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For example, the number 144 in base 10 is a sum-product number, because + + =, =, and =. A natural number n {\displaystyle n} is a sociable sum-product number if it is a periodic point for F b {\displaystyle F_{b}} , where F b p ( n ) = n {\displaystyle F_{b}^{p}(n)=n} for a positive integer p {\displaystyle p} , and forms a cycle of period p ...
In this example, the rule says: multiply 3 by 2, getting 6. The sets {A, B, C} and {X, Y} in this example are disjoint sets, but that is not necessary.The number of ways to choose a member of {A, B, C}, and then to do so again, in effect choosing an ordered pair each of whose components are in {A, B, C}, is 3 × 3 = 9.
The function is tacitly assumed to be real-valued: (,, …,). Factor graphs can be combined with message passing algorithms to efficiently compute certain characteristics of the function (,, …,), such as the marginal distributions.
A function, () , is said to be ... For example, an analytic function is the limit of its Taylor series, within its radius of convergence.
For example, given 100 binary variables, …,, computing a single marginal using and the above formula would involve summing over possible values for ′. If it is known that the probability mass function p {\displaystyle p} factors in a convenient way, belief propagation allows the marginals to be computed much more efficiently.
In number theory, an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to the product ab is the sum of the values of the function applied to a and b: [1] = + ().
Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
For example, if one starts with Euler's totient function φ, and repeatedly applies the transformation process, one obtains: φ the totient function; φ ∗ 1 = I, where I(n) = n is the identity function; I ∗ 1 = σ 1 = σ, the divisor function; If the starting function is the Möbius function itself, the list of functions is: μ, the Möbius ...