Toggl Track (formerly Toggl) is a time tracking software developed by Toggl OÜ which is headquartered in Tallinn, Estonia. The company offers online time tracking and reporting services through their website along with mobile and desktop applications. Time can be tracked through a start/stop button, manual entry, or dragging and resizing time blocks in a calendar view. == History == According to Alari Aho, Toggl's CEO and founder, the application has been fully self-funded from the start. The name was created using a random name generator.
Gradient vector flow
Gradient vector flow (GVF), a computer vision framework introduced by Chenyang Xu and Jerry L. Prince, is the vector field that is produced by a process that smooths and diffuses an input vector field. It is usually used to create a vector field from images that points to object edges from a distance. It is widely used in image analysis and computer vision applications for object tracking, shape recognition, segmentation, and edge detection. In particular, it is commonly used in conjunction with active contour model. == Background == Finding objects or homogeneous regions in images is a process known as image segmentation. In many applications, the locations of object edges can be estimated using local operators that yield a new image called an edge map. The edge map can then be used to guide a deformable model, sometimes called an active contour or a snake, so that it passes through the edge map in a smooth way, therefore defining the object itself. A common way to encourage a deformable model to move toward the edge map is to take the spatial gradient of the edge map, yielding a vector field. Since the edge map has its highest intensities directly on the edge and drops to zero away from the edge, these gradient vectors provide directions for the active contour to move. When the gradient vectors are zero, the active contour will not move, and this is the correct behavior when the contour rests on the peak of the edge map itself. However, because the edge itself is defined by local operators, these gradient vectors will also be zero far away from the edge and therefore the active contour will not move toward the edge when initialized far away from the edge. Gradient vector flow (GVF) is the process that spatially extends the edge map gradient vectors, yielding a new vector field that contains information about the location of object edges throughout the entire image domain. GVF is defined as a diffusion process operating on the components of the input vector field. It is designed to balance the fidelity of the original vector field, so it is not changed too much, with a regularization that is intended to produce a smooth field on its output. Although GVF was designed originally for the purpose of segmenting objects using active contours attracted to edges, it has been since adapted and used for many alternative purposes. Some newer purposes including defining a continuous medial axis representation, regularizing image anisotropic diffusion algorithms, finding the centers of ribbon-like objects, constructing graphs for optimal surface segmentations, creating a shape prior, and much more. == Theory == The theory of GVF was originally described by Xu and Prince. Let f ( x , y ) {\displaystyle \textstyle f(x,y)} be an edge map defined on the image domain. For uniformity of results, it is important to restrict the edge map intensities to lie between 0 and 1, and by convention f ( x , y ) {\displaystyle \textstyle f(x,y)} takes on larger values (close to 1) on the object edges. The gradient vector flow (GVF) field is given by the vector field v ( x , y ) = [ u ( x , y ) , v ( x , y ) ] {\displaystyle \textstyle \mathbf {v} (x,y)=[u(x,y),v(x,y)]} that minimizes the energy functional In this equation, subscripts denote partial derivatives and the gradient of the edge map is given by the vector field ∇ f = ( f x , f y ) {\displaystyle \textstyle \nabla f=(f_{x},f_{y})} . Figure 1 shows an edge map, the gradient of the (slightly blurred) edge map, and the GVF field generated by minimizing E {\displaystyle \textstyle {\mathcal {E}}} . Equation 1 is a variational formulation that has both a data term and a regularization term. The first term in the integrand is the data term. It encourages the solution v {\displaystyle \textstyle \mathbf {v} } to closely agree with the gradients of the edge map since that will make v − ∇ f {\displaystyle \textstyle \mathbf {v} -\nabla f} small. However, this only needs to happen when the edge map gradients are large since v − ∇ f {\displaystyle \textstyle \mathbf {v} -\nabla f} is multiplied by the square of the length of these gradients. The second term in the integrand is a regularization term. It encourages the spatial variations in the components of the solution to be small by penalizing the sum of all the partial derivatives of v {\displaystyle \textstyle \mathbf {v} } . As is customary in these types of variational formulations, there is a regularization parameter μ > 0 {\displaystyle \textstyle \mu >0} that must be specified by the user in order to trade off the influence of each of the two terms. If μ {\displaystyle \textstyle \mu } is large, for example, then the resulting field will be very smooth and may not agree as well with the underlying edge gradients. Theoretical Solution. Finding v ( x , y ) {\displaystyle \textstyle \mathbf {v} (x,y)} to minimize Equation 1 requires the use of calculus of variations since v ( x , y ) {\displaystyle \textstyle \mathbf {v} (x,y)} is a function, not a variable. Accordingly, the Euler equations, which provide the necessary conditions for v {\displaystyle \textstyle \mathbf {v} } to be a solution can be found by calculus of variations, yielding where ∇ 2 {\displaystyle \textstyle \nabla ^{2}} is the Laplacian operator. It is instructive to examine the form of the equations in (2). Each is a partial differential equation that the components u {\displaystyle u} and v {\displaystyle v} of v {\displaystyle \mathbf {v} } must satisfy. If the magnitude of the edge gradient is small, then the solution of each equation is guided entirely by Laplace's equation, for example ∇ 2 u = 0 {\displaystyle \textstyle \nabla ^{2}u=0} , which will produce a smooth scalar field entirely dependent on its boundary conditions. The boundary conditions are effectively provided by the locations in the image where the magnitude of the edge gradient is large, where the solution is driven to agree more with the edge gradients. Computational Solutions. There are two fundamental ways to compute GVF. First, the energy function E {\displaystyle {\mathcal {E}}} itself (1) can be directly discretized and minimized, for example, by gradient descent. Second, the partial differential equations in (2) can be discretized and solved iteratively. The original GVF paper used an iterative approach, while later papers introduced considerably faster implementations such as an octree-based method, a multi-grid method, and an augmented Lagrangian method. In addition, very fast GPU implementations have been developed in Extensions and Advances. GVF is easily extended to higher dimensions. The energy function is readily written in a vector form as which can be solved by gradient descent or by finding and solving its Euler equation. Figure 2 shows an illustration of a three-dimensional GVF field on the edge map of a simple object (see ). The data and regularization terms in the integrand of the GVF functional can also be modified. A modification described in , called generalized gradient vector flow (GGVF) defines two scalar functions and reformulates the energy as While the choices g ( ∇ f | ) = μ {\displaystyle \textstyle g(\nabla f|)=\mu } and h ( | ∇ f | ) = | ∇ f | 2 {\displaystyle \textstyle h(|\nabla f|)=|\nabla f|^{2}} reduce GGVF to GVF, the alternative choices g ( | ∇ f | ) = exp { − | ∇ f | / K } {\displaystyle \textstyle g(|\nabla f|)=\exp\{-|\nabla f|/K\}} and h ( ∇ f | ) = 1 − g ( | ∇ f | ) {\displaystyle \textstyle h(\nabla f|)=1-g(|\nabla f|)} , for K {\displaystyle K} a user-selected constant, can improve the tradeoff between the data term and its regularization in some applications. The GVF formulation has been further extended to vector-valued images in where a weighted structure tensor of a vector-valued image is used. A learning based probabilistic weighted GVF extension was proposed in to further improve the segmentation for images with severely cluttered textures or high levels of noise. The variational formulation of GVF has also been modified in motion GVF (MGVF) to incorporate object motion in an image sequence. Whereas the diffusion of GVF vectors from a conventional edge map acts in an isotropic manner, the formulation of MGVF incorporates the expected object motion between image frames. An alternative to GVF called vector field convolution (VFC) provides many of the advantages of GVF, has superior noise robustness, and can be computed very fast. The VFC field v V F C {\displaystyle \textstyle \mathbf {v} _{\mathrm {VFC} }} is defined as the convolution of the edge map f {\displaystyle f} with a vector field kernel k {\displaystyle \mathbf {k} } where The vector field kernel k {\displaystyle \textstyle \mathbf {k} } has vectors that always point toward the origin but their magnitudes, determined in detail by the function m {\displaystyle m} , decrease to zero with increasing distance from the origin. The beauty of VFC is that it can be computed very rapidly using a fast Fourier tra
List of library and information science journals
This list covers the journals, magazines, periodicals already published and continuing in the discipline of library and information science (LIS). It doesn't include ceased titles or predatory journals. Titles listed were taken from various scholarly sources, UGC Care and Wikipedia articles. == LIS journal prestige as assessed by LIS faculty == In a 2013 article by Laura Manzari, 232 LIS faculty members from ALA-accredited information science programs ranked the most prestigious journals in library and information science. The following journals were ranked in the top ten most prestigious: Journal of the Association for Information Science and Technology The Library Quarterly Annual Review of Information Science and Technology Journal of Documentation Library Trends Library and Information Science Research Information Processing and Management Journal of Education for Library and Information Science Education College & Research Libraries First Monday (journal) A subsequent study by Safón and Docampo in 2023 identified impactful LIS journals based on their influence on papers published in other LIS publications. Journals listed in the top ten in this study that did not appear in Manzari's list include: Scientometrics International Journal of Information Management Quantitative Science Studies MIS Quarterly Information and Management Journal of the Association for Information Systems Journal of Informetrics The Journal of Academic Librarianship == India == Annals of Library and Information Studies. (Pub: CSIR-NIScPR ), Formerly: Annals of Library Science. ISSN 0003-4835. (1954-) OPEN ACCESS Collnet Journal of Scientrometrics and Information Management (Pub: Taru Publications, Online through Taylor and Francis) ISSN: 0973-7766 Online 2168-930X. College Libraries (Pub: West Bengal College Librarians’ Association (WBCLA) ISSN 0972-1975, Quarterly DESIDOC Journal of Library and Information Technology (DJLIT) (Formerly: DESIDOC Bulletin 0970-8154, DESIDOC Bulletin of Information Technology. 0971-4383/0974-0643) (Pub: Defence Scientific Information & Documentation Centre) ISSN: 0974-0643, ISSN: 0976-4658 (O), Bi-monthly, OPEN ACCESS. Grandhalaya Sarvaswam (Bilingual: Telugu & English) [Pub: Andhra Pradesh Library Association, Vijayawada, Andhra Pradesh, India] (1915–) Gyankosh: Journal of Library and Information Management. (Pub: Integrated Academy Of Management And Technology. Through: Indian Journals.Com). ISSN: 2229-4023 (P), 2249-3182. Half yearly. IASLIC Bulletin (Pub: Indian Association of Special Libraries and Information Centres) ISSN: 0018-8411. Quarterly (1956-) IASLIC Newsletter (Pub: Indian Association of Special Libraries and Information Centres. (Pub: Indian Association of Special Libraries and Information Centres) ISSN 0018-845X. Monthly. (1966-) INFLIBNET Newsletter. (Pub: INFLIBNET). Monthly. Informatics Studies. (Pub: Centre For Informatics Research And Development). Quarterly. Through: Indian journals.com. ISSN: 2583-8994 (Online), 2320-530X (Print) ISST Journal of Advances in Librarianship (Pub:Intellectuals Society for Socio-Techno Welfare) ISSN: 0976-9021. Semiannual. Journal of Advanced Research in Library and Information Science. (JALIS Publishers). 4/year. ISSN 2277-2219. Journal of Indian Library Association (Pub: Indian Library Association). ISSN (P) 2277-5145 O) 2456-513X. Quarterly. (1965-). Journal of Scientometric Research. (Pub: Phcog.Net). ISSN (P) 2321-6654, (O) 2320-0057]; Frequency : Triannual. KELPRO Bulletin (Pub: Kerala Library Professionals' Organisation - KELPRO). ISSN 0975-4911( Print),2582-497X (O).(1993-) KIIT Journal of Library and Information Management (Pub: KIIT University, online through Indian Journals.com) Half yearly. ISSN: 2348-0858. Library Herald. (Pub: Delhi Library Association - DLA). Quarterly. ISSN: 0024-2292. Library Progress (International). (Pub: Bpas Publications, Through: ). Half yearly. ISSN: 0970-1052. (O) ISSN: 2320-317X. (1981-) Pearl: A Journal of Library and Information Science. (Pub: University Library Teacher's Association of Andhra Pradesh, Hyderabad), ISSN: 0973-7081 (print), 0975-6922 (online). Quarterly. RBU Journal of Library and Information Science. (Pub: Rabindra Bharati University).ISSN: 0972-2750. Annual. SALIS Journal of Information Management and Technology - SJIMT. (Pub: Society for the Advancement of Library and Information Science). Half-yearly. ISSN 0975-4105. SALIS Journal of Library and Information Science - SJLIS: an International Journal. (Pub: Society for the Advancement of Library and Information Science). Half-yearly. ISSN: 0973-3108. SRELS journal of Information and Knowledge (Formerly: Library Science with a Slant to Documentation, ISSN: 0024-2543; Library Science with a Slant to Documentation and Information Studies ISSN: 0970-6089; SRELS Journal of Information Management ISSN: ). Quarterly. ISSN: 2583-9314 (O) World Digital Libraries. Half yearly. ISSN: 0974-567X (P), 0975-7597 (O). == Other countries == African Journal of Library, Archives and Information Science Art Libraries Journal (Cambridge University Press) Bibliothèque de l'École des Chartes Canadian Journal of Information and Library Science Cataloging & Classification Quarterly Communications in Information Literacy Cataloging & Classification Quarterly Catholic Library Association Children and Libraries Code4Lib Journal College & Research Libraries Communications in Information Literacy Disability in Library and Information Studies Electronic Journal of Academic and Special Librarianship El Profesional de la Información (es) (EPI) (Formerly Information World en Español) Evidence Based Library and Information Practice (journal) Faslname-ye Ketab Florida Libraries. Florida Library Association. Georgia Library Quarterly. Quarterly. (Pub: Georgia Library Association). Hipertext.net IFLA Journal In the Library with the Lead Pipe Information & Culture International Journal of Information Retrieval Research (IJIRR) Information Processing and Management Information Research Information Sciences (journal) Information Visualization (journal) Information, Communication & Society International Journal of Geographical Information Science Information Research: An International Electronic Journal (IR) Internet Research (journal) Issues in Science and Technology Librarianship Italian Journal of Library and Information Studies (JLIS.it) JLIS.it Journal of Documentation (JDoc) Journal of Information Ethics Journal of Information Science (JIS) Journal of Information Technology Journal of Informetrics Journal of Librarianship and Information Science Journal of Library & Information Studies - JLIS. (Pub: National Taiwan University) Journal of Library Administration Journal of Religious & Theological Information Journal of the Association for Information Science and Technology (Formerly Journal of the American Society for Information Science and Technology) (JASIST) Journal of the Medical Library Association Journal of the Canadian Health Libraries Association (Pub: Canadian Health Libraries Association). Knowledge Organization (journal) Knowledge Quest. (Pub: American Association of School Librarians) Library and Information Science Abstracts Library Literature and Information Science Library, Information Science & Technology Abstracts Library Literature and Information Science Retrospective Library Review (journal) Library Trends Libri (journal) Malaysian Journal of Library and Information Science MLA Forum New Century Library New Review of Children's Literature and Librarianship Notes (journal) Portal – Libraries and the Academy Progressive Librarian, Progressive Librarians Guild Reference and User Services Quarterly Reference Services Review Research Evaluation (journal) Scientometrics (journal) Serials Review South African Journal of Libraries and Information Science The Charleston Advisor The Christian Librarian, from the Association of Christian Librarians The Journal of Academic Librarianship The Library Quarterly (LQ) The Public-Access Computer Systems Review TripleC Webolog
UI data binding
UI data binding is a software design pattern to simplify development of GUI applications. UI data binding binds UI elements to an application domain model. Most frameworks employ the Observer pattern as the underlying binding mechanism. To work efficiently, UI data binding has to address input validation and data type mapping. A bound control is a widget whose value is tied or bound to a field in a recordset (e.g., a column in a row of a table). Changes made to data within the control are automatically saved to the database when the control's exit event triggers. == Example == == Data binding frameworks and tools == === Delphi === DSharp third-party data binding tool OpenWire Visual Live Binding - third-party visual data binding tool === Java === JFace Data Binding JavaFX Property === .NET === Windows Forms data binding overview WPF data binding overview Avalonia Unity 3D data binding framework (available in modifications for NGUI, iGUI and EZGUI libraries) === JavaScript === Angular AngularJS Backbone.js Ember.js Datum.js knockout.js Meteor, via its Blaze live update engine OpenUI5 React Vue.js
Ubiquitous robot
Ubiquitous robot is a term used in an analogous way to ubiquitous computing. Software useful for "integrating robotic technologies with technologies from the fields of ubiquitous and pervasive computing, sensor networks, and ambient intelligence". The emergence of mobile phone, wearable computers and ubiquitous computing makes it likely that human beings will live in a ubiquitous world in which all devices are fully networked. The existence of ubiquitous space resulting from developments in computer and network technology will provide motivations to offer desired services by any IT device at any place and time through user interactions and seamless applications. This shift has hastened the ubiquitous revolution, which has further manifested itself in the new multidisciplinary research area, ubiquitous robotics. It initiates the third generation of robotics following the first generation of the industrial robot and the second generation of the personal robot. Ubiquitous robot (Ubibot) is a robot incorporating three components including virtual software robot or avatar, real-world mobile robot and embedded sensor system in surroundings. Software robot within a virtual world can control a real-world robot as a brain and interact with human beings. Researchers of KAIST, Korea describe these three components as a Sobot (Software robot), Mobot (Mobile robot), and Embot (Embedded robot).
Visual Expert
Visual Expert is a static code analysis tool, extracting design and technical information from software source code by reverse-engineering, used by programmers for software maintenance, modernization or optimization. It is designed to parse several programming languages at the same time (PL/SQL, Transact-SQL, PowerBuilder...) and analyze cross-language dependencies, in addition to each language's source code. Visual Expert checks source code against hundreds of code inspection rules for vulnerability assessment, bug fix, and maintenance issues. == Features == Cross-references exploration: Impact Analysis, E/R diagrams, call graphs, CRUD matrix, dependency graphs. Software documentation: a documentation generator produces technical documentation and low-level design descriptions. Inspect the code to detect bugs, security vulnerabilities and maintainability issues. Native integration with Jenkins. Reports on duplicate code, unused objects and methods and naming conventions. Calculates software metrics and source lines of code. Code comparison: finds differences between several versions of the same code. Performance analysis: identifies code parts that slow down the application because of their syntax - it extracts statistics about code execution from the database and combines it with the static analysis of the code. == Usage == Visual Expert is used in several contexts: Change impact analysis: evaluating the consequences of a change in the code or in a database. Avoiding negative side effects when evolving a system. Static Application Security Testing (SAST): detecting and removing security issues. Continuous Integration / Continuous Inspection : adding a static code analysis job in a CI/CD workflow to automatically verify the quality and security of a new build when it is released. Program comprehension: helping programmers understand and maintain existing code, or modernize legacy systems. Transferring knowledge of the code, from one programmer to another. Software sizing: calculating the size of an application, or a piece of code, in order to estimate development efforts. Code review: improving the code by finding and removing code smells, dead code, code causing poor performances or violations of coding conventions. == Limitations == As a static code analyzer, Visual Expert is limited to the programming languages supported by its code parsers - Oracle PL/SQL, SQL Server Transact-SQL, PowerBuilder. A preliminary reverse engineering is required. Visual Expert does it automatically, but its duration depends on the size of the code parsed. Users must wait for the parsing completion prior to using the features, or schedule it in advance. They must also allocate sufficient hardware resources to support their volume of code. Visual Expert is based on a client/server architecture: the code analysis is running on a Windows PC - preferably a server. The information extracted from the code is stored in a RDBMS, communicating with a client application installed on the programmer's computer - no web client is available. This requires that the code, the parsers, the RDBMS and the programmers’ computers are connected to the same LAN or VPN. == History == 1995- 1998 - Prog and Doc - Initial version distributed on the French market 2001 - Visual Expert 4.5 2003 - Visual Expert 5 2007 - Visual Expert 5.7 2010 - Visual Expert 6.0 2015 - Visual Expert 2015 - Server component added to schedule code analyses 2016 - Visual Expert 2016 - Oracle PL/SQL code parser, code inventory (lines of code, number of objects…) 2017 - Visual Expert 2017 - SQL Server T-SQL code parser, Code comparison, CRUD matrix 2018 - Visual Expert 2018 - DB Code Performance Analysis, integration with TFS 2019 - Visual Expert 2019 - Generation of E/R diagrams from the code 2020 - Visual Expert 2020 - Object dependency matrix, naming consistency verification, integration with GIT and SVN 2021 - Visual Expert 2021 - Continuous Code Inspection, integration with Jenkins 2022 - Visual Expert 2022 - Support for cloud-based repositories and large volumes of code 2023 - Visual Expert 2023 - Performance tuning for PowerBuilder 2024 - Visual Expert 2024 - New web UI to simplify deployment and use among large teams. 2025 - Visual Expert 2025 - AI-based features to explain code, generate comments, and optimize queries
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