Mixing
Why is $sup f_- (n) inf f_+ (m) = frac{5}{4} $?
$begingroup$
Let $f_- (n) = Pi_{i=0}^n ( sin(i) - frac{5}{4}) $
And let
$ f_+(m) = Pi_{i=0}^m ( sin(i) + frac{5}{4} ) $
It appears that
$$sup f_- (n) inf f_+ (m) = frac{5}{4} $$
Why is that so ?
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln(cos(x) - frac{5}{4}) dx = 0 $$
$$ int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) - frac{5}{4}) dx = 2 pi^2 i $$
That explains the finite values of $sup $ and $ inf $.. well almost. It can be proven that both are finite. But that does not explain the value of their product.
Update
This is probably not helpful at all , but it can be shown ( not easy ) that there exist a unique pair of functions $g_-(x) , g_+(x) $ , both entire and with period $2 pi $ such that
$$ g_-(n) = f_-(n) , g_+(m) = f_+(m) $$
However i have no closed form for any of those ...
As for the numerical test i got about $ln(u) (2 pi)^{-1}$ correct digits , where $u = m + n$ and the ratio $m/n$ is close to $1$.
Assuming no round-off errors i ended Up with $1.2499999999(?) $. That was enough to convince me.
I often get accused of " no context " or " no effort " but i have NOO idea how to even start here. I considered telescoping but failed and assumed it is not related. Since I also have no closed form for the product I AM STUCK.
I get upset when people assume this is homework.
It clearly is not imho ! What kind of teacher or book contains this ?
——-
Example :
Taking $m = n = 8000 $ we get
$$ max(f_-(1),f_-(2),...,f_-(8000)) = 1,308587092.. $$
$$ min(f_+(1),f_+(2),...,f_+(8000)) = 0,955226916.. $$
$$ 1.308587092.. X 0.955226916.. = 1.249997612208568.. $$
Supporting the claim.
Im not sure if $sup f_+ = 7,93.. $ or the average of $f_+ $ ( $ 3,57..$ ) relate to the above $1,308.. $ and $0,955..$ or the truth of the claimed value $5/4$.
In principe we could write the values $1,308..$ and $0,955..$ as complicated integrals.
By using the continuum product functions $f_-(v),f_+(w)$ where $v,w$ are positive reals.
This is by noticing $ sum^t sum_i a_i exp(t space i) = sum_i a_i ( exp((t+1)i) - 1)(exp(i) - 1)^{-1} $ and noticing the functions $f_+,f_-$ are periodic with $2 pi$.
Next with contour integration you can find min and max over that period $2 pi$ for the continuum product functions.
Then the product of those 2 integrals should give you $frac{5}{4}$.
—-
Maybe all of this is unnessarily complicated and some simple theorems from trigonometry or calculus could easily explain the conjectured value $frac{5}{4}$ .. but I do not see it.
——
——
Update
This conjecture is part of a more general phenomenon.
For example the second conjecture :
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16} ) $
$$ sup g(n) space inf g(n) = frac{9}{16} $$
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
And perhaps the first conjecture could also follow from this second one ?
Since these are additional questions and I can only accept one answer , I started a new thread with these additional questions :
Why is $inf g sup g = frac{9}{16} $?
calculus geometry fractions limsup-and-liminf products
asked Dec 28 '16 at 22:46
mickmick
5,18832165
$endgroup$
|
show 1 more comment
$begingroup$
Let $f_- (n) = Pi_{i=0}^n ( sin(i) - frac{5}{4}) $
And let
$ f_+(m) = Pi_{i=0}^m ( sin(i) + frac{5}{4} ) $
It appears that
$$sup f_- (n) inf f_+ (m) = frac{5}{4} $$
Why is that so ?
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln(cos(x) - frac{5}{4}) dx = 0 $$
$$ int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) - frac{5}{4}) dx = 2 pi^2 i $$
That explains the finite values of $sup $ and $ inf $.. well almost. It can be proven that both are finite. But that does not explain the value of their product.
Update
This is probably not helpful at all , but it can be shown ( not easy ) that there exist a unique pair of functions $g_-(x) , g_+(x) $ , both entire and with period $2 pi $ such that
$$ g_-(n) = f_-(n) , g_+(m) = f_+(m) $$
However i have no closed form for any of those ...
As for the numerical test i got about $ln(u) (2 pi)^{-1}$ correct digits , where $u = m + n$ and the ratio $m/n$ is close to $1$.
Assuming no round-off errors i ended Up with $1.2499999999(?) $. That was enough to convince me.
I often get accused of " no context " or " no effort " but i have NOO idea how to even start here. I considered telescoping but failed and assumed it is not related. Since I also have no closed form for the product I AM STUCK.
I get upset when people assume this is homework.
It clearly is not imho ! What kind of teacher or book contains this ?
——-
Example :
Taking $m = n = 8000 $ we get
$$ max(f_-(1),f_-(2),...,f_-(8000)) = 1,308587092.. $$
$$ min(f_+(1),f_+(2),...,f_+(8000)) = 0,955226916.. $$
$$ 1.308587092.. X 0.955226916.. = 1.249997612208568.. $$
Supporting the claim.
Im not sure if $sup f_+ = 7,93.. $ or the average of $f_+ $ ( $ 3,57..$ ) relate to the above $1,308.. $ and $0,955..$ or the truth of the claimed value $5/4$.
In principe we could write the values $1,308..$ and $0,955..$ as complicated integrals.
By using the continuum product functions $f_-(v),f_+(w)$ where $v,w$ are positive reals.
This is by noticing $ sum^t sum_i a_i exp(t space i) = sum_i a_i ( exp((t+1)i) - 1)(exp(i) - 1)^{-1} $ and noticing the functions $f_+,f_-$ are periodic with $2 pi$.
Next with contour integration you can find min and max over that period $2 pi$ for the continuum product functions.
Then the product of those 2 integrals should give you $frac{5}{4}$.
—-
Maybe all of this is unnessarily complicated and some simple theorems from trigonometry or calculus could easily explain the conjectured value $frac{5}{4}$ .. but I do not see it.
——
——
Update
This conjecture is part of a more general phenomenon.
For example the second conjecture :
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16} ) $
$$ sup g(n) space inf g(n) = frac{9}{16} $$
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
And perhaps the first conjecture could also follow from this second one ?
Since these are additional questions and I can only accept one answer , I started a new thread with these additional questions :
Why is $inf g sup g = frac{9}{16} $?
calculus geometry fractions limsup-and-liminf products
asked Dec 28 '16 at 22:46
mickmick
5,18832165
$endgroup$
$begingroup$
"It appears that": is it a conjecture or a claim?
$endgroup$
– A.Γ.
Dec 28 '16 at 23:22
$begingroup$
@A.G. : what is the difference between a conjecture , a guess , a test and a claim ?? Define those formally.
$endgroup$
– mick
Dec 29 '16 at 0:02
$begingroup$
Explain what makes you believe that it is true.
$endgroup$
– A.Γ.
Dec 29 '16 at 0:21
$begingroup$
Tested on a computer
$endgroup$
– mick
Dec 29 '16 at 1:26
$begingroup$
I edited the OP.
$endgroup$
– mick
Nov 12 '18 at 9:59
|
show 1 more comment
$begingroup$
Let $f_- (n) = Pi_{i=0}^n ( sin(i) - frac{5}{4}) $
And let
$ f_+(m) = Pi_{i=0}^m ( sin(i) + frac{5}{4} ) $
It appears that
$$sup f_- (n) inf f_+ (m) = frac{5}{4} $$
Why is that so ?
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln(cos(x) - frac{5}{4}) dx = 0 $$
$$ int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) - frac{5}{4}) dx = 2 pi^2 i $$
That explains the finite values of $sup $ and $ inf $.. well almost. It can be proven that both are finite. But that does not explain the value of their product.
Update
This is probably not helpful at all , but it can be shown ( not easy ) that there exist a unique pair of functions $g_-(x) , g_+(x) $ , both entire and with period $2 pi $ such that
$$ g_-(n) = f_-(n) , g_+(m) = f_+(m) $$
However i have no closed form for any of those ...
As for the numerical test i got about $ln(u) (2 pi)^{-1}$ correct digits , where $u = m + n$ and the ratio $m/n$ is close to $1$.
Assuming no round-off errors i ended Up with $1.2499999999(?) $. That was enough to convince me.
I often get accused of " no context " or " no effort " but i have NOO idea how to even start here. I considered telescoping but failed and assumed it is not related. Since I also have no closed form for the product I AM STUCK.
I get upset when people assume this is homework.
It clearly is not imho ! What kind of teacher or book contains this ?
——-
Example :
Taking $m = n = 8000 $ we get
$$ max(f_-(1),f_-(2),...,f_-(8000)) = 1,308587092.. $$
$$ min(f_+(1),f_+(2),...,f_+(8000)) = 0,955226916.. $$
$$ 1.308587092.. X 0.955226916.. = 1.249997612208568.. $$
Supporting the claim.
Im not sure if $sup f_+ = 7,93.. $ or the average of $f_+ $ ( $ 3,57..$ ) relate to the above $1,308.. $ and $0,955..$ or the truth of the claimed value $5/4$.
In principe we could write the values $1,308..$ and $0,955..$ as complicated integrals.
By using the continuum product functions $f_-(v),f_+(w)$ where $v,w$ are positive reals.
This is by noticing $ sum^t sum_i a_i exp(t space i) = sum_i a_i ( exp((t+1)i) - 1)(exp(i) - 1)^{-1} $ and noticing the functions $f_+,f_-$ are periodic with $2 pi$.
Next with contour integration you can find min and max over that period $2 pi$ for the continuum product functions.
Then the product of those 2 integrals should give you $frac{5}{4}$.
—-
Maybe all of this is unnessarily complicated and some simple theorems from trigonometry or calculus could easily explain the conjectured value $frac{5}{4}$ .. but I do not see it.
——
——
Update
This conjecture is part of a more general phenomenon.
For example the second conjecture :
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16} ) $
$$ sup g(n) space inf g(n) = frac{9}{16} $$
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
And perhaps the first conjecture could also follow from this second one ?
Since these are additional questions and I can only accept one answer , I started a new thread with these additional questions :
Why is $inf g sup g = frac{9}{16} $?
calculus geometry fractions limsup-and-liminf products
asked Dec 28 '16 at 22:46
mickmick
5,18832165
$endgroup$
Let $f_- (n) = Pi_{i=0}^n ( sin(i) - frac{5}{4}) $
And let
$ f_+(m) = Pi_{i=0}^m ( sin(i) + frac{5}{4} ) $
It appears that
$$sup f_- (n) inf f_+ (m) = frac{5}{4} $$
Why is that so ?
Notice
$$int_0^{2 pi} ln(sin(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) + frac{5}{4}) dx = Re int_0^{2 pi} ln(cos(x) - frac{5}{4}) dx = 0 $$
$$ int_0^{2 pi} ln (sin(x) - frac{5}{4}) dx = int_0^{2 pi} ln (cos(x) - frac{5}{4}) dx = 2 pi^2 i $$
That explains the finite values of $sup $ and $ inf $.. well almost. It can be proven that both are finite. But that does not explain the value of their product.
Update
This is probably not helpful at all , but it can be shown ( not easy ) that there exist a unique pair of functions $g_-(x) , g_+(x) $ , both entire and with period $2 pi $ such that
$$ g_-(n) = f_-(n) , g_+(m) = f_+(m) $$
However i have no closed form for any of those ...
As for the numerical test i got about $ln(u) (2 pi)^{-1}$ correct digits , where $u = m + n$ and the ratio $m/n$ is close to $1$.
Assuming no round-off errors i ended Up with $1.2499999999(?) $. That was enough to convince me.
I often get accused of " no context " or " no effort " but i have NOO idea how to even start here. I considered telescoping but failed and assumed it is not related. Since I also have no closed form for the product I AM STUCK.
I get upset when people assume this is homework.
It clearly is not imho ! What kind of teacher or book contains this ?
——-
Example :
Taking $m = n = 8000 $ we get
$$ max(f_-(1),f_-(2),...,f_-(8000)) = 1,308587092.. $$
$$ min(f_+(1),f_+(2),...,f_+(8000)) = 0,955226916.. $$
$$ 1.308587092.. X 0.955226916.. = 1.249997612208568.. $$
Supporting the claim.
Im not sure if $sup f_+ = 7,93.. $ or the average of $f_+ $ ( $ 3,57..$ ) relate to the above $1,308.. $ and $0,955..$ or the truth of the claimed value $5/4$.
In principe we could write the values $1,308..$ and $0,955..$ as complicated integrals.
By using the continuum product functions $f_-(v),f_+(w)$ where $v,w$ are positive reals.
This is by noticing $ sum^t sum_i a_i exp(t space i) = sum_i a_i ( exp((t+1)i) - 1)(exp(i) - 1)^{-1} $ and noticing the functions $f_+,f_-$ are periodic with $2 pi$.
Next with contour integration you can find min and max over that period $2 pi$ for the continuum product functions.
Then the product of those 2 integrals should give you $frac{5}{4}$.
—-
Maybe all of this is unnessarily complicated and some simple theorems from trigonometry or calculus could easily explain the conjectured value $frac{5}{4}$ .. but I do not see it.
——
——
Update
This conjecture is part of a more general phenomenon.
For example the second conjecture :
Let $g(n) = prod_{i=0}^n (sin^2(n) + frac{9}{16} ) $
$$ sup g(n) space inf g(n) = frac{9}{16} $$
It feels like this second conjecture could somehow follow from the first conjecture since
$$-(cos(n) + frac{5}{4})(cos(n) - frac{5}{4}) = - cos^2(n) + frac{25}{16} = sin^2(n) + frac{9}{16} $$
And perhaps the first conjecture could also follow from this second one ?
Since these are additional questions and I can only accept one answer , I started a new thread with these additional questions :
Why is $inf g sup g = frac{9}{16} $?
calculus geometry fractions limsup-and-liminf products
calculus geometry fractions limsup-and-liminf products
asked Dec 28 '16 at 22:46
mickmick
5,18832165
asked Dec 28 '16 at 22:46
mickmick
5,18832165
edited Feb 2 at 8:30
mick
asked Dec 28 '16 at 22:46
mickmick
5,18832165
asked Dec 28 '16 at 22:46
mickmick
5,18832165
asked Dec 28 '16 at 22:46
mickmick
5,18832165
5,18832165
$begingroup$
"It appears that": is it a conjecture or a claim?
$endgroup$
– A.Γ.
Dec 28 '16 at 23:22
$begingroup$
@A.G. : what is the difference between a conjecture , a guess , a test and a claim ?? Define those formally.
$endgroup$
– mick
Dec 29 '16 at 0:02
$begingroup$
Explain what makes you believe that it is true.
$endgroup$
– A.Γ.
Dec 29 '16 at 0:21
$begingroup$
Tested on a computer
$endgroup$
– mick
Dec 29 '16 at 1:26
$begingroup$
I edited the OP.
$endgroup$
– mick
Nov 12 '18 at 9:59
|
show 1 more comment
$begingroup$
"It appears that": is it a conjecture or a claim?
$endgroup$
– A.Γ.
Dec 28 '16 at 23:22
$begingroup$
@A.G. : what is the difference between a conjecture , a guess , a test and a claim ?? Define those formally.
$endgroup$
– mick
Dec 29 '16 at 0:02
$begingroup$
Explain what makes you believe that it is true.
$endgroup$
– A.Γ.
Dec 29 '16 at 0:21
$begingroup$
Tested on a computer
$endgroup$
– mick
Dec 29 '16 at 1:26
$begingroup$
I edited the OP.
$endgroup$
– mick
Nov 12 '18 at 9:59
$begingroup$
"It appears that": is it a conjecture or a claim?
$endgroup$
– A.Γ.
Dec 28 '16 at 23:22
$begingroup$
"It appears that": is it a conjecture or a claim?
$endgroup$
– A.Γ.
Dec 28 '16 at 23:22
$begingroup$
@A.G. : what is the difference between a conjecture , a guess , a test and a claim ?? Define those formally.
$endgroup$
– mick
Dec 29 '16 at 0:02
$begingroup$
@A.G. : what is the difference between a conjecture , a guess , a test and a claim ?? Define those formally.
$endgroup$
– mick
Dec 29 '16 at 0:02
$begingroup$
Explain what makes you believe that it is true.
$endgroup$
– A.Γ.
Dec 29 '16 at 0:21
$begingroup$
Explain what makes you believe that it is true.
$endgroup$
– A.Γ.
Dec 29 '16 at 0:21
$begingroup$
Tested on a computer
$endgroup$
– mick
Dec 29 '16 at 1:26
$begingroup$
Tested on a computer
$endgroup$
– mick
Dec 29 '16 at 1:26
$begingroup$
I edited the OP.
$endgroup$
– mick
Nov 12 '18 at 9:59
$begingroup$
I edited the OP.
$endgroup$
– mick
Nov 12 '18 at 9:59
|
show 1 more comment
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$begingroup$
"It appears that": is it a conjecture or a claim?
$endgroup$
– A.Γ.
Dec 28 '16 at 23:22
$begingroup$
@A.G. : what is the difference between a conjecture , a guess , a test and a claim ?? Define those formally.
$endgroup$
– mick
Dec 29 '16 at 0:02
$begingroup$
Explain what makes you believe that it is true.
$endgroup$
– A.Γ.
Dec 29 '16 at 0:21
$begingroup$
Tested on a computer
$endgroup$
– mick
Dec 29 '16 at 1:26
$begingroup$
I edited the OP.
$endgroup$
– mick
Nov 12 '18 at 9:59