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Theorems showing that certain objects of interest are the dual spaces (in the sense of linear algebra) of other objects of interest are often called dualities. Many of these dualities are given by a bilinear pairing of two K-vector spaces A ⊗ B → K. For perfect pairings, there is, therefore, an isomorphism of A to the dual of B.
In mathematics, a duality, generally speaking, 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.
Mathematics portal; The idea of mathematical duality was first noticed as projective duality.There it appears as the idea of interchanging dimension k and codimension k+1 in propositions of projective geometry.
Duality (mathematics), a mathematical concept Dual (category theory), a formalization of mathematical duality Duality (optimization) Duality (order theory), a concept regarding binary relations
A diagram of string theory dualities. Yellow arrows indicate S-duality. Blue arrows indicate T-duality. These dualities may be combined to obtain equivalences of any of the five theories with M-theory. [9] Theories that arise as different limits of M-theory turn out to be related in highly nontrivial ways.
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem.
Dual to the theory of equivalences is the theory of dualities between the module categories, where the functors used are contravariant rather than covariant. This theory, though similar in form, has significant differences because there is no duality between the categories of modules for any rings, although dualities may exist for subcategories.
In the philosophy of mind, mind–body dualism denotes either the view that mental phenomena are non-physical, [1] or that the mind and body are distinct and separable. [2] Thus, it encompasses a set of views about the relationship between mind and matter, as well as between subject and object, and is contrasted with other positions, such as physicalism and enactivism, in the mind–body problem.