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  Medical data breach

  Medical data, including patients' identity information, health status, disease diagnosis and treatment, and biogenetic information, not only involve patients' privacy but also have a special sensitivity and important value, which may bring physical and mental distress and property loss to patients and even negatively affect social stability and national security once leaked. However, the development and application of medical AI must rely on a large amount of medical data for algorithm training, and the larger and more diverse the amount of data, the more accurate the results of its analysis and prediction will be. However, the application of big data technologies such as data collection, analysis and processing, cloud storage, and information sharing has increased the risk of data leakage. In the United States, the rate of such breaches has increased over time, with 176 million records breached by the end of 2017. By 2024, the U.S. Department of Health and Human Services reported 725 large healthcare data breaches affecting approximately 275 million individual records in a single year, marking a significant escalation in both the frequency and scale of incidents. == Black market for health data == In February 2015 an NPR report claimed that organized crime networks had ways of selling health data in the black market. In 2015 a Beazley employee estimated that medical records could sell on the black market for US$40-50. == How data is lost == Theft, data loss, hacking, and unauthorized account access are ways in which medical data breaches happen. Among reported breaches of medical information in the United States networked information systems accounted for the largest number of records breached. There are many data breaches happening in the US health care system, among business associates of the health care providers that continuously gain access to patients' data. == List of data breaches == In February 2024, a ransomware attack on Change Healthcare, a subsidiary of UnitedHealth Group, compromised the protected health information of approximately 100 million individuals, making it the largest healthcare data breach in United States history. The attack disrupted claims processing for healthcare providers nationwide for several weeks. In May 2024, MediSecure suffered a cyberattack involving ransomware in Australia. In May 2021, the Health Service Executive in the Republic of Ireland was the victim of a cyberattack involving ransomware, in the Health Service Executive cyberattack, with admission records and test results present in a sample of the data reviewed by the Financial Times. In October 2018, the Centers for Medicare and Medicaid Services in the US reported that around 75,000 individual records had been affected by a data breach that took place through the ACA Agent and Broker Portal. In 2018, Social Indicators Research published the scientific evidence of 173,398,820 (over 173 million) individuals affected in USA from October 2008 (when the data were collected) to September 2017 (when the statistical analysis took place). In 2015, Anthem Inc. lost data for 37 million people in the Anthem medical data breach In 2014 4.5 million people using Complete Health Systems had their data stolen In 2013-14 1 million people using Montana Department of Public Health and Human Services had their data stolen In 2013 4 million people using Advocate Health and Hospitals Corporation had their data stolen In 2011 4.9 million users of Tricare services had their data stolen due to an employee error by Science Applications International Corporation In 2011 1.9 million people using Health Net had their data stolen In 2011 1 million people using Nemours Foundation had their data stolen In 2010 6800 people using New York-Presbyterian Hospital and Columbia University Medical Center had their data breached. In response, those organizations agreed to pay the United States Department of Health and Human Services a US$4.8 million dollar fine. In 2009 1 million people using BlueCross BlueShield of Tennessee had their data stolen == Regulation == In the United States, the Health Insurance Portability and Accountability Act and Health Information Technology for Economic and Clinical Health Act require companies to report data breaches to affected individuals and the federal government. Under the HIPAA Breach Notification Rule, covered entities must notify affected individuals without unreasonable delay and no later than 60 days after discovering a breach of unsecured protected health information. Breaches affecting 500 or more individuals must also be reported to the HHS Secretary and to prominent media outlets serving the affected state or jurisdiction within the same timeframe; HHS publicly lists these larger breaches on its breach portal, commonly known as the "wall of shame." Breaches affecting fewer than 500 individuals are reported to HHS annually, no later than 60 days after the end of the calendar year in which they were discovered. Health Information Privacy Health Insurance Portability and Accountability Act of 1996 (HIPAA). - 45 CFR Parts 160 and 164, Standards for Privacy of Individually Identifiable Health Information and Security Standards for the Protection of Electronic Protected Health Information. HIPAA includes provisions designed to save health care businesses money by encouraging electronic transactions, as well as regulations to protect the security and confidentiality of patient information. The Privacy Rule became effective April 14, 2001, and most covered entities (health plans, health care clearinghouses, and health care providers that conduct certain financial and administrative transactions electronically) had until April 2003 to comply. This security provision became effective April 21, 2003. The Health Insurance Portability and Accountability Act (HIPAA) is the baseline set of federal regulations governing medical information. It does three things: i. i. i.Establish a structure for how personal health information is disclosed and establish the rights of individuals with respect to health information; ii.Specify security standards for the retention and transmission of electronic patient information; iii.Need a common format and data structure for the electronic exchange of health information. California-Specific Laws California’s medical privacy laws, primarily the Confidentiality of Medical Information Act (CMIA), the data breach sections of the Civil Code, and sections of the Health and Safety Code, provide HIPAA-like protections, although the terminology is different. HIPAA establishes a federal "minimum standard" that applies where there are gaps in California law, and HIPAA also specifies that stricter state laws will override or supersede HIPAA. California's health care privacy laws apply to providers who provide personal health records (PHR), while HIPAA only applies when the provider providing the PHR is a business associate of a covered entity. Federal law does not grant individuals the right to file a lawsuit in the event of a data breach (only the Attorney General can file a lawsuit), but California law does. This means that California law sets a higher standard for medical privacy, and that individuals in California enjoy stronger legal protections and more ways to hold entities that violate their medical privacy accountable. In the UK, the legal framework for how patient data is cared for and processed is the Data Protection Act 2018 (DPA), which incorporates the EU General Data Protection Regulation (GDPR) into law, and the common law duty of confidentiality (CLDC). The data protection legislation requires that the collection and processing of personal data be fair, lawful and transparent. This means that the collection and processing of data as defined by data protection legislation must always have a valid lawful basis and must also meet the requirements of the CLDC. In the China, Article 18 of the "National Health Care Big Data Standards, Security and Services Management Measures (for Trial Implementation)" (National Health Planning and Development (2018) No. 23) promulgated by the National Health Care Commission in 2018 states, "The responsible unit shall adopt measures such as data classification, important data backup, and encryption authentication to guarantee the security of health care big data." However, the scope and definition of important data are not covered. Although the "Information Security Technology-Healthcare Data Security Guide" (the "Guide") issued by the National Standardization Committee also proposes that important data should be evaluated and approved in accordance with the regulations, there is likewise no definition of the connotation and definition of important data.

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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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  Information architecture

  Information architecture is the structural design of shared information environments, in particular the organisation of websites and software to support usability and findability. The term information architecture was coined by Richard Saul Wurman. Since its inception, information architecture has become an emerging community of practice focused on applying principles of design, architecture and information science in digital spaces. Typically, a model or concept of information is used and applied to activities which require explicit details of complex information systems. These activities include library systems and database development. == Definition == The term information architecture has different meanings in different branches of information systems or information technology. === User experience === In user experience design, information architecture has been described as the structural design of shared information environments, comprising the study and practice of organising and labelling web sites, intranets, online communities, and software to support user experience, in particular, the findability and usability of information. It has also been described as an emerging community of practice focused on bringing principles of design and architecture to the digital landscape. === Information systems === Technically speaking, information architecture comprises the combination of organization, labeling, search and navigation systems within websites and intranets, serving as a navigational aid to the content of information-rich systems. === Data architecture === Information architecture can be described as a subset of data architecture where usable data is constructed, designed, and arranged in a fashion most useful to the users of data. === Systems design === In the field of systems design, for example, information architecture is a component of enterprise architecture that deals with the information component when describing the structure of an enterprise. Some system design practitioners regard information architecture as strictly the application of information science to web design, which considers such issues as classification and information retrieval, and not factors like user experience and information design. == Principles == Principles of information architecture include the following: The principle of objects The principle of choices The principle of disclosure The principle of exemplars The principle of front doors The principle of multiple classification The principle of focused navigation The principle of growth == History == Richard Saul Wurman is credited with coining the term information architecture in relation to the design of information. From 1998 to 2015, Peter Morville and Louis Rosenfeld were co-authors of Information Architecture for the World Wide Web. Other authors include Jesse James Garrett and Christina Wodtke.

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  Relational data stream management system

  A relational data stream management system (RDSMS) is a distributed, in-memory data stream management system (DSMS) that is designed to use standards-compliant SQL queries to process unstructured and structured data streams in real-time. Unlike SQL queries executed in a traditional RDBMS, which return a result and exit, SQL queries executed in a RDSMS do not exit, generating results continuously as new data become available. Continuous SQL queries in a RDSMS use the SQL Window function to analyze, join and aggregate data streams over fixed or sliding windows. Windows can be specified as time-based or row-based. == RDSMS SQL Query Examples == Continuous SQL queries in a RDSMS conform to the ANSI SQL standards. The most common RDSMS SQL query is performed with the declarative SELECT statement. A continuous SQL SELECT operates on data across one or more data streams, with optional keywords and clauses that include FROM with an optional JOIN subclause to specify the rules for joining multiple data streams, the WHERE clause and comparison predicate to restrict the records returned by the query, GROUP BY to project streams with common values into a smaller set, HAVING to filter records resulting from a GROUP BY, and ORDER BY to sort the results. The following is an example of a continuous data stream aggregation using a SELECT query that aggregates a sensor stream from a weather monitoring station. The SELECTquery aggregates the minimum, maximum and average temperature values over a one-second time period, returning a continuous stream of aggregated results at one second intervals. RDSMS SQL queries also operate on data streams over time or row-based windows. The following example shows a second continuous SQL query using the WINDOW clause with a one-second duration. The WINDOW clause changes the behavior of the query, to output a result for each new record as it arrives. Hence the output is a stream of incrementally updated results with zero result latency.

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  North Atlantic Population Project

  The North Atlantic Population Project (NAPP) is a collaboration of historical demographers in Britain, Canada, Denmark, Germany, Iceland, Norway, and Sweden to produce a massive census microdata collection for the North Atlantic Region in the late-nineteenth century. The database includes complete individual-level census enumerations for each country, and provides information on over 110 million people. This large scale allows detailed analysis of small geographic areas and population subgroups. The NAPP database is designed to be compatible with the Integrated Public Use Microdata Series (IPUMS), and is disseminated through the IPUMS data-access system at the Minnesota Population Center, University of Minnesota. Major collaborators on the project include Lisa Dillon, University of Montreal; Chad Gaffield, University of Ottawa; Ólöf Garðarsdóttir, Statistics Iceland; Marianne Jarnes Erikstad, University of Tromsø; Jan Oldervall University of Bergen; Evan Roberts, University of Minnesota; Steven Ruggles, University of Minnesota; Kevin Schürer, UK Data Archive; Gunnar Thorvaldsen, University of Tromsø; and Matthew Woollard, UK Data Archive. The project is also coordinated by the Minnesota Population Center at the University of Minnesota.

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  Vinberg's algorithm

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

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  Query rewriting

  Query rewriting is a typically automatic transformation that takes a set of database tables, views, and/or queries, usually indices, often gathered data and query statistics, and other metadata, and yields a set of different queries, which produce the same results but execute with better performance (for example, faster, or with lower memory use). Query rewriting can be based on relational algebra or an extension thereof (e.g. multiset relational algebra with sorting, aggregation and three-valued predicates i.e. NULLs as in the case of SQL). The equivalence rules of relational algebra are exploited, in other words, different query structures and orderings can be mathematically proven to yield the same result. For example, filtering on fields A and B, or cross joining R and S can be done in any order, but there can be a performance difference. Multiple operations may be combined, and operation orders may be altered. The result of query rewriting may not be at the same abstraction level or application programming interface (API) as the original set of queries (though often is). For example, the input queries may be in relational algebra or SQL, and the rewritten queries may be closer to the physical representation of the data, e.g. array operations. Query rewriting can also involve materialization of views and other subqueries; operations that may or may not be available to the API user. The query rewriting transformation can be aided by creating indices from which the optimizer can choose (some database systems create their own indexes if deemed useful), mandating the use of specific indices, creating materialized and/or denormalized views, or helping a database system gather statistics on the data and query use, as the optimality depends on patterns in data and typical query usage. Query rewriting may be rule based or optimizer based. Some sources discuss query rewriting as a distinct step prior to optimization, operating at the level of the user accessible algebra API (e.g. SQL). There are other, largely unrelated concepts also named similarly, for example, query rewriting by search engines.

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  Sardinas–Patterson algorithm

  In coding theory, the Sardinas–Patterson algorithm is a classical algorithm for determining in polynomial time whether a given variable-length code is uniquely decodable, named after August Albert Sardinas and George W. Patterson, who published it in 1953. The algorithm carries out a systematic search for a string which admits two different decompositions into codewords. As Knuth reports, the algorithm was rediscovered about ten years later in 1963 by Floyd, despite the fact that it was at the time already well known in coding theory. == Idea of the algorithm == Consider the code { a ↦ 1 , b ↦ 011 , c ↦ 01110 , d ↦ 1110 , e ↦ 10011 } {\displaystyle \{\,{\texttt {a}}\mapsto {\texttt {1}},{\texttt {b}}\mapsto {\texttt {011}},{\texttt {c}}\mapsto {\texttt {01110}},{\texttt {d}}\mapsto {\texttt {1110}},{\texttt {e}}\mapsto {\texttt {10011}}\,\}} . This code, which is based on an example by Berstel, is an example of a code which is not uniquely decodable, since the string 011101110011 can be interpreted as the sequence of codewords 01110 – 1110 – 011, but also as the sequence of codewords 011 – 1 – 011 – 10011. Two possible decodings of this encoded string are thus given by cdb and babe. In general, a codeword can be found by the following idea: In the first round, we choose two codewords x 1 {\displaystyle x_{1}} and y 1 {\displaystyle y_{1}} such that x 1 {\displaystyle x_{1}} is a prefix of y 1 {\displaystyle y_{1}} , that is, x 1 w = y 1 {\displaystyle x_{1}w=y_{1}} for some "dangling suffix" w {\displaystyle w} . If one tries first x 1 = 011 {\displaystyle x_{1}={\texttt {011}}} and y 1 = 01110 {\displaystyle y_{1}={\texttt {01110}}} , the dangling suffix is w = 10 {\displaystyle {\texttt {w}}={\texttt {10}}} . If we manage to find two sequences x 2 , … , x p {\displaystyle x_{2},\ldots ,x_{p}} and y 2 , … , y q {\displaystyle y_{2},\ldots ,y_{q}} of codewords such that x 2 ⋯ x p = w y 2 ⋯ y q {\displaystyle x_{2}\cdots x_{p}=wy_{2}\cdots y_{q}} , then we are finished: For then the string x = x 1 x 2 ⋯ x p {\displaystyle x=x_{1}x_{2}\cdots x_{p}} can alternatively be decomposed as y 1 y 2 ⋯ y q {\displaystyle y_{1}y_{2}\cdots y_{q}} , and we have found the desired string having at least two different decompositions into codewords. In the second round, we try out two different approaches: the first trial is to look for a codeword that has w as prefix. Then we obtain a new dangling suffix w, with which we can continue our search. If we eventually encounter a dangling suffix that is itself a codeword (or the empty word), then the search will terminate, as we know there exists a string with two decompositions. The second trial is to seek for a codeword that is itself a prefix of w. In our example, we have w = 10 {\displaystyle w={\texttt {10}}} , and the sequence 1 is a codeword. We can thus also continue with w = 0 {\displaystyle w={\texttt {0}}} as the new dangling suffix. == Precise description of the algorithm == The algorithm is described most conveniently using quotients of formal languages. In general, for two sets of strings D and N, the (left) quotient N − 1 D {\displaystyle N^{-1}D} is defined as the residual words obtained from D by removing some prefix in N. Formally, N − 1 D = { y ∣ x y ∈ D and x ∈ N } {\displaystyle N^{-1}D=\{\,y\mid xy\in D~{\textrm {and}}~x\in N\,\}} . Now let C {\displaystyle C} denote the (finite) set of codewords in the given code. The algorithm proceeds in rounds, where we maintain in each round not only one dangling suffix as described above, but the (finite) set of all potential dangling suffixes. Starting with round i = 1 {\displaystyle i=1} , the set of potential dangling suffixes will be denoted by S i {\displaystyle S_{i}} . The sets S i {\displaystyle S_{i}} are defined inductively as follows: S 1 = C − 1 C ∖ { ε } {\displaystyle S_{1}=C^{-1}C\setminus \{\varepsilon \}} . Here, the symbol ε {\displaystyle \varepsilon } denotes the empty word. S i + 1 = C − 1 S i ∪ S i − 1 C {\displaystyle S_{i+1}=C^{-1}S_{i}\cup S_{i}^{-1}C} , for all i ≥ 1 {\displaystyle i\geq 1} . The algorithm computes the sets S i {\displaystyle S_{i}} in increasing order of i {\displaystyle i} . As soon as one of the S i {\displaystyle S_{i}} contains a word from C or the empty word, then the algorithm terminates and answers that the given code is not uniquely decodable. Otherwise, once a set S i {\displaystyle S_{i}} equals a previously encountered set S j {\displaystyle S_{j}} with j < i {\displaystyle j Read more →

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  List of Ruby software and tools

  This is a list of software and programming tools for the Ruby programming language, which includes libraries, web frameworks, implementations, tools, and related projects. == Web tools == Capistrano (software) – remote server automation tool Mongrel – Ruby web server Rack – interface between web servers and web applications Ruby on Rails – full-stack web application framework Sinatra – lightweight Ruby web application framework Spree Commerce – e-commerce platform WEBrick – Ruby HTTP server toolkit == Libraries == BioRuby – bioinformatics and computational biology library for Ruby Bogus – Ruby library for creating reliable test doubles with contract verification ERuby – embedded Ruby templating EventMachine – event-driven I/O library Factory Bot – test fixtures library Fat comma – Ruby library for JSON-like hash syntax Geocoder – Ruby library for geocoding and reverse geocoding addresses Haml – HTML templating engine Markaby – HTML generation via Ruby Nokogiri – XML/HTML parsing library RSpec – behavior-driven testing framework for Ruby RubyGems – package manager for Ruby libraries and applications Sass – CSS preprocessor Sidekiq – background job framework for Ruby, used to handle asynchronous tasks. Uconv – Unicode text conversion library Watir – web application testing framework == Ruby implementations == HotRuby – Ruby interpreter implemented in JavaScript, enabling Ruby code to run in web browsers. IronRuby – Ruby for .NET platform JRuby – Ruby on the Java Virtual Machine MacRuby – Ruby implementation for macOS Mod ruby – Apache module that embeds the Ruby interpreter to improve performance of Ruby web applications Mruby – lightweight Ruby interpreter Rubinius – alternative Ruby implementation, based loosely on the Smalltalk-80 Blue Book design. Ruby MRI – the standard Ruby interpreter YARV – "Yet Another Ruby VM," the bytecode interpreter used in modern Ruby implementations == Tools == Homebrew – package manager for macOS and Linux written in Ruby Pry – interactive Ruby shell Rake – build and task management Ruby Version Manager – environment manager RubyCocoa – bridge between Ruby and Cocoa RubyForge – project hosting site RubyMotion – for iOS/macOS development RubySpec – language specification tests == Integrated Development Environments == Aptana Studio — integrated RadRails plugin for Ruby on Rails development Eclipse DLTK Ruby Plugin — Ruby development plugin for Eclipse Eric — open-source Python-based IDE with Ruby support Komodo IDE — commercial cross-platform IDE with Ruby support RubyMine — commercial IDE for Ruby and Rails by JetBrains SlickEdit — commercial cross-platform IDE with Ruby support == List of websites using Ruby on Rails == Airbnb Basecamp Diaspora – decentralized social network application built with Ruby on Rails Discourse – open-source discussion platform built with Ruby on Rails Fiverr GitHub Hulu Shopify SoundCloud Twitch Zendesk

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  Organizational information theory

  Organizational Information Theory (OIT) is a communication theory, developed by Karl Weick, offering systemic insight into the processing and exchange of information within organizations and among its members. Unlike the past structure-centered theory, OIT focuses on the process of organizing in dynamic, information-rich environments. Given that, it contends that the main activity of organizations is the process of making sense of equivocal information. Organizational members are instrumental to reduce equivocality and achieve sensemaking through some strategies — enactment, selection, and retention of information. With a framework that is interdisciplinary in nature, organizational information theory's desire to eliminate both ambiguity and complexity from workplace messaging builds upon earlier findings from general systems theory and phenomenology. == Inspiration and influence of pre-existing theories == 1. General Systems Theory The General Systems Theory, on its most basic premise, describes the phenomenon of a cohesive group of interrelated parts. When one part of the system is changed or affected, it will affect the system as a whole. Weick uses this theoretical framework from 1950 to influence his organizational information theory. Likewise, organizations can be viewed as a system of related parts that work together towards a common goal or vision. Applying this to Weick's organizational information theory, organizations must work to reduce ambiguity and complexity in the workplace to maximize cohesiveness and efficiency. Weick uses the term, coupling, to describe how organizations, like a system, can be composed of interrelated and dependent parts. Coupling looks at the relationship between people and work. There are two types of coupling: 1. Loose coupling Loose coupling describes that while people within the organization or system are connected and often work together, they do not depend on one another to continue or fully complete individual work. The dependencies are weak and workflow is flexible. For example, "if the whole Science department completely shuts down because all of teachers are sick or for whatsoever reason, the school can still continue to operate because other departments are still present." 2. Tight coupling Tight coupling describes when connections within an organization are strong and dependent. If one part of the organization is not operating correctly, the organization as a whole cannot continue to their fullest potential. " For instance, the format and ink section completely shuts down hence the succeeding steps cannot be continued, so the whole process of the organization will be dropped. Thus, components of a system are directly dependent on one another." 2. Theory of evolution The theory of evolution, by Charles Darwin, is a framework for survival of the fittest. According to Darwin, organisms attempt to adapt and live in an unforgiving environment. Those that are unsuccessful in adaptation do not survive, while the strong organisms continue to thrive and reproduce. Weick invokes inspiration from Darwin, to incorporate a biological perspective to his theory. It is natural for organizations to have to adapt to incoming information that often interfere with the preexisting environment. Organizations that are able to plan and alter strategies in accordance with their constant need of organizing and sense making, will survive and be the most successful. However, there is a notable difference between animal evolution and survival of the fittest in organizations, "A given animal is what it is; variation comes through mutation. But the nature of an organization can change when its members alter their behavior." == Assumptions == 1. Human organizations exist in an information environment Unlike senders and receivers models, OIT stands on the situational perspective. Karl Weick views a human organization as an open social system. People in that system develop a mechanism to establish goals, obtain and process information, or perceive the environment. In this process, people and the environment come to conclusions on "what's going on here?". Colville believes that this attributional process is retrospective. Take an education institution as an example. A university can obtain information regarding students' needs in numerous ways. It might create feedback section in its website. It could organize alumni panels or academic affairs to attract prospective students and collect concrete questions they are interested in. It may also conduct the survey or host focus group to get the information. After that, the staff of the university have to decide how to deal with these information, based on which, it has to set and accomplish its goals for current and prospective students. 2. The information an organization receives differs in terms of equivocality Weick posits that numerous feasible interpretations of reality exist when organizations process information. Their varying levels of understandability lead to different outcomes of information inputs. In other academic works, scholars tend to say that messages are uncertain or ambiguous. While according to OIT, messages are described to be equivocal. believes that people proactively exclude a number of possibilities to perceive what is going on in the environment. Due to OIT's situational perspective, the meanings of messages consist of the messages, the interpretations of receivers, and the interactional context. However, ambiguity and uncertainty can mean that a standard answer - the only one true objective interpretation - exists. Also, Weick emphasizes that "the equivocality is the engine that motivates people to organize". Maitlis and Christianson states that the equivocality trigger sensemaking for three reasons: environment jolts and organizational crises, threats to identity, and planned change interventions. 3. Human organizations engage in information processing to reduce equivocality of information Based upon the first two assumption, OIT proposes that information processing within organizations is a social activity. Sharing is the key feature of organizational information processing. In that particular context, members jointly make sense the reality by reducing equivocality. It other words, the sensemaking is a joint responsibility which includes numerous interdependent people to accomplish. In this process, organizations and its members combine actions and attributions together in order to find the balance between the complexity of thoughts and the simplicity of actions. Weick also proposes that people create their own environment though enactment, which is the action of making sense. This is because people have different perceptual schemas and selective perception, so people create different information environments. In creating different information environments, people can arrive at the same or close to the same understanding or solution through different thought processes and overall understanding. == Key concepts == === The organization === In order to place Weick's vision regarding Organizational Information Theory into proper working context, exploring his view regarding what constitutes the organization and how its individuals embody that construct might yield significant insights. From a fundamental standpoint, he shared a belief that organizational validation is derived---not through bricks and mortar, or locale—but from a series of events which enable entities to "collect, manage and use the information they receive." In elaborating further on what constitutes an organization during early writings outlining OIT, Weick said, "The word organization is a noun and it is also a myth. if one looks for an organization, one will not find it. What will be found is that there are events linked together, that transpire within concrete walls and these sequences, their pathways, their timing, are the forms we erroneously make into substances when we talk about an organization". When viewed in this modular fashion, the organization meets Weick's theoretical vision by encompassing parameters that are less bound by concrete, wood, and structural restraints and more by an ability to serve as a repository where information can be consistently and effectively channeled. Taking these defining characteristics into account, proper channel execution relies on maximization of messaging clarity, context, delivery and evolution through any system. One example as to how these interactions might unfold on a more granular level within these confines can be gleaned through Weick's double interact loop, which he considers the "building blocks of every organization". Simply put, double interacts describe interpersonal exchanges that, inherently, occur across the organizational chain of command and in life, itself. Thus: "An act occurs when you say something (Can I have a Popsicle?). An interact occurs when you say something and I respond ("No, it will spoil your dinner

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  Evidence-based library and information practice

  Evidence-based library and information practice (EBLIP) or evidence-based librarianship (EBL) is the use of evidence-based practices (EBP) in the field of library and information science (LIS). This means that all practical decisions made within LIS should 1) be based on research studies and 2) that these research studies are selected and interpreted according to some specific norms characteristic for EBP. Typically such norms disregard theoretical studies and qualitative studies and consider quantitative studies according to a narrow set of criteria of what counts as evidence. If such a narrow set of methodological criteria are not applied, it is better instead to speak of research based library and information practice. == Characteristics == Evidence-based practice in general has been characterised as a positivist approach; EBLIP is therefore also a positivist approach to LIS. As such, EBLIP is an approach in contrast to other approaches to LIS. The use of statistical approaches known as meta-analysis to conclude what evidence has been reported in the literature is one among other methods which is typical for the evidence-based approach. In 2002, Booth noted the three schools of EBILP had some commonalities, including the context of day-to-day decision-making, an emphasis on improving the quality of professional practice, a pragmatic focus on the 'best available evidence', incorporation of the user perspective, the acceptance of a broad range of quantitative and qualitative research designs, and access, either first-hand or second-hand, to the (process of) evidence-based practice and its products. He added one more, that EBILP is concerned with getting the best value for money. == The role of library and information science in EBP == Evidence-based practice in general is based on a very thorough search of the scientific literature and a very thorough selection and analysis of the retrieved literature. A close familiarity with database searching is needed, and library and information professionals have important roles to play in this respect. Therefore LIS professionals should be well suited to help professionals in other disciplines doing EBP. EBLIP is the application of this approach on LIS itself. It should be mentioned, however, that EBP started in medicine as evidence-based medicine (EBM) from which it spread to other fields. Only slowly and to a limited extent has EBP moved on to LIS. The EBLIP process can be applied to a variety of scenarios in LIS, including customer service, collection development, library management and information literacy instruction. In general, quantitative methods are used in LIS research. A 2010 study revealed five categories that capture the different ways library and information professionals experience evidence-based practice: Evidence-based practice is experienced as irrelevant; Evidence-based practice is experienced as learning from published research; Evidence-based practice is experienced as service improvement; Evidence-based practice is experienced as a way of being; Evidence-based practice is experienced as a weapon.

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  Virtual data room

  A virtual data room (sometimes called a VDR or Deal Room) is an online repository of information that is used for the storing and distribution of documents. In many cases, a virtual data room is used to facilitate the due diligence process during an M&A transaction, loan syndication, or private equity and venture capital transactions. This due diligence process has traditionally used a physical data room to accomplish the disclosure of documents. For reasons of cost, efficiency and security, virtual data rooms have widely replaced the more traditional physical data room. A virtual data room is an extranet to which the bidders and their advisers are given access via the internet. An extranet is essentially a website with limited controlled access, using a secure log-on supplied by the vendor, which can be disabled at any time, by the vendor, if a bidder withdraws. Much of the information released is confidential and restrictions are applied to the viewer's ability to release this to third parties (by means of forwarding, copying or printing). This can be effectively applied to protect the data using digital rights management. The virtual data room provides access to secure documents for authorized users through a dedicated web site, or through secure agent applications. In the process of mergers and acquisitions the data room is set up as part of the central repository of data relating to companies or divisions being acquired or sold. The data room enables the interested parties to view information relating to the business in a controlled environment where confidentiality can be preserved. Conventionally this was achieved by establishing a supervised, physical data room in secure premises with controlled access. In most cases, with a physical data room, only one bidder team can access the room at a time. A virtual data room is designed to have the same advantages as a conventional data room (controlling access, viewing, copying and printing, etc.) with fewer disadvantages. Due to their increased efficiency, many businesses and industries have moved to using virtual data rooms instead of physical data rooms. In 2006, a spokesperson for a company which sets up virtual deal rooms was reported claiming that the process reduced the bidding process by about thirty days compared to physical data rooms. In the process of startup fundraising, a virtual data room is set up to be a central location for key data, documents, and financials. These are shared with venture capital and angel investors and allows them to streamline due diligence. == Application == Any business dealing with private data can apply VDRs when secure transaction processing is required. This includes financial institutions that need to negotiate confidential customer information without involving third parties. VDRs have traditionally been used for IPOs and real estate asset management. Technology companies may use them to exchange and review code or confidential data needed for operations. The same is true for clients, who entrust their valuable code only to the most qualified people in the organisation. The code is not something that can be printed out and brought in a folder. It resides on a computer and must be used together. VDR can find application in any business that manages data in the form of documents, especially law firms, financial advisers or the B2B sector. The latter work with documents that must always be handled and controlled confidentially, and it is difficult to store them securely when they are on a server that other people can access. In addition, in B2B, it is important to close the deal as quickly as possible: the average sales cycle is one to three months. VDR can be compared to a locked filing cabinet where all those folders and documents are kept. It automates the mathematics of pricing to prevent revenue leakage, and initially integrates CRM to ensure accurate synchronisation of all account data, which is important for B2B in particular and sales in general. While virtual data rooms offer many advantages, they are not suitable for every industry. For example, some governments may decide to continue using physical data rooms for highly confidential information sharing. The damage from potential cyberattacks and data breaches exceeds the benefits offered by virtual data rooms. In such cases, the use of VDRs is not considered. Data breaches have particularly affected the US healthcare system from March 2021 to March 2022 - according to IBM Security the cost of the breach was a record high of $10.1 million.

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  Cloud-based design and manufacturing

  Cloud-based design and manufacturing (CBDM) refers to a service-oriented networked product development model in which service consumers are able to configure products or services and reconfigure manufacturing systems through Infrastructure-as-a-Service (IaaS), Platform-as-a-Service (PaaS), Hardware-as-a-Service (HaaS), and Software-as-a-Service (SaaS). Adapted from the original cloud computing paradigm and introduced into the realm of computer-aided product development, Cloud-Based Design and Manufacturing is gaining significant momentum and attention from both academia and industry. Cloud-based design and manufacturing includes two aspects: cloud-based design and cloud-based manufacturing. Another related concept is cloud manufacturing that is more general and popular. Cloud-Based Design (CBD) refers to a networked design model that leverages cloud computing, service-oriented architecture (SOA), Web 2.0 (e.g., social network sites), and semantic web technologies to support cloud-based engineering design services in distributed and collaborative environments. Cloud-Based Manufacturing (CBM) refers to a networked manufacturing model that exploits on-demand access to a shared collection of diversified and distributed manufacturing resources to form temporary, reconfigurable production lines which enhance efficiency, reduce product lifecycle costs, and allow for optimal resource allocation in response to variable-demand customer generated tasking. The enabling technologies for Cloud-Based Design and Manufacturing include cloud computing, Web 2.0, Internet of Things (IoT), and service-oriented architecture (SOA). == History == The term cloud-based design and manufacturing (CBDM) was initially coined by Dazhong Wu, David Rosen, and Dirk Schaefer at Georgia Tech in 2012 for the purpose of articulating a new paradigm for digital manufacturing and design innovation in distributed and collaborative settings. The main objective of CBDM is to further reduce time and cost associated with maintaining information and communication technology (ICT) infrastructures for design and manufacturing, enhancing digital manufacturing and design innovation in distributed and collaborative environments, and adapting to rapidly changing market demands. In 2014, the same research group also published the worldwide first two books on the subjects of Cloud-Based Design and Manufacturing (CBDM) and Social Product Development (SPD) with Springer, edited by Dirk Schaefer. == Characteristics == CBDM exhibits the following key characteristics: Cloud-based distributed file system High performance computing Cloud-based social collaboration Ubiquitous access to distributed big data Rapid manufacturing scalability Agility On-demand self-service Semantic Web Real-time request for quotation Pay-per-use pricing model Multi-tenancy CBDM differs from traditional collaborative and distributed design and manufacturing systems such as web-based systems and agent-based systems from a number of perspectives, including (1) computing architecture, (2) data storage, (3) sourcing process, (4) information and communication technology infrastructure, (5) business model, (6) programming model, and (7) communication. == Service models == Infrastructure as a service (IaaS) Platform as a service (PaaS) Hardware as a service (HaaS) Software as a service (SaaS) Similar to cloud computing, CBDM services can be categorized into four major deployment models: the public cloud, private cloud, hybrid cloud, and community cloud. == Research progress in Academia == The Defense Advanced Research Projects Agency (DARPA) MENTOR program Engineering and Physical Sciences Research Council cloud manufacturing program European Commission's Seventh Framework Program (EC FP7)

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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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  Algorithmic mechanism design

  Algorithmic mechanism design (AMD) lies at the intersection of economic game theory, optimization, and computer science. The prototypical problem in mechanism design is to design a system for multiple self-interested participants, such that the participants' self-interested actions at equilibrium lead to good system performance. Typical objectives studied include revenue maximization and social welfare maximization. Algorithmic mechanism design differs from classical economic mechanism design in several respects. It typically employs the analytic tools of theoretical computer science, such as worst case analysis and approximation ratios, in contrast to classical mechanism design in economics which often makes distributional assumptions about the agents. It also considers computational constraints to be of central importance: mechanisms that cannot be efficiently implemented in polynomial time are not considered to be viable solutions to a mechanism design problem. This often, for example, rules out the classic economic mechanism, the Vickrey–Clarke–Groves auction. == History == Noam Nisan and Amir Ronen first coined "Algorithmic mechanism design" in a research paper published in 1999.

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