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The stationary points are the solutions of the above system of equations plus the constraint = . Note that the 1 2 m ( m − 1 ) {\displaystyle \ {\tfrac {1}{2}}m(m-1)\ } equations are not independent, since the left-hand side of the equation belongs to the subvariety of Λ 2 ( T x ∗ M ) {\displaystyle \ \Lambda ^{2}(T_{x}^{\ast }M ...
In fact computability can itself be defined via the lambda calculus: a function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N, F(x)=y if and only if f x = β y, where x and y are the Church numerals corresponding to x and y, respectively and = β ...
In the following example the single occurrence of in the expression is bound by the second lambda: . (. ) The set of free variables of a lambda expression, M {\displaystyle M} , is denoted as FV ( M ) {\displaystyle \operatorname {FV} (M)} and is defined by recursion on the structure of the terms, as follows:
The Y combinator is an implementation of a fixed-point combinator in lambda calculus. Fixed-point combinators may also be easily defined in other functional and imperative languages. The implementation in lambda calculus is more difficult due to limitations in lambda calculus. The fixed-point combinator may be used in a number of different areas:
For example, a list of three elements x, y and z can be encoded by a higher-order function that when applied to a combinator c and a value n returns c x (c y (c z n)). Equivalently, it is an application of the chain of functional compositions of partial applications, (c x ∘ c y ∘ c z) n.
Linear interpolation on a data set (red points) consists of pieces of linear interpolants (blue lines). Linear interpolation on a set of data points (x 0, y 0), (x 1, y 1), ..., (x n, y n) is defined as piecewise linear, resulting from the concatenation of linear segment interpolants between each pair of data points.
The Carmichael lambda function of a prime power can be expressed in terms of the Euler totient. Any number that is not 1 or a prime power can be written uniquely as the product of distinct prime powers, in which case λ of the product is the least common multiple of the λ of the prime power factors.
The golden-section search is a technique for finding an extremum (minimum or maximum) of a function inside a specified interval. For a strictly unimodal function with an extremum inside the interval, it will find that extremum, while for an interval containing multiple extrema (possibly including the interval boundaries), it will converge to one of them.