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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. Specifically, λ(n) is given by the recurrence
Dirichlet lambda function, λ(s) = (1 – 2 −s)ζ(s) where ζ is the Riemann zeta function; Liouville function, λ(n) = (–1) Ω(n) Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p; Modular lambda function, λ(τ), a highly symmetric holomorphic function on the complex upper half-plane
Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. [3] Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.
In this case particular lambda terms (which define functions) are considered as values. "Running" (beta reducing) the fixed-point combinator on the encoding gives a lambda term for the result which may then be interpreted as fixed-point value. Alternately, a function may be considered as a lambda term defined purely in lambda calculus.
Meta-functions will be given that describe the conversion between lambda and let expressions. A meta-function is a function that takes a program as a parameter. The program is data for the meta-program. The program and the meta program are at different meta-levels. The following conventions will be used to distinguish program from the meta program,
Lambda lifts may also be repeated, in order to transform the program. Repeated lifts may be used to convert a program written in lambda calculus into a set of recursive functions, without lambdas. This demonstrates the equivalence of programs written in lambda calculus and programs written as functions. [1]
Although Goodman and Kruskal's lambda is a simple way to assess the association between variables, it yields a value of 0 (no association) whenever two variables are in accord—that is, when the modal category is the same for all values of the independent variable, even if the modal frequencies or percentages vary. As an example, consider the ...
Umbrella sampling is another free-energy calculation technique that is typically used for calculating the free-energy change associated with a change in "position" coordinates as opposed to "chemical" coordinates, although umbrella sampling can also be used for a chemical transformation when the "chemical" coordinate is treated as a dynamic ...