Representation theoryRepresentation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).
Induced representationIn group theory, the induced representation is a representation of a group, G, which is constructed using a known representation of a subgroup H. Given a representation of H, the induced representation is, in a sense, the "most general" representation of G that extends the given one. Since it is often easier to find representations of the smaller group H than of G, the operation of forming induced representations is an important tool to construct new representations.
Computer-generated imageryComputer-generated imagery (CGI) is a specific-technology or application of computer graphics for creating or improving images in art, printed media, simulators, videos and video games. These images are either static (i.e. s) or dynamic (i.e. moving images). CGI both refers to 2D computer graphics and (more frequently) 3D computer graphics with the purpose of designing characters, virtual worlds, or scenes and special effects (in films, television programs, commercials, etc.).
Penrose tilingA Penrose tiling is an example of an aperiodic tiling. Here, a tiling is a covering of the plane by non-overlapping polygons or other shapes, and a tiling is aperiodic if it does not contain arbitrarily large periodic regions or patches. However, despite their lack of translational symmetry, Penrose tilings may have both reflection symmetry and fivefold rotational symmetry. Penrose tilings are named after mathematician and physicist Roger Penrose, who investigated them in the 1970s.
Mathematical modelA mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science).
Machine toolA machine tool is a machine for handling or machining metal or other rigid materials, usually by cutting, boring, grinding, shearing, or other forms of deformations. Machine tools employ some sort of tool that does the cutting or shaping. All machine tools have some means of constraining the workpiece and provide a guided movement of the parts of the machine. Thus, the relative movement between the workpiece and the cutting tool (which is called the toolpath) is controlled or constrained by the machine to at least some extent, rather than being entirely "offhand" or "freehand".
Symmetry (physics)In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries.
Unitary representationIn mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous. The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book Gruppentheorie und Quantenmechanik.
Weight (representation theory)In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.
Behavioral patternIn software engineering, behavioral design patterns are design patterns that identify common communication patterns among objects. By doing so, these patterns increase flexibility in carrying out communication.
