MidpointIn geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. The midpoint of a segment in n-dimensional space whose endpoints are and is given by That is, the ith coordinate of the midpoint (i = 1, 2, ..., n) is Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction.
Regular expressionA regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a match pattern in text. Usually such patterns are used by string-searching algorithms for "find" or "find and replace" operations on strings, or for input validation. Regular expression techniques are developed in theoretical computer science and formal language theory. The concept of regular expressions began in the 1950s, when the American mathematician Stephen Cole Kleene formalized the concept of a regular language.
Top-level domainA top-level domain (TLD) is one of the domains at the highest level in the hierarchical Domain Name System of the Internet after the root domain. The top-level domain names are installed in the root zone of the name space. For all domains in lower levels, it is the last part of the domain name, that is, the last non empty label of a fully qualified domain name. For example, in the domain name www.example.com, the top-level domain is .com. .arpaThe domain name arpa is a top-level domain (TLD) in the Domain Name System (DNS) of the Internet. It is used predominantly for the management of technical network infrastructure. Prominent among such functions are the subdomains in-addr.arpa and ip6.arpa, which provide namespaces for reverse DNS lookup of IPv4 and IPv6 addresses, respectively. The name originally was the acronym for the Advanced Research Projects Agency (ARPA), the funding organization in the United States that developed the ARPANET, the precursor of the Internet.
Computer simulationComputer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determined by comparing their results to the real-world outcomes they aim to predict. Computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics (computational physics), astrophysics, climatology, chemistry, biology and manufacturing, as well as human systems in economics, psychology, social science, health care and engineering.
Moduli schemeIn mathematics, a moduli scheme is a moduli space that exists in the developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept (algebraic spaces, algebraic stacks of Michael Artin). Work of Grothendieck and David Mumford (see geometric invariant theory) opened up this area in the early 1960s.
Scheme (mathematics)In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).
Program optimizationIn computer science, program optimization, code optimization, or software optimization, is the process of modifying a software system to make some aspect of it work more efficiently or use fewer resources. In general, a computer program may be optimized so that it executes more rapidly, or to make it capable of operating with less memory storage or other resources, or draw less power. Although the word "optimization" shares the same root as "optimal", it is rare for the process of optimization to produce a truly optimal system.
Regular grammarIn theoretical computer science and formal language theory, a regular grammar is a grammar that is right-regular or left-regular. While their exact definition varies from textbook to textbook, they all require that all production rules have at most one non-terminal symbol; that symbol is either always at the end or always at the start of the rule's right-hand side. Every regular grammar describes a regular language.
Moduli spaceIn mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space.
Regular languageIn theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expression engines, which are augmented with features that allow the recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognized by a finite automaton.
Displacement (geometry)In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point trajectory. A displacement may be identified with the translation that maps the initial position to the final position.
