Closed-Form for RD(f(n)): Finding k(n)

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SUMMARY

This discussion focuses on deriving a closed-form expression, k(n), for the rounding-down function RD(f(n)) where f(n) is defined for positive integers n. The key findings indicate that for even n, RD(f(n)) equals f(n), while for odd n, RD(f(n)) can be expressed as f(n) minus a constant fraction x, which varies by function. Specifically, the constants are x=0.5 for the first function, x=0.75 for the second, and x=0.25 for the third. The proposed solution involves defining a function g(n) that differentiates between even and odd n to facilitate the closed-form expression.

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  • Understanding of rounding functions, specifically RD(f(n)) and the floor function.
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  • Experience with function manipulation and algebraic expressions.
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  • Research the properties and applications of the floor function, particularly in mathematical analysis.
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Mathematicians, computer scientists, and students studying discrete mathematics or numerical methods who are interested in function analysis and closed-form solutions.

Big-Daddy
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I have 3 functions in n which are rounded down to the nearest integer, and I need a closed-form way of writing each function. n will be a positive integer in all cases.

(Let the notation RD(f(n)) denote rounding-down the value of f(n) for a certain n, to the nearest integer.) What I have found with each function is that f(n)=RD(f(n)) for all even values of n, and RD(f(n))=f(n)-x where x is a fraction which varies from function to function but, for a given function, is constant across all values of n. For the first function, x=0.5, for the second, x=0.75, for the third, x=0.25.

Knowing this, how can I rewrite RD(f(n)) as a closed-form function, k(n), of n and f(n)? The key lies in finding some g(n) so that g(n)=-1 if n is odd and g(n)=0 if n is even; then I could write k(n)=RD(f(n))=f(n)+g(n)*x.
 
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Can't you use the "floor" function, f(x)= \lfloor x \rfloor which is defined as "the largest integer less than or equal to x"?
 
HallsofIvy said:
Can't you use the "floor" function, f(x)= \lfloor x \rfloor which is defined as "the largest integer less than or equal to x"?

No, that's the very definition of my problem. I need a k(x) such that k(x) = RD(f(x)) = \lfloor f(x) \rfloor to use your notation.
 

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