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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 ...
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.
The modified Dietz method [1] [2] [3] is a measure of the ex post (i.e. historical) performance of an investment portfolio in the presence of external flows. (External flows are movements of value such as transfers of cash, securities or other instruments in or out of the portfolio, with no equal simultaneous movement of value in the opposite direction, and which are not income from the ...
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.
The formula for an integration by parts is () ′ = [() ()] ′ (). Beside the boundary conditions , we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g ′ {\displaystyle g'} becomes g {\displaystyle g} ) and one which is differentiated ( f {\displaystyle f} becomes f ...
Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y 1/2 /log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x.
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function . The various studies of the behaviour of the divisor function are sometimes called divisor problems .
The principle can be viewed as an example of the sieve method extensively used in number theory and is sometimes referred to as the sieve formula. [ 4 ] As finite probabilities are computed as counts relative to the cardinality of the probability space , the formulas for the principle of inclusion–exclusion remain valid when the cardinalities ...
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