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Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves , space curves , polyhedra , ordinary differential equations , partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such as soap films ...
In mathematics education, a representation is a way of encoding an idea or a relationship, and can be both internal (e.g., mental construct) and external (e.g., graph). Thus multiple representations are ways to symbolize, to describe and to refer to the same mathematical entity. They are used to understand, to develop, and to communicate ...
Penrose graphical notation (tensor diagram notation) of a matrix product state of five particles. In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. [1]
In mathematics, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures.Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the representing objects y i conform, in some consistent way, to ...
Proof without words of the Nicomachus theorem (Gulley (2010)) that the sum of the first n cubes is the square of the n th triangular number. In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text.
Let be a vector space over a field. [6] For instance, suppose is or , the standard n-dimensional space of column vectors over the real or complex numbers, respectively.In this case, the idea of representation theory is to do abstract algebra concretely by using matrices of real or complex numbers.
In the mathematical fields of representation theory and group theory, a linear representation of a group is a monomial representation if there is a finite-index subgroup and a one-dimensional linear representation of , such that is equivalent to the induced representation.
Further examples of real representations are the spinor representations of the spin groups in 8k−1, 8k, and 8k+1 dimensions for k = 1, 2, 3 ... This periodicity modulo 8 is known in mathematics not only in the theory of Clifford algebras, but also in algebraic topology, in KO-theory; see spin representation and Bott periodicity.