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Well-Defined
In mathematics, an expression is called well-defined or unambiguous if its definition assigns it a unique interpretation or value. Otherwise, the expression is said to be not well-defined, ill-defined or ambiguous. A function is well-defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well-defined (and thus not a function). The term well-defined can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well-defined is not the same as a function that is undefined. For example, if f(x) = 1/x, then the fact that f(0) is undefined does not mean that the f is not well-defined — but that 0 is simply not in the domain of f.
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  • 01 Dec 2022
Topic Review
DNS Zone
A DNS zone is a specific portion of the DNS namespace in the Domain Name System (DNS), which is managed by a specific organization or administrator. A DNS zone is an administrative space that allows for more granular control of the DNS components, such as authoritative nameserver. The DNS is broken up into many different zones, which are distinctly managed areas in the DNS namespace. DNS zones are not necessarily physically separated from one another, however, a DNS zone can contain multiple subdomains and multiple zones can exist on the same server. The domain namespace of the Internet is organized into a hierarchical layout of subdomains below the DNS root domain. The individual domains of this tree may serve as delegation points for administrative authority and management. However, usually, it is furthermore desirable to implement fine-grained boundaries of delegation, so that multiple sub-levels of a domain may be managed independently. Therefore, the domain name space is partitioned into areas (zones) for this purpose. A zone starts at a domain and extends downward in the tree to the leaf nodes or to the top-level of subdomains where other zones start. A DNS zone is implemented in the configuration system of a domain name server. Historically, it is defined in the zone file, an operating system text file that starts with the special DNS record type Start of Authority (SOA) and contains all records for the resources described within the zone. This format was originally used by the Berkeley Internet Name Domain Server (BIND) software package and is defined in RFC 1034 and RFC 1035.
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Topic Review
Sigma-Algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections. The pair (X, Σ) is called a measurable space or Borel space. A σ-algebra is a type of algebra of sets. An algebra of sets needs only to be closed under the union or intersection of finitely many subsets, which is a weaker condition. The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation. In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic, particularly when the statistic is a function or a random process and the notion of conditional density is not applicable. If X = {a, b, c, d}, one possible σ-algebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is the empty set. In general, a finite algebra is always a σ-algebra. If {A1, A2, A3, …} is a countable partition of X then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra. A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy).
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Topic Review
Unfollow Everything
Facebook (and parent company Meta Platforms) has been the subject of criticism and legal action. Criticisms include the outsize influence Facebook has on the lives and health of its users and employees, as well as Facebook's influence on the way media, specifically news, is reported and distributed. Notable issues include Internet privacy, such as use of a widespread "like" button on third-party websites tracking users, possible indefinite records of user information, automatic facial recognition software, and its role in the workplace, including employer-employee account disclosure. The use of Facebook can have negative psychological effects that include feelings of sexual jealousy, stress, lack of attention, and social media addiction that in some cases is comparable to drug addiction. Facebook's operations have also received coverage. The company's electricity usage, tax avoidance, real-name user requirement policies, censorship policies, handling of user data, and its involvement in the United States PRISM surveillance program and Facebook–Cambridge Analytica data scandal have been highlighted by the media and by critics. Facebook has come under scrutiny for 'ignoring' or shirking its responsibility for the content posted on its platform, including copyright and intellectual property infringement, hate speech, incitement of rape, violence against minorities, terrorism, fake news, Facebook murder, crimes, and violent incidents live-streamed through its Facebook Live functionality. The company and its employees have also been subject to litigation cases over the years, with its most prominent case concerning allegations that CEO Mark Zuckerberg broke an oral contract with Cameron Winklevoss, Tyler Winklevoss, and Divya Narendra to build the then-named "HarvardConnection" social network in 2004, instead allegedly opting to steal the idea and code to launch Facebook months before HarvardConnection began. The original lawsuit was eventually settled in 2009, with Facebook paying approximately $20 million in cash and 1.25 million shares. A new lawsuit in 2011 was dismissed. Some critics point to problems which they say will result in the demise of Facebook. Facebook has been banned by several governments for various reasons, including Syria, China, Iran and Russia.
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Topic Review
Anki
Anki (/ˈɒŋkiː/; Japanese: [aŋki]) is a free and open-source flashcard program using spaced repetition, a technique from cognitive science for fast and long-lasting memorization. "Anki" (暗記) is the Japanese word for "memorization". The SM-2 algorithm, created for SuperMemo in the late 1980s, forms the basis of the spaced repetition methods employed in the program. Anki's implementation of the algorithm has been modified to allow priorities on cards and to show flashcards in order of their urgency. The cards are presented using HTML and may include text, images, sounds, videos, and LaTeX equations. The decks of cards, along with the user's statistics, are stored in the open SQLite format.
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  • 01 Dec 2022
Topic Review
Term (Logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constructed from constant symbols, variables and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, [math]\displaystyle{ (x+1)*(x+1) }[/math] is a term built from the constant 1, the variable x, and the binary function symbols [math]\displaystyle{ + }[/math] and [math]\displaystyle{ * }[/math]; it is part of the atomic formula [math]\displaystyle{ (x+1)*(x+1) \ge 0 }[/math] which evaluates to true for each real-numbered value of x. Besides in logic, terms play important roles in universal algebra, and rewriting systems.
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Topic Review
Environment Modules
The Environment Modules system is a tool to help users manage their Unix or Linux shell environment, by allowing groups of related environment-variable settings to be made or removed dynamically. Modules has been around since the early 1990s and is used at some of the largest computer centers to deploy multiple versions of different software tools to users. The National Energy Research Scientific Computing Center (NERSC) reports that they use Environment Modules to manage nearly all software. Environment Modules is specified as a Baseline Configuration requirement of the DoD High Performance Computing Modernization Program (HPCMP) Project Baseline Configuration team for participating DoD Supercomputing Resource Centers (DSRCs).
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Topic Review
Microsoft Azure
Microsoft Azure, often referred to as Azure (/ˈæʒər, ˈeɪʒər/ AZH-ər, AY-zhər, UK also /ˈæzjʊər, ˈeɪzjʊər/ AZ-ure, AY-zure), is a cloud computing service operated by Microsoft for application management via Microsoft-managed data centers. It provides software as a service (SaaS), platform as a service (PaaS) and infrastructure as a service (IaaS) and supports many different programming languages, tools, and frameworks, including both Microsoft-specific and third-party software and systems. Azure, announced at Microsoft's Professional Developers Conference (PDC) in October 2008, went by the internal project codename "Project Red Dog", and was formally released in February 2010 as Windows Azure, before being renamed Microsoft Azure on March 25, 2014.
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Topic Review
Geolocation Software
In computing, geolocation software is software that is capable of deducing the geolocation of a device connected to the Internet. The identification of a device's IP address can be used to determine the country, city, or post/ZIP code, determining an object's geographical location. Other methods include examination of a MAC address, image metadata, or credit card information.
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Topic Review
Proofs Involving the Moore–Penrose Inverse
In linear algebra, the Moore–Penrose inverse is a matrix that satisfies some but not necessarily all of the properties of an inverse matrix. This article collects together a variety of proofs involving the Moore–Penrose inverse.
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  • 01 Dec 2022
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