Is it possible to represent this function as a polynomial, by removing the ceiling function?
$begingroup$
I've been working through a derivation and have arrived at the following expression:
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
where $x,y in mathbb{R^+}$.
I would like to know whether this can be reduced further, by converting the floor function to a polynomial to simplify expression?
furthermore, lets say there was a constant k, where $k in mathbb{N^+}$ hence,
$$E = 1 - frac{x}{k.y} left( bigglceil dfrac{x}{k.y} biggrceil right)^{-1}$$
could this be reduced to,
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
Kind regards!
polynomials floor-function ceiling-function
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
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add a comment |
$begingroup$
I've been working through a derivation and have arrived at the following expression:
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
where $x,y in mathbb{R^+}$.
I would like to know whether this can be reduced further, by converting the floor function to a polynomial to simplify expression?
furthermore, lets say there was a constant k, where $k in mathbb{N^+}$ hence,
$$E = 1 - frac{x}{k.y} left( bigglceil dfrac{x}{k.y} biggrceil right)^{-1}$$
could this be reduced to,
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
Kind regards!
polynomials floor-function ceiling-function
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
$endgroup$
add a comment |
$begingroup$
I've been working through a derivation and have arrived at the following expression:
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
where $x,y in mathbb{R^+}$.
I would like to know whether this can be reduced further, by converting the floor function to a polynomial to simplify expression?
furthermore, lets say there was a constant k, where $k in mathbb{N^+}$ hence,
$$E = 1 - frac{x}{k.y} left( bigglceil dfrac{x}{k.y} biggrceil right)^{-1}$$
could this be reduced to,
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
Kind regards!
polynomials floor-function ceiling-function
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
$endgroup$
I've been working through a derivation and have arrived at the following expression:
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
where $x,y in mathbb{R^+}$.
I would like to know whether this can be reduced further, by converting the floor function to a polynomial to simplify expression?
furthermore, lets say there was a constant k, where $k in mathbb{N^+}$ hence,
$$E = 1 - frac{x}{k.y} left( bigglceil dfrac{x}{k.y} biggrceil right)^{-1}$$
could this be reduced to,
$$E = 1 - frac{x}y left( bigglceil dfrac{x}{y} biggrceil right)^{-1}$$
Kind regards!
polynomials floor-function ceiling-function
polynomials floor-function ceiling-function
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
edited Jan 9 at 12:57
Afroeskimo
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
asked Jan 9 at 10:56
AfroeskimoAfroeskimo
115
115
add a comment |
add a comment |
1 Answer
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If I understand your question correctly then the answer is No. The function
$$
f(x)=frac{x}{lceil x rceil}
$$
has infinitely many (jump) discontinuities.
If we can somehow convert $f(x)$ into a polynomial or a rational function then it would be continuous on the whole $Bbb R$ except for only finitely many points. This is impossible since it'd contradict the fact that $f(x)$ has infinitely many discontinuity.
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
$endgroup$
$begingroup$
Understood! I would like to know whether it would be possible to remove a positive integer that resides in the ceil function, in a way that I have mentioned in the edited version of my question?
$endgroup$
– Afroeskimo
Jan 9 at 12:29
$begingroup$
@Afroeskimo In general you cannot pull a constant out of the floor/ceiling function so I'm afraid the answer is also No.
$endgroup$
– BigbearZzz
Jan 9 at 12:49
$begingroup$
even if the constant is an integer?
$endgroup$
– Afroeskimo
Jan 9 at 13:03
$begingroup$
@Afroeskimo Nope, you can consider $x=1.5$ and $k=2$, for example.
$endgroup$
– BigbearZzz
Jan 9 at 13:09
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
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oldest
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active
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votes
$begingroup$
If I understand your question correctly then the answer is No. The function
$$
f(x)=frac{x}{lceil x rceil}
$$
has infinitely many (jump) discontinuities.
If we can somehow convert $f(x)$ into a polynomial or a rational function then it would be continuous on the whole $Bbb R$ except for only finitely many points. This is impossible since it'd contradict the fact that $f(x)$ has infinitely many discontinuity.
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
$endgroup$
$begingroup$
Understood! I would like to know whether it would be possible to remove a positive integer that resides in the ceil function, in a way that I have mentioned in the edited version of my question?
$endgroup$
– Afroeskimo
Jan 9 at 12:29
$begingroup$
@Afroeskimo In general you cannot pull a constant out of the floor/ceiling function so I'm afraid the answer is also No.
$endgroup$
– BigbearZzz
Jan 9 at 12:49
$begingroup$
even if the constant is an integer?
$endgroup$
– Afroeskimo
Jan 9 at 13:03
$begingroup$
@Afroeskimo Nope, you can consider $x=1.5$ and $k=2$, for example.
$endgroup$
– BigbearZzz
Jan 9 at 13:09
add a comment |
$begingroup$
If I understand your question correctly then the answer is No. The function
$$
f(x)=frac{x}{lceil x rceil}
$$
has infinitely many (jump) discontinuities.
If we can somehow convert $f(x)$ into a polynomial or a rational function then it would be continuous on the whole $Bbb R$ except for only finitely many points. This is impossible since it'd contradict the fact that $f(x)$ has infinitely many discontinuity.
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
$endgroup$
$begingroup$
Understood! I would like to know whether it would be possible to remove a positive integer that resides in the ceil function, in a way that I have mentioned in the edited version of my question?
$endgroup$
– Afroeskimo
Jan 9 at 12:29
$begingroup$
@Afroeskimo In general you cannot pull a constant out of the floor/ceiling function so I'm afraid the answer is also No.
$endgroup$
– BigbearZzz
Jan 9 at 12:49
$begingroup$
even if the constant is an integer?
$endgroup$
– Afroeskimo
Jan 9 at 13:03
$begingroup$
@Afroeskimo Nope, you can consider $x=1.5$ and $k=2$, for example.
$endgroup$
– BigbearZzz
Jan 9 at 13:09
add a comment |
$begingroup$
If I understand your question correctly then the answer is No. The function
$$
f(x)=frac{x}{lceil x rceil}
$$
has infinitely many (jump) discontinuities.
If we can somehow convert $f(x)$ into a polynomial or a rational function then it would be continuous on the whole $Bbb R$ except for only finitely many points. This is impossible since it'd contradict the fact that $f(x)$ has infinitely many discontinuity.
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
$endgroup$
If I understand your question correctly then the answer is No. The function
$$
f(x)=frac{x}{lceil x rceil}
$$
has infinitely many (jump) discontinuities.
If we can somehow convert $f(x)$ into a polynomial or a rational function then it would be continuous on the whole $Bbb R$ except for only finitely many points. This is impossible since it'd contradict the fact that $f(x)$ has infinitely many discontinuity.
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
answered Jan 9 at 11:11
BigbearZzzBigbearZzz
9,07421753
9,07421753
$begingroup$
Understood! I would like to know whether it would be possible to remove a positive integer that resides in the ceil function, in a way that I have mentioned in the edited version of my question?
$endgroup$
– Afroeskimo
Jan 9 at 12:29
$begingroup$
@Afroeskimo In general you cannot pull a constant out of the floor/ceiling function so I'm afraid the answer is also No.
$endgroup$
– BigbearZzz
Jan 9 at 12:49
$begingroup$
even if the constant is an integer?
$endgroup$
– Afroeskimo
Jan 9 at 13:03
$begingroup$
@Afroeskimo Nope, you can consider $x=1.5$ and $k=2$, for example.
$endgroup$
– BigbearZzz
Jan 9 at 13:09
add a comment |
$begingroup$
Understood! I would like to know whether it would be possible to remove a positive integer that resides in the ceil function, in a way that I have mentioned in the edited version of my question?
$endgroup$
– Afroeskimo
Jan 9 at 12:29
$begingroup$
@Afroeskimo In general you cannot pull a constant out of the floor/ceiling function so I'm afraid the answer is also No.
$endgroup$
– BigbearZzz
Jan 9 at 12:49
$begingroup$
even if the constant is an integer?
$endgroup$
– Afroeskimo
Jan 9 at 13:03
$begingroup$
@Afroeskimo Nope, you can consider $x=1.5$ and $k=2$, for example.
$endgroup$
– BigbearZzz
Jan 9 at 13:09
$begingroup$
Understood! I would like to know whether it would be possible to remove a positive integer that resides in the ceil function, in a way that I have mentioned in the edited version of my question?
$endgroup$
– Afroeskimo
Jan 9 at 12:29
$begingroup$
Understood! I would like to know whether it would be possible to remove a positive integer that resides in the ceil function, in a way that I have mentioned in the edited version of my question?
$endgroup$
– Afroeskimo
Jan 9 at 12:29
$begingroup$
@Afroeskimo In general you cannot pull a constant out of the floor/ceiling function so I'm afraid the answer is also No.
$endgroup$
– BigbearZzz
Jan 9 at 12:49
$begingroup$
@Afroeskimo In general you cannot pull a constant out of the floor/ceiling function so I'm afraid the answer is also No.
$endgroup$
– BigbearZzz
Jan 9 at 12:49
$begingroup$
even if the constant is an integer?
$endgroup$
– Afroeskimo
Jan 9 at 13:03
$begingroup$
even if the constant is an integer?
$endgroup$
– Afroeskimo
Jan 9 at 13:03
$begingroup$
@Afroeskimo Nope, you can consider $x=1.5$ and $k=2$, for example.
$endgroup$
– BigbearZzz
Jan 9 at 13:09
$begingroup$
@Afroeskimo Nope, you can consider $x=1.5$ and $k=2$, for example.
$endgroup$
– BigbearZzz
Jan 9 at 13:09
add a comment |
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