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Hi. I asked this question about Boolean function minimization a while ago and got no answer. I would greatly appreciate any help with it

@ThomasAndrews well, now I do understand. Thank you for that

@ThomasAndrews i don't understand. After my edit this is literally the problem VTand gave a link to

There is a handful of useful comments there already, so I tend towards editing the existing one being a better choice

Or should I close this question and make another one?

I edited the question as I still think the problem with positive reals deserves attention and may be useful to someone

@VTand I see. Indeed I tried to modify my question just because of curiosity. Didn;t take the time to think about what I was doing

@ThomasAndrews what about now?

@ThomasAndrews I made an edit with an additional restriction

The only reason I believe it is true is because I have seen it on a math olympiad. And I answered to VTand's comment

@ThomasAndrews I rather tend towards thinking this question is poorly stated yet an interesting concept lies behind it

@VTand Good question. However, I simply found this problem without any more details than what I gave you. I assume that only division by $0$ is illegal for $B$ and the elements may not be distinct

@ThomasAndrews No, by no means, any finite subset of $\Bbb{R}$

Hi. Can anyone help me with this question about Boolean function minimization I posted a while ago? math.stackexchange.com/questions/4484690/…

In fact, I got $\frac{q}{C_0} - \frac{q_0 - q}{C} = - L q'' = V_0 cos(\omega t)$

And even though it gives different results, the physical implications are the same

Current reference direction can be chosen arbitrarily

the equation $\frac{q}{C_0} - \frac{q_0 - q}{C} = - L q''$ is just as correct as $\frac{q}{C_0} - \frac{q_0 - q}{C} = L q''$

After all, either I wasn't able to understand you the whole time, or I simply tend to disagree with you

Hi

I do not understand how to determine the E fields' direction in advance, before solving the circuit. The very porpoise of solving the circuit analysis problem is to determine the charge distribution, E fields and all that stuff. How can I use any of that to solve the circuit?

4) And you state that both capasitor plates connected to K are negatively charged. Do you mean at any time? Why is that so?

1) "That it equals $V_0 cosωt$ is another physical insight" - explain this please. 2) The very reason of implementing the passive sign convention is that we cannot know in advance the direction of the E field, so we just assume the current and voltage direction and then see if the negative sign shows up. In fact, the E field can oscillate and change direction. 3) I still don't understand those $V_\infty$ and short circuit arguments. How did you obtain this from some Maxwell's/Kirchhoff's or other fundamental equations and how exactly it influences the capacitot voltage at any given time?

1) As I understand, you defined capacitor voltages as $\varphi_\text{positively charged plate} - \varphi_\text{negatively charged plate}$, right? 2) What do you mean by "short-circuited case" and what is $V_\infty$ 3) In your convention, voltage having positive sign means that while traversing the chosen loop, we encounter the positive terminal first? 4) How to determine the sign of $L\dot{I}$ and that it equals $V_0 cos(\omega t)$? 5) Why can't I just assume the direction of current and use the passive sign convention?

Passive sign convention implies choosing an arbitraty current direction, then assuming that current always enters the positive terminal. This way, $U_R = IR$, $U_C = \frac{q}{C}$ and $U_L = L \frac{dI}{dt}$. For charge conservation I'm not sure how to determine the sign and I may be wrong there, since I can't know for sure which terminal of a capacitor is positively charged. That's why I need help with understanding where to get these equation

I know where the charge conservation comes from. As I stated, the equation out of nowhere was this: $$V_{C_0} - V_C = L \frac{dI}{dt} = V_0 cos(\omega t)$$ I don't understand where this comes from. And I don't understand why this: $$\frac{q_{c_0}}{C_0} + \frac{q_{c}}{C} + L q'' = 0$$ is wrong, as it is the mere application of KVL with passive sign convention, which accounts for magnetic flux stuff in the inductor. And also my equation reduces to $q'' = - \frac{1}{LC_{equiv}} q$, which is the expected result

I do know the charge of each capacitor, but that charge is different. Thus, the charge does not depend only on time, but also on what part of the circuit we are considering

it seems like you think I was assigned this problem by a teacher and all I want is an answer. And this is likely the reason why my question was closed despite me showing my work and asking about the physical concepts, not the mere answer. I want to understand the behaviour of the system. I want to know how to rigorously obtain the answer from the very postulates. I have NO interest in doing anything with some "magical equations" untill I completely understand where they come from

I literally do not have the innitial condition for charge as the charge is by the very specification of the problem different in each capacitor. So there is no "circuit charge function"

What "equations" are you talking about? I told you why I can't solve this: $$\frac{q_{c_0}}{C_0} + \frac{q_{c}}{C} + L q'' = 0$$ due to not having the innitial charge condition

And where am I wrong in the part that we have a wire with potential difference between the capacitors and thus a short circuit?

The inductance term must be with negative sign if one is to treat it as EMF of self-inductance. Also, if we use passive sign convention, the KVL (or Faraday's law to be precise) equation is $V_{C_0} + V_C + V_L = \frac{q_{c_0}}{C0}+\frac{q_c}{C}+Lq′′ = 0$, which gives the correct expression for frequency, actually. And I don't understand how the part $L \frac{dI}{dt} = V_0 cos(\omega t)$ is obtained. The only way is to find current and it is the derivative of charge. The only way to obtain charge is to solve the differential equation I got. But for this I need the innitial conditions