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).
Scale modelA scale model is a physical model which is geometrically similar to an object (known as the prototype). Scale models are generally smaller than large prototypes such as vehicles, buildings, or people; but may be larger than small prototypes such as anatomical structures or subatomic particles. Models built to the same scale as the prototype are called mockups. Scale models are used as tools in engineering design and testing, promotion and sales, filmmaking special effects, military strategy, and hobbies such as rail transport modeling, wargaming and racing; and as toys.
Algebra representationIn abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital) ring. If the algebra is not unital, it may be made so in a standard way (see the adjoint functors page); there is no essential difference between modules for the resulting unital ring, in which the identity acts by the identity mapping, and representations of the algebra.
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 .
Rail transport modelling scalesRail transport modelling uses a variety of scales (ratio between the real world and the model) to ensure scale models look correct when placed next to each other. Model railway scales are standardized worldwide by many organizations and hobbyist groups. Some of the scales are recognized globally, while others are less widespread and, in many cases, virtually unknown outside their circle of origin. Scales may be expressed as a numeric ratio (e.g. 1/87 or 1:87) or as letters defined in rail transport modelling standards (e.
O scaleO scale (or O gauge) is a scale commonly used for toy trains and rail transport modelling. Introduced by German toy manufacturer Märklin around 1900, by the 1930s three-rail alternating current O gauge was the most common model railroad scale in the United States and remained so until the early 1960s. In Europe, its popularity declined before World War II due to the introduction of smaller scales. O gauge had its heyday when model railroads were considered toys, with more emphasis placed on cost, durability, and the ability to be easily handled and operated by pre-adult hands.
Architectural modelAn architectural model is a type of scale model made to study aspects of an architectural design or to communicate design intent. They are made using a variety of materials including paper, plaster, plastic, resin, wood, glass, and metal. Models are built either with traditional handcraft techniques or via 3D printing technologies such as stereolithography, fused filament fabrication, and selective laser sintering. The use of architectural models dates to pre-history. Some of the oldest standing models were found in Malta at Tarxien Temples.
Scale (ratio)The scale ratio of a model represents the proportional ratio of a linear dimension of the model to the same feature of the original. Examples include a 3-dimensional scale model of a building or the scale drawings of the elevations or plans of a building. In such cases the scale is dimensionless and exact throughout the model or drawing. The scale can be expressed in four ways: in words (a lexical scale), as a ratio, as a fraction and as a graphical (bar) scale.
ApprenticeshipApprenticeship is a system for training a new generation of practitioners of a trade or profession with on-the-job training and often some accompanying study (classroom work and reading). Apprenticeships can also enable practitioners to gain a license to practice in a regulated occupation. Most of their training is done while working for an employer who helps the apprentices learn their trade or profession, in exchange for their continued labor for an agreed period after they have achieved measurable competencies.
Lie algebra representationIn the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is given by the commutator. In the language of physics, one looks for a vector space together with a collection of operators on satisfying some fixed set of commutation relations, such as the relations satisfied by the angular momentum operators.
