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An algebra for this endofunctor is a set X together with a function 1 + N × X → X. To define such a function, we need a point x ∈ X and a function N × X → X. The set of finite lists of natural numbers is an initial algebra for this functor.
The algebra (, [,]) in the above example is an initial algebra. Various finite data structures used in programming , such as lists and trees , can be obtained as initial algebras of specific endofunctors.
Re – real part of a complex number. [2] (Also written.) resp – respectively. RHS – right-hand side of an equation. rk – rank. (Also written as rank.) RMS, rms – root mean square. rng – non-unital ring. rot – rotor of a vector field. (Also written as curl.) rowsp – row space of a matrix. RTP – required to prove.
In mathematics, specifically category theory, a functor is a mapping between categories.Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces.
See History of algebra: The symbol x. 1637 [2] René Descartes (La Géométrie) √ ̅ . radical symbol (for square root) 1637 (with the vinculum above the radicand)
The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem.
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Hence the primary algebra and 2 are isomorphic but for one detail: primary algebra complementation can be nullary, in which case it denotes a primitive value. Modulo this detail, 2 is a model of the primary algebra. The primary arithmetic suggests the following arithmetic axiomatization of 2: 1+1=1+0=0+1=1=~0, and 0+0=0=~1.
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