How can we find some radius of circle which fully contains $xarctan(x)-ax+yarctan(y)-by=0$?












6












$begingroup$


How can we find some radius of circle with center at origin which contains $xarctan(x)-ax+yarctan(y)-by=0$,



where $pi/2>a>0$ and $pi/2>b>0$.



I'm not sure how can we prove that these inequalities should hold so we can have closed curve: $pi/2>a$ and $pi/2>b$ .



Also very interesting similar equation $xarctan(x)-ax+(-x+y)arctan(-x+y)-b(-x+y)+yarctan(y)-cy=0$.



I need some approximate estimation so I can prove that for some radius this estimation will be correct.



For example I've found such a circle for $a=1.5$ and $b=1.5$:



enter image description here



And for $xarctan(x)-1.5x+(-x+y)arctan(-x+y)-1.5(-x+y)+yarctan(y)-1.5y=0$:



enter image description here



Maybe Lagrange multipliers can help?
I'm also interested in higher dimensions where we can add $z$ and find some radius of a sphere which fully contains $xarctan(x)-ax+yarctan(y)-by+zarctan(z)-cz=0$. But I think it can be done in similar way as for two dimensions.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Are you looking for the smallest circle, or the smallest one centred on $O$?
    $endgroup$
    – J.G.
    Dec 10 '18 at 17:09










  • $begingroup$
    I need a circle centred on $O$ (origin). And not necessarily the smallest (but the smallest will be good, of course).
    $endgroup$
    – Tag
    Dec 10 '18 at 17:11
















6












$begingroup$


How can we find some radius of circle with center at origin which contains $xarctan(x)-ax+yarctan(y)-by=0$,



where $pi/2>a>0$ and $pi/2>b>0$.



I'm not sure how can we prove that these inequalities should hold so we can have closed curve: $pi/2>a$ and $pi/2>b$ .



Also very interesting similar equation $xarctan(x)-ax+(-x+y)arctan(-x+y)-b(-x+y)+yarctan(y)-cy=0$.



I need some approximate estimation so I can prove that for some radius this estimation will be correct.



For example I've found such a circle for $a=1.5$ and $b=1.5$:



enter image description here



And for $xarctan(x)-1.5x+(-x+y)arctan(-x+y)-1.5(-x+y)+yarctan(y)-1.5y=0$:



enter image description here



Maybe Lagrange multipliers can help?
I'm also interested in higher dimensions where we can add $z$ and find some radius of a sphere which fully contains $xarctan(x)-ax+yarctan(y)-by+zarctan(z)-cz=0$. But I think it can be done in similar way as for two dimensions.










share|cite|improve this question











$endgroup$












  • $begingroup$
    Are you looking for the smallest circle, or the smallest one centred on $O$?
    $endgroup$
    – J.G.
    Dec 10 '18 at 17:09










  • $begingroup$
    I need a circle centred on $O$ (origin). And not necessarily the smallest (but the smallest will be good, of course).
    $endgroup$
    – Tag
    Dec 10 '18 at 17:11














6












6








6


0



$begingroup$


How can we find some radius of circle with center at origin which contains $xarctan(x)-ax+yarctan(y)-by=0$,



where $pi/2>a>0$ and $pi/2>b>0$.



I'm not sure how can we prove that these inequalities should hold so we can have closed curve: $pi/2>a$ and $pi/2>b$ .



Also very interesting similar equation $xarctan(x)-ax+(-x+y)arctan(-x+y)-b(-x+y)+yarctan(y)-cy=0$.



I need some approximate estimation so I can prove that for some radius this estimation will be correct.



For example I've found such a circle for $a=1.5$ and $b=1.5$:



enter image description here



And for $xarctan(x)-1.5x+(-x+y)arctan(-x+y)-1.5(-x+y)+yarctan(y)-1.5y=0$:



enter image description here



Maybe Lagrange multipliers can help?
I'm also interested in higher dimensions where we can add $z$ and find some radius of a sphere which fully contains $xarctan(x)-ax+yarctan(y)-by+zarctan(z)-cz=0$. But I think it can be done in similar way as for two dimensions.










share|cite|improve this question











$endgroup$




How can we find some radius of circle with center at origin which contains $xarctan(x)-ax+yarctan(y)-by=0$,



where $pi/2>a>0$ and $pi/2>b>0$.



I'm not sure how can we prove that these inequalities should hold so we can have closed curve: $pi/2>a$ and $pi/2>b$ .



Also very interesting similar equation $xarctan(x)-ax+(-x+y)arctan(-x+y)-b(-x+y)+yarctan(y)-cy=0$.



I need some approximate estimation so I can prove that for some radius this estimation will be correct.



For example I've found such a circle for $a=1.5$ and $b=1.5$:



enter image description here



And for $xarctan(x)-1.5x+(-x+y)arctan(-x+y)-1.5(-x+y)+yarctan(y)-1.5y=0$:



enter image description here



Maybe Lagrange multipliers can help?
I'm also interested in higher dimensions where we can add $z$ and find some radius of a sphere which fully contains $xarctan(x)-ax+yarctan(y)-by+zarctan(z)-cz=0$. But I think it can be done in similar way as for two dimensions.







maxima-minima implicit-function






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 10 '18 at 17:03







Tag

















asked Dec 10 '18 at 15:22









TagTag

696




696












  • $begingroup$
    Are you looking for the smallest circle, or the smallest one centred on $O$?
    $endgroup$
    – J.G.
    Dec 10 '18 at 17:09










  • $begingroup$
    I need a circle centred on $O$ (origin). And not necessarily the smallest (but the smallest will be good, of course).
    $endgroup$
    – Tag
    Dec 10 '18 at 17:11


















  • $begingroup$
    Are you looking for the smallest circle, or the smallest one centred on $O$?
    $endgroup$
    – J.G.
    Dec 10 '18 at 17:09










  • $begingroup$
    I need a circle centred on $O$ (origin). And not necessarily the smallest (but the smallest will be good, of course).
    $endgroup$
    – Tag
    Dec 10 '18 at 17:11
















$begingroup$
Are you looking for the smallest circle, or the smallest one centred on $O$?
$endgroup$
– J.G.
Dec 10 '18 at 17:09




$begingroup$
Are you looking for the smallest circle, or the smallest one centred on $O$?
$endgroup$
– J.G.
Dec 10 '18 at 17:09












$begingroup$
I need a circle centred on $O$ (origin). And not necessarily the smallest (but the smallest will be good, of course).
$endgroup$
– Tag
Dec 10 '18 at 17:11




$begingroup$
I need a circle centred on $O$ (origin). And not necessarily the smallest (but the smallest will be good, of course).
$endgroup$
– Tag
Dec 10 '18 at 17:11










1 Answer
1






active

oldest

votes


















5












$begingroup$

Definition. Given $ainmathbb R$, define the function $f_a(x)=x(arctan x-a)$.



Definition. Given $a,binmathbb R$, consider the closed set
$$
begin{split}
E_{a,b} &= {(x,y)inmathbb R^2 : x(arctan x-a)+y(arctan y-b)=0} \
&= {(x,y)inmathbb R^2 : f_a(x)+f_b(y)=0}
end{split}
$$



Lemma 1. For every $ainmathbb R$, the set $[0,infty)$ is in the image of $f_a$.



Proof. Assume $ageq0$. Then $f_a(0)=0$ and
$$
lim_{xto-infty}x(arctan x-a)=(-pi/2-a) lim_{xto-infty}x=infty.
$$

Since $f_a$ is continous, these limits imply the thesis. The case $aleq0$ is analogous, with the signs changed. □



Lemma 2. If $|a|>pi/2$, then the function $f_a:mathbb Rtomathbb R$ is surjective.



Proof. Assume $a>pi/2$. We have $pmpi/2-a < 0$, so
$$
lim_{xto+infty} x(arctan x-a)
= (pi/2-a)lim_{xtoinfty} x = -infty
$$

and
$$
lim_{xto-infty} x(arctan x-a)
= (-pi/2-a)lim_{xto-infty} x = infty.
$$

Since $f_a$ is continuous, these limits imply that it is surjective. The case $a<-pi/2$ is analogous. □



Lemma 3. If $|a|<pi/2$, then
$$
lim_{xtopminfty} f_a(x) = infty.
$$



Proof. A direct computation shows
$$
lim_{xtoinfty} f_a(x) = (pi/2-a)lim_{xtoinfty}x = infty
$$

and the same holds for $xto-infty$. □



Proposition. If $|a|>pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Since $f_a$ is surjective by Lemma 2, for every $yinmathbb R$ there exists $xinmathbb R$ such that $f_a(x)=-f_b(y)$. This means that $(x,y)in E_{a,b}$. Therefore the set $E_{a,b}$ is unbounded because we can find points with arbitrary large $y$.



Proposition. If $|a|=pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Assume $a=pi/2$. The other case is analogous. Recall the well known limit
$$
lim_{xtoinfty} f_{pi/2}(x) = lim_{xtoinfty} x(arctan x-pi/2) = -1.
$$

For every $x$ sufficiently large we have that $f_{pi/2}(x)leq0$, so by Lemma 1 there exists $yinmathbb R$ such that $f_b(y)=-f_a(x)in[0,infty)$. Therefore we can find points $(x,y)in E_{pi/2,b}$ with arbitrary large $x$. □



Proposition. If $|a|<pi/2$ and $|b|<pi/2$, then $E_{a,b}$ is bounded.



Proof. By Lemma 3, the function $f_a(x)+f_b(y)$ is coercive, meaning that $f_a(x)+f_b(y)toinfty$ if $|(x,y)|toinfty$, therefore its sublevel sets are bounded. In particular $E_{a,b}$ is bounded as a consequence. □



Corollary. $E_{a,b}$ is bounded if and only if $|a|<pi/2$ and $|b|<pi/2$.





Now, given $a,bin(-pi/2,pi/2)$, how can we find an estimate on $max_{(x,y)in E_{a,b}} x^2+y^2$? We could try the Lagrange multipliers approach again, similarly to what we did here.
The stationary points must satisfy
$$
bigl(f'_a(x), f'_b(y)bigr) = lambda (x, y) qquad text{for some $lambdainmathbb R$},
$$

which is equivalent to
$$
frac1{1+x^2} + frac{arctan x-a}x
= frac{f'_a(x)}{x} = lambda
= frac{f'_b(y)}{y}
=frac1{1+y^2} + frac{arctan y-b}y.
$$



Unfortunately, this time I'm not able to find a closed form solution to the system of equations
$$
left{begin{array}{l}
f_a(x)+f_b(y)=0 , \
frac{f'_a(x)}x=frac{f'_b(y)}y .
end{array}right.
$$



One can of course fall back to numerical solutions. I'm using Mathematica for that.
Here I fix the values $a=3/2$ and $b=5/4$, then I find extremal points numerically both with the built-in NMaximize function and by solving the Lagrange multiplier system with FindRoot. The two solutions are the same, up to machine precision. Then I plot the set $E_{a,b}$ in blue, the root locus of the Lagrange multiplier equation in orange, and the smallest fitting circle in gray.



enter image description here



enter image description here






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Well, that falls in the broad question of numerical optimization. I proved that the set is bounded. Therefore the problem $max_{(x,y)in E_{a,b}} x^2+y^2$ is well posed. Then there are plenty of algorithms to perform numerical maximization. Here everything is smooth, so there should be no problems with running any numerical algorithm. There cannot be a divergence, because the set is bounded!
    $endgroup$
    – Federico
    Dec 11 '18 at 18:05






  • 1




    $begingroup$
    Ok, so for the $n$-dimensional version, the same statement holds. Namely, if you define $E_{a_1,dots,a_n}={f_{a_1}(x_1)+dots+f_{a_n}(x_n)=0}$, then $E_{a_1,dots,a_n}$ is bounded if and only if $|a_i|<pi/2$ for all $i=1,dots,n$. The proof is also the same
    $endgroup$
    – Federico
    Dec 13 '18 at 14:33








  • 1




    $begingroup$
    Once you have this condition ($|a_i|<pi/2 forall i$), you can apply again a numerical scheme to maximize $x_1^2+dots+x_n^2$ over $E_{a_1,dots,a_n}$ and it should be able to find the optimum
    $endgroup$
    – Federico
    Dec 13 '18 at 14:35








  • 1




    $begingroup$
    You mention the symmetry in the case $a_1=dots=a_n$. While it is true that in this case $E_{a_1,dots,a_n}$ is symmetric w.r.t. permutations of the variables, be aware that the maximum of $x_1^2+dots+x_n^2$ is not achieved when all $x_i$'s except one are zero, or when $x_1=dots=x_n$. The maximum is achieved at a point which is not particularly special from the point of view of symmetries. This is something different from your other problem
    $endgroup$
    – Federico
    Dec 13 '18 at 14:38






  • 1




    $begingroup$
    @Tag The set ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is equal to the projection on the $(x,y)$ plane of $E_{a,b,c}cap{z=y-x}={(x,y,z):f_a(x)+f_b(y)+f_c(z)=0}cap{z=y-x}$. So, if $a,b,cin(-pi/2,pi/2)$, then $E_{a,b,c}$ is bounded and therefore slicing and projecting it gives a bounded set. Notice that in this case it is no longer an "if and only if". I'm not sure we can say that if ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is bounded then $a,b,cin(-pi/2,pi/2)$. I don't know the answer because I haven't analyzed this problem in detail. But at least the other implication is true.
    $endgroup$
    – Federico
    Dec 17 '18 at 16:55













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1 Answer
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active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









5












$begingroup$

Definition. Given $ainmathbb R$, define the function $f_a(x)=x(arctan x-a)$.



Definition. Given $a,binmathbb R$, consider the closed set
$$
begin{split}
E_{a,b} &= {(x,y)inmathbb R^2 : x(arctan x-a)+y(arctan y-b)=0} \
&= {(x,y)inmathbb R^2 : f_a(x)+f_b(y)=0}
end{split}
$$



Lemma 1. For every $ainmathbb R$, the set $[0,infty)$ is in the image of $f_a$.



Proof. Assume $ageq0$. Then $f_a(0)=0$ and
$$
lim_{xto-infty}x(arctan x-a)=(-pi/2-a) lim_{xto-infty}x=infty.
$$

Since $f_a$ is continous, these limits imply the thesis. The case $aleq0$ is analogous, with the signs changed. □



Lemma 2. If $|a|>pi/2$, then the function $f_a:mathbb Rtomathbb R$ is surjective.



Proof. Assume $a>pi/2$. We have $pmpi/2-a < 0$, so
$$
lim_{xto+infty} x(arctan x-a)
= (pi/2-a)lim_{xtoinfty} x = -infty
$$

and
$$
lim_{xto-infty} x(arctan x-a)
= (-pi/2-a)lim_{xto-infty} x = infty.
$$

Since $f_a$ is continuous, these limits imply that it is surjective. The case $a<-pi/2$ is analogous. □



Lemma 3. If $|a|<pi/2$, then
$$
lim_{xtopminfty} f_a(x) = infty.
$$



Proof. A direct computation shows
$$
lim_{xtoinfty} f_a(x) = (pi/2-a)lim_{xtoinfty}x = infty
$$

and the same holds for $xto-infty$. □



Proposition. If $|a|>pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Since $f_a$ is surjective by Lemma 2, for every $yinmathbb R$ there exists $xinmathbb R$ such that $f_a(x)=-f_b(y)$. This means that $(x,y)in E_{a,b}$. Therefore the set $E_{a,b}$ is unbounded because we can find points with arbitrary large $y$.



Proposition. If $|a|=pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Assume $a=pi/2$. The other case is analogous. Recall the well known limit
$$
lim_{xtoinfty} f_{pi/2}(x) = lim_{xtoinfty} x(arctan x-pi/2) = -1.
$$

For every $x$ sufficiently large we have that $f_{pi/2}(x)leq0$, so by Lemma 1 there exists $yinmathbb R$ such that $f_b(y)=-f_a(x)in[0,infty)$. Therefore we can find points $(x,y)in E_{pi/2,b}$ with arbitrary large $x$. □



Proposition. If $|a|<pi/2$ and $|b|<pi/2$, then $E_{a,b}$ is bounded.



Proof. By Lemma 3, the function $f_a(x)+f_b(y)$ is coercive, meaning that $f_a(x)+f_b(y)toinfty$ if $|(x,y)|toinfty$, therefore its sublevel sets are bounded. In particular $E_{a,b}$ is bounded as a consequence. □



Corollary. $E_{a,b}$ is bounded if and only if $|a|<pi/2$ and $|b|<pi/2$.





Now, given $a,bin(-pi/2,pi/2)$, how can we find an estimate on $max_{(x,y)in E_{a,b}} x^2+y^2$? We could try the Lagrange multipliers approach again, similarly to what we did here.
The stationary points must satisfy
$$
bigl(f'_a(x), f'_b(y)bigr) = lambda (x, y) qquad text{for some $lambdainmathbb R$},
$$

which is equivalent to
$$
frac1{1+x^2} + frac{arctan x-a}x
= frac{f'_a(x)}{x} = lambda
= frac{f'_b(y)}{y}
=frac1{1+y^2} + frac{arctan y-b}y.
$$



Unfortunately, this time I'm not able to find a closed form solution to the system of equations
$$
left{begin{array}{l}
f_a(x)+f_b(y)=0 , \
frac{f'_a(x)}x=frac{f'_b(y)}y .
end{array}right.
$$



One can of course fall back to numerical solutions. I'm using Mathematica for that.
Here I fix the values $a=3/2$ and $b=5/4$, then I find extremal points numerically both with the built-in NMaximize function and by solving the Lagrange multiplier system with FindRoot. The two solutions are the same, up to machine precision. Then I plot the set $E_{a,b}$ in blue, the root locus of the Lagrange multiplier equation in orange, and the smallest fitting circle in gray.



enter image description here



enter image description here






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Well, that falls in the broad question of numerical optimization. I proved that the set is bounded. Therefore the problem $max_{(x,y)in E_{a,b}} x^2+y^2$ is well posed. Then there are plenty of algorithms to perform numerical maximization. Here everything is smooth, so there should be no problems with running any numerical algorithm. There cannot be a divergence, because the set is bounded!
    $endgroup$
    – Federico
    Dec 11 '18 at 18:05






  • 1




    $begingroup$
    Ok, so for the $n$-dimensional version, the same statement holds. Namely, if you define $E_{a_1,dots,a_n}={f_{a_1}(x_1)+dots+f_{a_n}(x_n)=0}$, then $E_{a_1,dots,a_n}$ is bounded if and only if $|a_i|<pi/2$ for all $i=1,dots,n$. The proof is also the same
    $endgroup$
    – Federico
    Dec 13 '18 at 14:33








  • 1




    $begingroup$
    Once you have this condition ($|a_i|<pi/2 forall i$), you can apply again a numerical scheme to maximize $x_1^2+dots+x_n^2$ over $E_{a_1,dots,a_n}$ and it should be able to find the optimum
    $endgroup$
    – Federico
    Dec 13 '18 at 14:35








  • 1




    $begingroup$
    You mention the symmetry in the case $a_1=dots=a_n$. While it is true that in this case $E_{a_1,dots,a_n}$ is symmetric w.r.t. permutations of the variables, be aware that the maximum of $x_1^2+dots+x_n^2$ is not achieved when all $x_i$'s except one are zero, or when $x_1=dots=x_n$. The maximum is achieved at a point which is not particularly special from the point of view of symmetries. This is something different from your other problem
    $endgroup$
    – Federico
    Dec 13 '18 at 14:38






  • 1




    $begingroup$
    @Tag The set ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is equal to the projection on the $(x,y)$ plane of $E_{a,b,c}cap{z=y-x}={(x,y,z):f_a(x)+f_b(y)+f_c(z)=0}cap{z=y-x}$. So, if $a,b,cin(-pi/2,pi/2)$, then $E_{a,b,c}$ is bounded and therefore slicing and projecting it gives a bounded set. Notice that in this case it is no longer an "if and only if". I'm not sure we can say that if ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is bounded then $a,b,cin(-pi/2,pi/2)$. I don't know the answer because I haven't analyzed this problem in detail. But at least the other implication is true.
    $endgroup$
    – Federico
    Dec 17 '18 at 16:55


















5












$begingroup$

Definition. Given $ainmathbb R$, define the function $f_a(x)=x(arctan x-a)$.



Definition. Given $a,binmathbb R$, consider the closed set
$$
begin{split}
E_{a,b} &= {(x,y)inmathbb R^2 : x(arctan x-a)+y(arctan y-b)=0} \
&= {(x,y)inmathbb R^2 : f_a(x)+f_b(y)=0}
end{split}
$$



Lemma 1. For every $ainmathbb R$, the set $[0,infty)$ is in the image of $f_a$.



Proof. Assume $ageq0$. Then $f_a(0)=0$ and
$$
lim_{xto-infty}x(arctan x-a)=(-pi/2-a) lim_{xto-infty}x=infty.
$$

Since $f_a$ is continous, these limits imply the thesis. The case $aleq0$ is analogous, with the signs changed. □



Lemma 2. If $|a|>pi/2$, then the function $f_a:mathbb Rtomathbb R$ is surjective.



Proof. Assume $a>pi/2$. We have $pmpi/2-a < 0$, so
$$
lim_{xto+infty} x(arctan x-a)
= (pi/2-a)lim_{xtoinfty} x = -infty
$$

and
$$
lim_{xto-infty} x(arctan x-a)
= (-pi/2-a)lim_{xto-infty} x = infty.
$$

Since $f_a$ is continuous, these limits imply that it is surjective. The case $a<-pi/2$ is analogous. □



Lemma 3. If $|a|<pi/2$, then
$$
lim_{xtopminfty} f_a(x) = infty.
$$



Proof. A direct computation shows
$$
lim_{xtoinfty} f_a(x) = (pi/2-a)lim_{xtoinfty}x = infty
$$

and the same holds for $xto-infty$. □



Proposition. If $|a|>pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Since $f_a$ is surjective by Lemma 2, for every $yinmathbb R$ there exists $xinmathbb R$ such that $f_a(x)=-f_b(y)$. This means that $(x,y)in E_{a,b}$. Therefore the set $E_{a,b}$ is unbounded because we can find points with arbitrary large $y$.



Proposition. If $|a|=pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Assume $a=pi/2$. The other case is analogous. Recall the well known limit
$$
lim_{xtoinfty} f_{pi/2}(x) = lim_{xtoinfty} x(arctan x-pi/2) = -1.
$$

For every $x$ sufficiently large we have that $f_{pi/2}(x)leq0$, so by Lemma 1 there exists $yinmathbb R$ such that $f_b(y)=-f_a(x)in[0,infty)$. Therefore we can find points $(x,y)in E_{pi/2,b}$ with arbitrary large $x$. □



Proposition. If $|a|<pi/2$ and $|b|<pi/2$, then $E_{a,b}$ is bounded.



Proof. By Lemma 3, the function $f_a(x)+f_b(y)$ is coercive, meaning that $f_a(x)+f_b(y)toinfty$ if $|(x,y)|toinfty$, therefore its sublevel sets are bounded. In particular $E_{a,b}$ is bounded as a consequence. □



Corollary. $E_{a,b}$ is bounded if and only if $|a|<pi/2$ and $|b|<pi/2$.





Now, given $a,bin(-pi/2,pi/2)$, how can we find an estimate on $max_{(x,y)in E_{a,b}} x^2+y^2$? We could try the Lagrange multipliers approach again, similarly to what we did here.
The stationary points must satisfy
$$
bigl(f'_a(x), f'_b(y)bigr) = lambda (x, y) qquad text{for some $lambdainmathbb R$},
$$

which is equivalent to
$$
frac1{1+x^2} + frac{arctan x-a}x
= frac{f'_a(x)}{x} = lambda
= frac{f'_b(y)}{y}
=frac1{1+y^2} + frac{arctan y-b}y.
$$



Unfortunately, this time I'm not able to find a closed form solution to the system of equations
$$
left{begin{array}{l}
f_a(x)+f_b(y)=0 , \
frac{f'_a(x)}x=frac{f'_b(y)}y .
end{array}right.
$$



One can of course fall back to numerical solutions. I'm using Mathematica for that.
Here I fix the values $a=3/2$ and $b=5/4$, then I find extremal points numerically both with the built-in NMaximize function and by solving the Lagrange multiplier system with FindRoot. The two solutions are the same, up to machine precision. Then I plot the set $E_{a,b}$ in blue, the root locus of the Lagrange multiplier equation in orange, and the smallest fitting circle in gray.



enter image description here



enter image description here






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Well, that falls in the broad question of numerical optimization. I proved that the set is bounded. Therefore the problem $max_{(x,y)in E_{a,b}} x^2+y^2$ is well posed. Then there are plenty of algorithms to perform numerical maximization. Here everything is smooth, so there should be no problems with running any numerical algorithm. There cannot be a divergence, because the set is bounded!
    $endgroup$
    – Federico
    Dec 11 '18 at 18:05






  • 1




    $begingroup$
    Ok, so for the $n$-dimensional version, the same statement holds. Namely, if you define $E_{a_1,dots,a_n}={f_{a_1}(x_1)+dots+f_{a_n}(x_n)=0}$, then $E_{a_1,dots,a_n}$ is bounded if and only if $|a_i|<pi/2$ for all $i=1,dots,n$. The proof is also the same
    $endgroup$
    – Federico
    Dec 13 '18 at 14:33








  • 1




    $begingroup$
    Once you have this condition ($|a_i|<pi/2 forall i$), you can apply again a numerical scheme to maximize $x_1^2+dots+x_n^2$ over $E_{a_1,dots,a_n}$ and it should be able to find the optimum
    $endgroup$
    – Federico
    Dec 13 '18 at 14:35








  • 1




    $begingroup$
    You mention the symmetry in the case $a_1=dots=a_n$. While it is true that in this case $E_{a_1,dots,a_n}$ is symmetric w.r.t. permutations of the variables, be aware that the maximum of $x_1^2+dots+x_n^2$ is not achieved when all $x_i$'s except one are zero, or when $x_1=dots=x_n$. The maximum is achieved at a point which is not particularly special from the point of view of symmetries. This is something different from your other problem
    $endgroup$
    – Federico
    Dec 13 '18 at 14:38






  • 1




    $begingroup$
    @Tag The set ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is equal to the projection on the $(x,y)$ plane of $E_{a,b,c}cap{z=y-x}={(x,y,z):f_a(x)+f_b(y)+f_c(z)=0}cap{z=y-x}$. So, if $a,b,cin(-pi/2,pi/2)$, then $E_{a,b,c}$ is bounded and therefore slicing and projecting it gives a bounded set. Notice that in this case it is no longer an "if and only if". I'm not sure we can say that if ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is bounded then $a,b,cin(-pi/2,pi/2)$. I don't know the answer because I haven't analyzed this problem in detail. But at least the other implication is true.
    $endgroup$
    – Federico
    Dec 17 '18 at 16:55
















5












5








5





$begingroup$

Definition. Given $ainmathbb R$, define the function $f_a(x)=x(arctan x-a)$.



Definition. Given $a,binmathbb R$, consider the closed set
$$
begin{split}
E_{a,b} &= {(x,y)inmathbb R^2 : x(arctan x-a)+y(arctan y-b)=0} \
&= {(x,y)inmathbb R^2 : f_a(x)+f_b(y)=0}
end{split}
$$



Lemma 1. For every $ainmathbb R$, the set $[0,infty)$ is in the image of $f_a$.



Proof. Assume $ageq0$. Then $f_a(0)=0$ and
$$
lim_{xto-infty}x(arctan x-a)=(-pi/2-a) lim_{xto-infty}x=infty.
$$

Since $f_a$ is continous, these limits imply the thesis. The case $aleq0$ is analogous, with the signs changed. □



Lemma 2. If $|a|>pi/2$, then the function $f_a:mathbb Rtomathbb R$ is surjective.



Proof. Assume $a>pi/2$. We have $pmpi/2-a < 0$, so
$$
lim_{xto+infty} x(arctan x-a)
= (pi/2-a)lim_{xtoinfty} x = -infty
$$

and
$$
lim_{xto-infty} x(arctan x-a)
= (-pi/2-a)lim_{xto-infty} x = infty.
$$

Since $f_a$ is continuous, these limits imply that it is surjective. The case $a<-pi/2$ is analogous. □



Lemma 3. If $|a|<pi/2$, then
$$
lim_{xtopminfty} f_a(x) = infty.
$$



Proof. A direct computation shows
$$
lim_{xtoinfty} f_a(x) = (pi/2-a)lim_{xtoinfty}x = infty
$$

and the same holds for $xto-infty$. □



Proposition. If $|a|>pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Since $f_a$ is surjective by Lemma 2, for every $yinmathbb R$ there exists $xinmathbb R$ such that $f_a(x)=-f_b(y)$. This means that $(x,y)in E_{a,b}$. Therefore the set $E_{a,b}$ is unbounded because we can find points with arbitrary large $y$.



Proposition. If $|a|=pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Assume $a=pi/2$. The other case is analogous. Recall the well known limit
$$
lim_{xtoinfty} f_{pi/2}(x) = lim_{xtoinfty} x(arctan x-pi/2) = -1.
$$

For every $x$ sufficiently large we have that $f_{pi/2}(x)leq0$, so by Lemma 1 there exists $yinmathbb R$ such that $f_b(y)=-f_a(x)in[0,infty)$. Therefore we can find points $(x,y)in E_{pi/2,b}$ with arbitrary large $x$. □



Proposition. If $|a|<pi/2$ and $|b|<pi/2$, then $E_{a,b}$ is bounded.



Proof. By Lemma 3, the function $f_a(x)+f_b(y)$ is coercive, meaning that $f_a(x)+f_b(y)toinfty$ if $|(x,y)|toinfty$, therefore its sublevel sets are bounded. In particular $E_{a,b}$ is bounded as a consequence. □



Corollary. $E_{a,b}$ is bounded if and only if $|a|<pi/2$ and $|b|<pi/2$.





Now, given $a,bin(-pi/2,pi/2)$, how can we find an estimate on $max_{(x,y)in E_{a,b}} x^2+y^2$? We could try the Lagrange multipliers approach again, similarly to what we did here.
The stationary points must satisfy
$$
bigl(f'_a(x), f'_b(y)bigr) = lambda (x, y) qquad text{for some $lambdainmathbb R$},
$$

which is equivalent to
$$
frac1{1+x^2} + frac{arctan x-a}x
= frac{f'_a(x)}{x} = lambda
= frac{f'_b(y)}{y}
=frac1{1+y^2} + frac{arctan y-b}y.
$$



Unfortunately, this time I'm not able to find a closed form solution to the system of equations
$$
left{begin{array}{l}
f_a(x)+f_b(y)=0 , \
frac{f'_a(x)}x=frac{f'_b(y)}y .
end{array}right.
$$



One can of course fall back to numerical solutions. I'm using Mathematica for that.
Here I fix the values $a=3/2$ and $b=5/4$, then I find extremal points numerically both with the built-in NMaximize function and by solving the Lagrange multiplier system with FindRoot. The two solutions are the same, up to machine precision. Then I plot the set $E_{a,b}$ in blue, the root locus of the Lagrange multiplier equation in orange, and the smallest fitting circle in gray.



enter image description here



enter image description here






share|cite|improve this answer











$endgroup$



Definition. Given $ainmathbb R$, define the function $f_a(x)=x(arctan x-a)$.



Definition. Given $a,binmathbb R$, consider the closed set
$$
begin{split}
E_{a,b} &= {(x,y)inmathbb R^2 : x(arctan x-a)+y(arctan y-b)=0} \
&= {(x,y)inmathbb R^2 : f_a(x)+f_b(y)=0}
end{split}
$$



Lemma 1. For every $ainmathbb R$, the set $[0,infty)$ is in the image of $f_a$.



Proof. Assume $ageq0$. Then $f_a(0)=0$ and
$$
lim_{xto-infty}x(arctan x-a)=(-pi/2-a) lim_{xto-infty}x=infty.
$$

Since $f_a$ is continous, these limits imply the thesis. The case $aleq0$ is analogous, with the signs changed. □



Lemma 2. If $|a|>pi/2$, then the function $f_a:mathbb Rtomathbb R$ is surjective.



Proof. Assume $a>pi/2$. We have $pmpi/2-a < 0$, so
$$
lim_{xto+infty} x(arctan x-a)
= (pi/2-a)lim_{xtoinfty} x = -infty
$$

and
$$
lim_{xto-infty} x(arctan x-a)
= (-pi/2-a)lim_{xto-infty} x = infty.
$$

Since $f_a$ is continuous, these limits imply that it is surjective. The case $a<-pi/2$ is analogous. □



Lemma 3. If $|a|<pi/2$, then
$$
lim_{xtopminfty} f_a(x) = infty.
$$



Proof. A direct computation shows
$$
lim_{xtoinfty} f_a(x) = (pi/2-a)lim_{xtoinfty}x = infty
$$

and the same holds for $xto-infty$. □



Proposition. If $|a|>pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Since $f_a$ is surjective by Lemma 2, for every $yinmathbb R$ there exists $xinmathbb R$ such that $f_a(x)=-f_b(y)$. This means that $(x,y)in E_{a,b}$. Therefore the set $E_{a,b}$ is unbounded because we can find points with arbitrary large $y$.



Proposition. If $|a|=pi/2$. Then $E_{a,b}$ is unbounded.



Proof. Assume $a=pi/2$. The other case is analogous. Recall the well known limit
$$
lim_{xtoinfty} f_{pi/2}(x) = lim_{xtoinfty} x(arctan x-pi/2) = -1.
$$

For every $x$ sufficiently large we have that $f_{pi/2}(x)leq0$, so by Lemma 1 there exists $yinmathbb R$ such that $f_b(y)=-f_a(x)in[0,infty)$. Therefore we can find points $(x,y)in E_{pi/2,b}$ with arbitrary large $x$. □



Proposition. If $|a|<pi/2$ and $|b|<pi/2$, then $E_{a,b}$ is bounded.



Proof. By Lemma 3, the function $f_a(x)+f_b(y)$ is coercive, meaning that $f_a(x)+f_b(y)toinfty$ if $|(x,y)|toinfty$, therefore its sublevel sets are bounded. In particular $E_{a,b}$ is bounded as a consequence. □



Corollary. $E_{a,b}$ is bounded if and only if $|a|<pi/2$ and $|b|<pi/2$.





Now, given $a,bin(-pi/2,pi/2)$, how can we find an estimate on $max_{(x,y)in E_{a,b}} x^2+y^2$? We could try the Lagrange multipliers approach again, similarly to what we did here.
The stationary points must satisfy
$$
bigl(f'_a(x), f'_b(y)bigr) = lambda (x, y) qquad text{for some $lambdainmathbb R$},
$$

which is equivalent to
$$
frac1{1+x^2} + frac{arctan x-a}x
= frac{f'_a(x)}{x} = lambda
= frac{f'_b(y)}{y}
=frac1{1+y^2} + frac{arctan y-b}y.
$$



Unfortunately, this time I'm not able to find a closed form solution to the system of equations
$$
left{begin{array}{l}
f_a(x)+f_b(y)=0 , \
frac{f'_a(x)}x=frac{f'_b(y)}y .
end{array}right.
$$



One can of course fall back to numerical solutions. I'm using Mathematica for that.
Here I fix the values $a=3/2$ and $b=5/4$, then I find extremal points numerically both with the built-in NMaximize function and by solving the Lagrange multiplier system with FindRoot. The two solutions are the same, up to machine precision. Then I plot the set $E_{a,b}$ in blue, the root locus of the Lagrange multiplier equation in orange, and the smallest fitting circle in gray.



enter image description here



enter image description here







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 11 '18 at 17:37

























answered Dec 11 '18 at 15:04









FedericoFederico

5,034514




5,034514








  • 1




    $begingroup$
    Well, that falls in the broad question of numerical optimization. I proved that the set is bounded. Therefore the problem $max_{(x,y)in E_{a,b}} x^2+y^2$ is well posed. Then there are plenty of algorithms to perform numerical maximization. Here everything is smooth, so there should be no problems with running any numerical algorithm. There cannot be a divergence, because the set is bounded!
    $endgroup$
    – Federico
    Dec 11 '18 at 18:05






  • 1




    $begingroup$
    Ok, so for the $n$-dimensional version, the same statement holds. Namely, if you define $E_{a_1,dots,a_n}={f_{a_1}(x_1)+dots+f_{a_n}(x_n)=0}$, then $E_{a_1,dots,a_n}$ is bounded if and only if $|a_i|<pi/2$ for all $i=1,dots,n$. The proof is also the same
    $endgroup$
    – Federico
    Dec 13 '18 at 14:33








  • 1




    $begingroup$
    Once you have this condition ($|a_i|<pi/2 forall i$), you can apply again a numerical scheme to maximize $x_1^2+dots+x_n^2$ over $E_{a_1,dots,a_n}$ and it should be able to find the optimum
    $endgroup$
    – Federico
    Dec 13 '18 at 14:35








  • 1




    $begingroup$
    You mention the symmetry in the case $a_1=dots=a_n$. While it is true that in this case $E_{a_1,dots,a_n}$ is symmetric w.r.t. permutations of the variables, be aware that the maximum of $x_1^2+dots+x_n^2$ is not achieved when all $x_i$'s except one are zero, or when $x_1=dots=x_n$. The maximum is achieved at a point which is not particularly special from the point of view of symmetries. This is something different from your other problem
    $endgroup$
    – Federico
    Dec 13 '18 at 14:38






  • 1




    $begingroup$
    @Tag The set ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is equal to the projection on the $(x,y)$ plane of $E_{a,b,c}cap{z=y-x}={(x,y,z):f_a(x)+f_b(y)+f_c(z)=0}cap{z=y-x}$. So, if $a,b,cin(-pi/2,pi/2)$, then $E_{a,b,c}$ is bounded and therefore slicing and projecting it gives a bounded set. Notice that in this case it is no longer an "if and only if". I'm not sure we can say that if ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is bounded then $a,b,cin(-pi/2,pi/2)$. I don't know the answer because I haven't analyzed this problem in detail. But at least the other implication is true.
    $endgroup$
    – Federico
    Dec 17 '18 at 16:55
















  • 1




    $begingroup$
    Well, that falls in the broad question of numerical optimization. I proved that the set is bounded. Therefore the problem $max_{(x,y)in E_{a,b}} x^2+y^2$ is well posed. Then there are plenty of algorithms to perform numerical maximization. Here everything is smooth, so there should be no problems with running any numerical algorithm. There cannot be a divergence, because the set is bounded!
    $endgroup$
    – Federico
    Dec 11 '18 at 18:05






  • 1




    $begingroup$
    Ok, so for the $n$-dimensional version, the same statement holds. Namely, if you define $E_{a_1,dots,a_n}={f_{a_1}(x_1)+dots+f_{a_n}(x_n)=0}$, then $E_{a_1,dots,a_n}$ is bounded if and only if $|a_i|<pi/2$ for all $i=1,dots,n$. The proof is also the same
    $endgroup$
    – Federico
    Dec 13 '18 at 14:33








  • 1




    $begingroup$
    Once you have this condition ($|a_i|<pi/2 forall i$), you can apply again a numerical scheme to maximize $x_1^2+dots+x_n^2$ over $E_{a_1,dots,a_n}$ and it should be able to find the optimum
    $endgroup$
    – Federico
    Dec 13 '18 at 14:35








  • 1




    $begingroup$
    You mention the symmetry in the case $a_1=dots=a_n$. While it is true that in this case $E_{a_1,dots,a_n}$ is symmetric w.r.t. permutations of the variables, be aware that the maximum of $x_1^2+dots+x_n^2$ is not achieved when all $x_i$'s except one are zero, or when $x_1=dots=x_n$. The maximum is achieved at a point which is not particularly special from the point of view of symmetries. This is something different from your other problem
    $endgroup$
    – Federico
    Dec 13 '18 at 14:38






  • 1




    $begingroup$
    @Tag The set ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is equal to the projection on the $(x,y)$ plane of $E_{a,b,c}cap{z=y-x}={(x,y,z):f_a(x)+f_b(y)+f_c(z)=0}cap{z=y-x}$. So, if $a,b,cin(-pi/2,pi/2)$, then $E_{a,b,c}$ is bounded and therefore slicing and projecting it gives a bounded set. Notice that in this case it is no longer an "if and only if". I'm not sure we can say that if ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is bounded then $a,b,cin(-pi/2,pi/2)$. I don't know the answer because I haven't analyzed this problem in detail. But at least the other implication is true.
    $endgroup$
    – Federico
    Dec 17 '18 at 16:55










1




1




$begingroup$
Well, that falls in the broad question of numerical optimization. I proved that the set is bounded. Therefore the problem $max_{(x,y)in E_{a,b}} x^2+y^2$ is well posed. Then there are plenty of algorithms to perform numerical maximization. Here everything is smooth, so there should be no problems with running any numerical algorithm. There cannot be a divergence, because the set is bounded!
$endgroup$
– Federico
Dec 11 '18 at 18:05




$begingroup$
Well, that falls in the broad question of numerical optimization. I proved that the set is bounded. Therefore the problem $max_{(x,y)in E_{a,b}} x^2+y^2$ is well posed. Then there are plenty of algorithms to perform numerical maximization. Here everything is smooth, so there should be no problems with running any numerical algorithm. There cannot be a divergence, because the set is bounded!
$endgroup$
– Federico
Dec 11 '18 at 18:05




1




1




$begingroup$
Ok, so for the $n$-dimensional version, the same statement holds. Namely, if you define $E_{a_1,dots,a_n}={f_{a_1}(x_1)+dots+f_{a_n}(x_n)=0}$, then $E_{a_1,dots,a_n}$ is bounded if and only if $|a_i|<pi/2$ for all $i=1,dots,n$. The proof is also the same
$endgroup$
– Federico
Dec 13 '18 at 14:33






$begingroup$
Ok, so for the $n$-dimensional version, the same statement holds. Namely, if you define $E_{a_1,dots,a_n}={f_{a_1}(x_1)+dots+f_{a_n}(x_n)=0}$, then $E_{a_1,dots,a_n}$ is bounded if and only if $|a_i|<pi/2$ for all $i=1,dots,n$. The proof is also the same
$endgroup$
– Federico
Dec 13 '18 at 14:33






1




1




$begingroup$
Once you have this condition ($|a_i|<pi/2 forall i$), you can apply again a numerical scheme to maximize $x_1^2+dots+x_n^2$ over $E_{a_1,dots,a_n}$ and it should be able to find the optimum
$endgroup$
– Federico
Dec 13 '18 at 14:35






$begingroup$
Once you have this condition ($|a_i|<pi/2 forall i$), you can apply again a numerical scheme to maximize $x_1^2+dots+x_n^2$ over $E_{a_1,dots,a_n}$ and it should be able to find the optimum
$endgroup$
– Federico
Dec 13 '18 at 14:35






1




1




$begingroup$
You mention the symmetry in the case $a_1=dots=a_n$. While it is true that in this case $E_{a_1,dots,a_n}$ is symmetric w.r.t. permutations of the variables, be aware that the maximum of $x_1^2+dots+x_n^2$ is not achieved when all $x_i$'s except one are zero, or when $x_1=dots=x_n$. The maximum is achieved at a point which is not particularly special from the point of view of symmetries. This is something different from your other problem
$endgroup$
– Federico
Dec 13 '18 at 14:38




$begingroup$
You mention the symmetry in the case $a_1=dots=a_n$. While it is true that in this case $E_{a_1,dots,a_n}$ is symmetric w.r.t. permutations of the variables, be aware that the maximum of $x_1^2+dots+x_n^2$ is not achieved when all $x_i$'s except one are zero, or when $x_1=dots=x_n$. The maximum is achieved at a point which is not particularly special from the point of view of symmetries. This is something different from your other problem
$endgroup$
– Federico
Dec 13 '18 at 14:38




1




1




$begingroup$
@Tag The set ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is equal to the projection on the $(x,y)$ plane of $E_{a,b,c}cap{z=y-x}={(x,y,z):f_a(x)+f_b(y)+f_c(z)=0}cap{z=y-x}$. So, if $a,b,cin(-pi/2,pi/2)$, then $E_{a,b,c}$ is bounded and therefore slicing and projecting it gives a bounded set. Notice that in this case it is no longer an "if and only if". I'm not sure we can say that if ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is bounded then $a,b,cin(-pi/2,pi/2)$. I don't know the answer because I haven't analyzed this problem in detail. But at least the other implication is true.
$endgroup$
– Federico
Dec 17 '18 at 16:55






$begingroup$
@Tag The set ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is equal to the projection on the $(x,y)$ plane of $E_{a,b,c}cap{z=y-x}={(x,y,z):f_a(x)+f_b(y)+f_c(z)=0}cap{z=y-x}$. So, if $a,b,cin(-pi/2,pi/2)$, then $E_{a,b,c}$ is bounded and therefore slicing and projecting it gives a bounded set. Notice that in this case it is no longer an "if and only if". I'm not sure we can say that if ${(x,y):f_a(x)+f_b(y)+f_c(y-x)=0}$ is bounded then $a,b,cin(-pi/2,pi/2)$. I don't know the answer because I haven't analyzed this problem in detail. But at least the other implication is true.
$endgroup$
– Federico
Dec 17 '18 at 16:55




















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Required, but never shown







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