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Wave-particle duality is the concept in quantum mechanics that quantum entities exhibit particle or wave properties according to the experimental circumstances. [ 1 ] : 59 It expresses the inability of the classical concepts such as particle or wave to fully describe the behavior of quantum objects. [ 2 ] :
The notion of cofibration is distinguished from these because its homotopy-theoretic definition is more amenable to formal analysis and generalization. If f : X → Y {\displaystyle f:X\to Y} is a continuous map between topological spaces, there is an associated topological space M f {\displaystyle Mf} called the mapping cylinder of f ...
The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained by replacing the category of abelian groups with the category of modules.
Duality can be generalized to n-dimensional space and dual polytopes; in two dimension these are called dual polygons. The vertices of one polytope correspond to the ( n − 1)-dimensional elements, or facets, of the other, and the j points that define a ( j − 1)-dimensional element will correspond to j hyperplanes that intersect to give a ...
e. Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the moon. Classical physics is still used in much of modern ...
Let be an abelian category. (Examples include the category of modules over a ring and the category of sheaves of abelian groups on a topological space.) The derived category () is defined by a universal property with respect to the category of cochain complexes with terms in .
A toroidal polyhedron. In geometry, a polyhedron ( pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices . A convex polyhedron is a polyhedron that bounds a convex set.
Duality (mathematics) In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. In other cases the dual of the dual – the double dual or ...