created to counter this limitation. For qualitative reviews, reporting guidelines include ENTREQ (Enhancing transparency in reporting the synthesis of qualitative research) for qualitative evidence syntheses; RAMESES (Realist And MEta-narrative Evidence Syntheses: Evolving Standards) for meta-narrative and realist reviews; and eMERGe (Improving reporting of Meta-Ethnography) for meta-ethnograph. Developments in systematic reviews during the 21st century included realist reviews and the meta-narrative approach, both of which addressed problems of variation in methods and heterogeneity existing on some subjects. == Types == There are over 30 types of systematic review and Table 1 below non-exhaustingly summarises some of these. There is not always consensus on the boundaries and distinctions between the approaches described below. === Scoping reviews === Scoping reviews are distinct from systematic reviews in several ways. A scoping review is an attempt to search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry. This can mean that the concept search and method (including data extraction, organisation and analysis) are refined throughout the process, sometimes requiring deviations from any protocol or original research plan. A scoping review may often be a preliminary stage before a systematic review, which 'scopes' out an area of inquiry and maps the language and key concepts to determine if a systematic review is possible or appropriate, or to lay the groundwork for a full systematic review. The goal can be to assess how much data or evidence is available regarding a certain area of interest. This process is further complicated if it is mapping concepts across multiple languages or cultures. As a scoping review should be systematically conducted and reported (with a transparent and repeatable method), some academic publishers categorize them as a kind of 'systematic review', which may cause confusion. Scoping reviews are helpful when it is not possible to carry out a systematic synthesis of research findings, for example, when there are no published clinical trials in the area of inquiry. Scoping reviews are helpful when determining if it is possible or appropriate to carry out a systematic review, and are a useful method when an area of inquiry is very broad, for example, exploring how the public are involved in all stages systematic reviews. There is still a lack of clarity when defining the exact method of a scoping review as it is both an iterative process and is still relatively new. There have been several attempts to improve the standardisation of the method, for example via a Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guideline extension for scoping reviews (PRISMA-ScR). PROSPERO (the International Prospective Register of Systematic Reviews) does not permit the submission of protocols of scoping reviews, although some journals will publish protocols for scoping reviews. == Stages == While there are multiple kinds of systematic review methods, the main stages of a review can be summarised as follows: === Defining the research question === Some reported that the 'best practices' involve 'defining an answerable question' and publishing the protocol of the review before initiating it to reduce the risk of unplanned research duplication and to enable transparency and consistency between methodology and protocol. Clinical reviews of quantitative data are often structured using the mnemonic PICO, which stands for 'Population or Problem', 'Intervention or Exposure', 'Comparison', and 'Outcome', with other variations existing for other kinds of research. For qualitative reviews, PICo is 'Population or Problem', 'Interest', and 'Context'. === Searching for sources === Relevant criteria can include selecting research that is of good quality and answers the defined question. The search strategy should be designed to retrieve literature that matches the protocol's specified inclusion and exclusion criteria. The methodology section of a systematic review should list all of the databases and citation indices that were searched. The titles and abstracts of identified articles can be checked against predetermined criteria for eligibility and r

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  Randomized rounding

  In computer science and operations research, randomized rounding is a widely used approach for designing and analyzing approximation algorithms. Many combinatorial optimization problems are computationally intractable to solve exactly (to optimality). For such problems, randomized rounding can be used to design fast (polynomial time) approximation algorithms—that is, algorithms that are guaranteed to return an approximately optimal solution given any input. The basic idea of randomized rounding is to convert an optimal solution of a relaxation of the problem into an approximately-optimal solution to the original problem. The resulting algorithm is usually analyzed using the probabilistic method. == Overview == The basic approach has three steps: Formulate the problem to be solved as an integer linear program (ILP). Compute an optimal fractional solution x {\displaystyle x} to the linear programming relaxation (LP) of the ILP. Round the fractional solution x {\displaystyle x} of the LP to an integer solution x ′ {\displaystyle x'} of the ILP. (Although the approach is most commonly applied with linear programs, other kinds of relaxations are sometimes used. For example, see Goemans' and Williamson's semidefinite programming-based Max-Cut approximation algorithm.) In the first step, the challenge is to choose a suitable integer linear program. Familiarity with linear programming, in particular modelling using linear programs and integer linear programs, is required. For many problems, there is a natural integer linear program that works well, such as in the Set Cover example below. (The integer linear program should have a small integrality gap; indeed randomized rounding is often used to prove bounds on integrality gaps.) In the second step, the optimal fractional solution can typically be computed in polynomial time using any standard linear programming algorithm. In the third step, the fractional solution must be converted into an integer solution (and thus a solution to the original problem). This is called rounding the fractional solution. The resulting integer solution should (provably) have cost not much larger than the cost of the fractional solution. This will ensure that the cost of the integer solution is not much larger than the cost of the optimal integer solution. The main technique used to do the third step (rounding) is to use randomization, and then to use probabilistic arguments to bound the increase in cost due to the rounding (following the probabilistic method from combinatorics). Therein, probabilistic arguments are used to show the existence of discrete structures with desired properties. In this context, one uses such arguments to show the following: Given any fractional solution x {\displaystyle x} of the LP, with positive probability the randomized rounding process produces an integer solution x ′ {\displaystyle x'} that approximates x {\displaystyle x} according to some desired criterion. Finally, to make the third step computationally efficient, one either shows that x ′ {\displaystyle x'} approximates x {\displaystyle x} with high probability (so that the step can remain randomized) or one derandomizes the rounding step, typically using the method of conditional probabilities. The latter method converts the randomized rounding process into an efficient deterministic process that is guaranteed to reach a good outcome. == Example: the set cover problem == The following example illustrates how randomized rounding can be used to design an approximation algorithm for the set cover problem. Fix any instance ⟨ c , S ⟩ {\displaystyle \langle c,{\mathcal {S}}\rangle } of set cover over a universe U {\displaystyle {\mathcal {U}}} . === Computing the fractional solution === For step 1, let IP be the standard integer linear program for set cover for this instance. For step 2, let LP be the linear programming relaxation of IP, and compute an optimal solution x ∗ {\displaystyle x^{}} to LP using any standard linear programming algorithm. This takes time polynomial in the input size. The feasible solutions to LP are the vectors x {\displaystyle x} that assign each set s ∈ S {\displaystyle s\in {\mathcal {S}}} a non-negative weight x s {\displaystyle x_{s}} , such that, for each element e ∈ U {\displaystyle e\in {\mathcal {U}}} , x ′ {\displaystyle x'} covers e {\displaystyle e} —the total weight assigned to the sets containing e {\displaystyle e} is at least 1, that is, ∑ s ∋ e x s ≥ 1. {\displaystyle \sum _{s\ni e}x_{s}\geq 1.} The optimal solution x ∗ {\displaystyle x^{}} is a feasible solution whose cost ∑ s ∈ S c ( S ) x s ∗ {\displaystyle \sum _{s\in {\mathcal {S}}}c(S)x_{s}^{}} is as small as possible. Note that any set cover C {\displaystyle {\mathcal {C}}} for S {\displaystyle {\mathcal {S}}} gives a feasible solution x {\displaystyle x} (where x s = 1 {\displaystyle x_{s}=1} for s ∈ C {\displaystyle s\in {\mathcal {C}}} , x s = 0 {\displaystyle x_{s}=0} otherwise). The cost of this C {\displaystyle {\mathcal {C}}} equals the cost of x {\displaystyle x} , that is, ∑ s ∈ C c ( s ) = ∑ s ∈ S c ( s ) x s . {\displaystyle \sum _{s\in {\mathcal {C}}}c(s)=\sum _{s\in {\mathcal {S}}}c(s)x_{s}.} In other words, the linear program LP is a relaxation of the given set-cover problem. Since x ∗ {\displaystyle x^{}} has minimum cost among feasible solutions to the LP, the cost of x ∗ {\displaystyle x^{}} is a lower bound on the cost of the optimal set cover. === Randomized rounding step === In step 3, we must convert the minimum-cost fractional set cover x ∗ {\displaystyle x^{}} into a feasible integer solution x ′ {\displaystyle x'} (corresponding to a true set cover). The rounding step should produce an x ′ {\displaystyle x'} that, with positive probability, has cost within a small factor of the cost of x ∗ {\displaystyle x^{}} .Then (since the cost of x ∗ {\displaystyle x^{}} is a lower bound on the cost of the optimal set cover), the cost of x ′ {\displaystyle x'} will be within a small factor of the optimal cost. As a starting point, consider the most natural rounding scheme: For each set s ∈ S {\displaystyle s\in {\mathcal {S}}} in turn, take x s ′ = 1 {\displaystyle x'_{s}=1} with probability min ( 1 , x s ∗ ) {\displaystyle \min(1,x_{s}^{})} , otherwise take x s ′ = 0 {\displaystyle x'_{s}=0} . With this rounding scheme, the expected cost of the chosen sets is at most ∑ s c ( s ) x s ∗ {\displaystyle \sum _{s}c(s)x_{s}^{}} , the cost of the fractional cover. This is good. Unfortunately the coverage is not good. When the variables x s ∗ {\displaystyle x_{s}^{}} are small, the probability that an element e {\displaystyle e} is not covered is about ∏ s ∋ e 1 − x s ∗ ≈ ∏ s ∋ e exp ⁡ ( − x s ∗ ) = exp ⁡ ( − ∑ s ∋ e x s ∗ ) ≈ exp ⁡ ( − 1 ) . {\displaystyle \prod _{s\ni e}1-x_{s}^{}\approx \prod _{s\ni e}\exp(-x_{s}^{})=\exp {\Big (}-\sum _{s\ni e}x_{s}^{}{\Big )}\approx \exp(-1).} So only a constant fraction of the elements will be covered in expectation. To make x ′ {\displaystyle x'} cover every element with high probability, the standard rounding scheme first scales up the rounding probabilities by an appropriate factor λ > 1 {\displaystyle \lambda >1} . Here is the standard rounding scheme: Fix a parameter λ ≥ 1 {\displaystyle \lambda \geq 1} . For each set s ∈ S {\displaystyle s\in {\mathcal {S}}} in turn, take x s ′ = 1 {\displaystyle x'_{s}=1} with probability min ( λ x s ∗ , 1 ) {\displaystyle \min(\lambda x_{s}^{},1)} , otherwise take x s ′ = 0 {\displaystyle x'_{s}=0} . Scaling the probabilities up by λ {\displaystyle \lambda } increases the expected cost by λ {\displaystyle \lambda } , but makes coverage of all elements likely. The idea is to choose λ {\displaystyle \lambda } as small as possible so that all elements are provably covered with non-zero probability. Here is a detailed analysis. ==== Lemma (approximation guarantee for rounding scheme) ==== Fix λ = ln ⁡ ( 2 | U | ) {\displaystyle \lambda =\ln(2|{\mathcal {U}}|)} . With positive probability, the rounding scheme returns a set cover x ′ {\displaystyle x'} of cost at most 2 ln ⁡ ( 2 | U | ) c ⋅ x ∗ {\displaystyle 2\ln(2|{\mathcal {U}}|)c\cdot x^{}} (and thus of cost O ( log ⁡ | U | ) {\displaystyle O(\log |{\mathcal {U}}|)} times the cost of the optimal set cover). (Note: with care the O ( log ⁡ | U | ) {\displaystyle O(\log |{\mathcal {U}}|)} can be reduced to ln ⁡ ( | U | ) + O ( log ⁡ log ⁡ | U | ) {\displaystyle \ln(|{\mathcal {U}}|)+O(\log \log |{\mathcal {U}}|)} .) ==== Proof ==== The output x ′ {\displaystyle x'} of the random rounding scheme has the desired properties as long as none of the following "bad" events occur: the cost c ⋅ x ′ {\displaystyle c\cdot x'} of x ′ {\displaystyle x'} exceeds 2 λ c ⋅ x ∗ {\displaystyle 2\lambda c\cdot x^{}} , or for some element e {\displaystyle e} , x ′ {\displaystyle x'} fails to cover e {\displaystyle e} . The expectation of each x s ′ {\displaystyle x'_{s}} is at most λ x s ∗ {\displaystyle \lambda x_{s

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