# An example where the obvious does not work# find eta such that f <= bound# assumptions: L,R,eta >0f:=(eta*L+R)/(1-exp(-eta));

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bound:=L+sqrt(2*L*R)+R;

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#idea divide both by L to reduce number of varsff:=(eta+r)/(1-exp(-eta));

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bbound:=1+sqrt(2*r)+r;

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#visual checkrset:=3;

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plot({subs(r=rset,ff),subs(r=rset,bbound)},eta=0.2..10);

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#for all rset > 0 there always seems to be a choice for eta minimize(ff,eta); #does not work

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sol:=solve(factor(diff(ff,eta)),eta) assuming eta>0;

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?LambertWplot(sol,r=0..10); #maple bug, eta should be positive

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# Explore using a different branch of the LambertW function#new idea for proceeding: lower bound exp(eta)taylor(exp(eta),eta=0,5);

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plot([exp(eta),1+eta,1+eta+eta^2/2],eta=-2..3,color=[black,red,green]); #lower bounds

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

plot([1/(1-exp(-eta)),1/(1-1/(1+eta)),1/(1-1/(1+eta+eta^2/2))],eta=0.1..10,color=[black,red,green]);#upper bounds

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

ff1:=(eta+r)/(1-1/(1+eta)); #1.order taylor

NiM+SSRmZjFHNiIqJiwmSSRldGFHRiUiIiJJInJHRiVGKUYpLCZGKUYpKiQsJkYpRilGKEYpISIiRi5GLg==

sol1:=solve(diff(ff1,eta),eta);

NiM+SSVzb2wxRzYiNiQqJEkickdGJSMiIiIiIiMsJEYnISIi

gap1:=bbound-subs(eta=sol1[1],ff);#plug in solution

NiM+SSVnYXAxRzYiLCoiIiJGJyomIiIjI0YnRilJInJHRiVGKkYnRitGJyomLCYqJEYrRipGJ0YrRidGJywmRidGJy1JJGV4cEc2JEkqcHJvdGVjdGVkR0YzSShfc3lzbGliR0YlNiMsJEYuISIiRjdGN0Y3

plot(gap1,r=0..200);

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

limit(gap1,r=0);limit(gap1,r=infinity);#looks good

NiMiIiE=NiNJKWluZmluaXR5R0kqcHJvdGVjdGVkR0Yk

limit(diff(gap1,r),r=0,right); #bends the wrong way

NiMsJEkpaW5maW5pdHlHSSpwcm90ZWN0ZWRHRiUhIiI=

plot(gap1,r=0..0.2);#you see that it dips below zero

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

ff2:=(eta+r)/(1-1/(1+eta+eta^2/2)); #2nd order approx. of exponential

NiM+SSRmZjJHNiIqJiwmSSRldGFHRiUiIiJJInJHRiVGKUYpLCZGKUYpKiQsKEYpRilGKEYpKiRGKCIiIyNGKUYvISIiRjFGMQ==

sol2:=solve(diff(ff2,eta),eta);

NiM+SSVzb2wyRzYiLUknUm9vdE9mRzYkSSpwcm90ZWN0ZWRHRilJKF9zeXNsaWJHRiU2IywsKiRJI19aR0YlIiIjRi8qJEYuIiIkIiIlKiRGLkYyIiIiSSJyR0YlISIlKiZGNUY0Ri5GNEY2

plot(sol2,r=0..10);

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

sol2guess:=sqrt(2*r);

NiM+SSpzb2wyZ3Vlc3NHNiIqJiIiIyMiIiJGJ0kickdGJUYo

plot([sol2,sol2guess,sqrt(r)],r=0..10,color=[black,red,green]);

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

gap2:=factor(simplify(bbound-subs(eta=sol2guess,ff)));#should be >0

NiM+SSVnYXAyRzYiKiYsKiEiIiIiIi1JJGV4cEc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YlNiMsJComIiIjI0YpRjJJInJHRiVGM0YoRikqKEYyRjNGNEYzRipGKUYpKiZGNEYpRipGKUYpRiksJkYoRilGKkYpRig=

plot(gap2,r=0..200);plot(gap2,r=0..0.2);

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

limit(gap2,r=infinity);limit(gap2,r=0);limit(diff(gap2,r),r=0);

NiMiIiI=NiMiIiE=NiMjIiIiIiIk

simplify(subs(eta=sol2guess,ff2));#upper bound on ff with guess of eta hits bound

NiMsKCIiIkYkKiYiIiMjRiRGJkkickc2IkYnRiRGKEYk

plot(subs(eta=sol2guess,ff2)-subs(eta=sol2,ff2),r=0..100);

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

#We lower bounded exp(eta), which led to the upper bound ff2 of ff#We guessed a solution sol2eta for eta #Plugging this solution into the upper bound ff2 produced bboundlimit(subs(eta=sol2guess,ff2)-subs(eta=sol2,ff2),r=infinity);

NiNJKWluZmluaXR5R0kqcHJvdGVjdGVkR0Yk

#So there is a better choice of eta for minimizing ff2#Is there one in closed form???plot(subs(eta=sol2guess,ff2)-subs(eta=sol2,ff2),r=0..0.2);

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plot(subs(eta=sol2guess,ff2)-subs(eta=sol2,ff2),r=0..10000);

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