Regular representationIn mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation. One distinguishes the left regular representation λ given by left translation and the right regular representation ρ given by the inverse of right translation. Representation theory of finite groups#Left- and right-regular representation For a finite group G, the left regular representation λ (over a field K) is a linear representation on the K-vector space V freely generated by the elements of G, i.
Adjoint representationIn mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: . For any Lie group, this natural representation is obtained by linearizing (i.
Middle EastThe Middle East (term originally coined in English ) is a geopolitical region encompassing the Arabian Peninsula, the Levant, Turkey, Egypt, Iran, and Iraq. The term came into widespread usage as a replacement of the term Near East (as opposed to the Far East) beginning in the early 20th century. The term "Middle East" has led to some confusion over its changing definitions, and being seen as too Eurocentric.
Projective representationIn the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group where GL(V) is the general linear group of invertible linear transformations of V over F, and F∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). In more concrete terms, a projective representation of is a collection of operators satisfying the homomorphism property up to a constant: for some constant .
Xi JinpingXi Jinping (; born 15 June 1953) is a Chinese politician who has been serving as the general secretary of the Chinese Communist Party (since 2012), chairman of the Central Military Commission (CMC, since 2012) and the president of the People's Republic of China (PRC, since 2013). He is the fifth paramount leader of the PRC since its foundation in 1949. The son of Chinese Communist veteran Xi Zhongxun, Xi was exiled to rural Yanchuan County as a teenager following his father's purge during the Cultural Revolution.
Error functionIn mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: Some authors define without the factor of . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.
Gamma functionIn mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
Google PixelGoogle Pixel is a brand of portable consumer electronic devices developed by Google that run either ChromeOS or the Android operating system. The main line of Pixel products consist of Android-powered smartphones, which have been produced since October 2016 as the replacement of the older Nexus, and of which the Pixel 7 is the current model. The Pixel brand also includes laptop and tablet computers, as well as several accessories, and was originally introduced in February 2013 with the Chromebook Pixel.
Parametric equationIn mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called parametric curve and parametric surface, respectively. In such cases, the equations are collectively called a parametric representation, or parametric system, or parameterization (alternatively spelled as parametrisation) of the object.
Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
Line segmentIn geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ).
Pixel 4The Pixel 4 and Pixel 4 XL are a pair of Android smartphones designed, developed, and marketed by Google as part of the Google Pixel product line. They collectively serve as the successors to the Pixel 3 and Pixel 3 XL. They were officially announced on October 15, 2019 at the Made by Google event and released in the United States on October 24, 2019. On September 30, 2020, they were succeeded by the Pixel 5. Google confirmed the device's design in June 2019 after renders of it were leaked online.
