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## Inductor Current Equation

i recently came across an exercise involving solving for the energy released in a freewheeling diode. This is the circuit: The problem is that i’m not understanding the current equations for the inductor(L) when the transistor is on and off: Transistor ON: Transistor OFF: This is because when the diode resistance is really small, the […]

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## Tricky question in Calculus

I have studied calulus a long time ago but I have seen this question and I just can’t think of a counter example: Let $f$ be a function s.t $\frac{f(x+1)}{f(x)}=3$ , $f(x)>0$ for every $x\in[0,\infty)$ i) Proof / disproof : $\lim_{x\rightarrow\infty}f(x) = \infty$ ii) Now assume $f$ is continuous in $[0, \infty)$ . Prove that […]

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## Insert specific non sequential values into a serial field (ID) of a postgres table

Context On the one hand, I have a PostgreSQL data model with a table that has one ID field defined as a serial (which relies on a sequence). On the other hand, I first had some data that have their own unique IDs, but which were not starting from 1, e.g.: 1001, 1002, 1003, 1004, […]

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## Using transfinite induction to split $R$ to continuum many pairwise disjoint subsets of $R$

I am looking for different ways to partition $R$. I know some like : (1) Define a relation as following $$x\sim y \ \text{iff} \ x-y\in\mathbb Q(x,y\in\mathbb R)$$. The equivalence classes have the form $[r]=r+\mathbb Q$ and clear they are countable dense and pairwise disjoint and $\mathbb R=\bigcup_{r\in\mathbb R} [r]$. (2) Let $P$ be the […]

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## Is this sentence’s tense correct?

Thus, Alexander’s untimely death at the age of 32 naturally begs the question of would could have been had he been given the chance to live a normal age. I’m not sure whether “would could have been had he” is correct, help?

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## Is this sentence’s tense correct?

Thus, Alexander’s untimely death at the age of 32 naturally begs the question of would could have been had he been given the chance to live a normal age. I’m not sure whether “would could have been had he” is correct, help?

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## Relation between enthalpy, entropy and Gibbs energy at thermal equilibrium

$${\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr}}$$ … Equation (i) If system is in thermal equilibrium with the surrounding i.e. ${T_{sys} = T_{surr} = T}$ Increase/ decrease in enthalpy of surrounding = decrease/ increase in enthalpy of system $${\Delta H_{surr} = -\Delta H_{sys}}$$ Therefore, entropy change of surroundings, {\Delta S_{surr} = \frac{\Delta H_{surr}}{T}= \frac{-\Delta […]

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## Understanding zorn’s lemma

right now i am taking a ring theory class, (the class is online…) and we just proved that every ring has a maximal ideal. Anyone who is familiar with the proof knows that in order to prove it, we have to use zorn’s lemma. Basically proofs that are using zorn’s lemma have the same “plan” […]

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## Show that if $(b^n-1)/(b-1)$ is a power of prime numbers, where $b,n>1$ are positive integers, then $n$ must be a prime number.

Show that if $(b^n-1)/(b-1)$ is the power of a prime number, where $b,n>1$ are positive integers, then $n$ must be a prime number. My solution: If $n$ is composite, then let $n=mk$, $m,k>1$, \begin{align*} \frac{b^n-1}{b-1} &= 1+b+\cdots+b^{n-1} \\ &=(1+b+\cdots+b^{k-1} )+(b^k+b^{k+1}+\cdots+b^{2k-1}) \\ &\quad\,+\cdots+(b^{(m-1)k}+b^{(m-1)k+1}+\cdots+b^{mk-1}) \\ &=(1+b+\cdots+b^{k-1})(1+b^k+\cdots+b^{(m-1)k}) \end{align*} Which is composite and distinct, thus, for $(b^n-1)/(b-1)$ to be […]

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## Show that if (b^n-1)/(b-1) is a power of prime numbers, where b,n>1 are positive integers, then n must be a prime number.

Show that if (b^n-1)/(b-1) is a power of prime numbers, where b,n>1 are positive integers, then n must be a prime number. my solution: If n is composite, then let n=mk,m,k>1, (b^n-1)/(b-1)=1+b+⋯+b^(n-1)=(1+b+⋯+b^(k-1) )+(b^k+b^(k+1)+⋯+b^(2k-1) )+⋯+(b^(m-1)k+b^((m-1)k+1)+⋯+b^(mk-1) )=(1+b+⋯+b^(k-1) )(1+b^k+⋯+b^(m-1)k ) Which is composite and distinct, thus, for (b^n-1)/(b-1) to be a power of primes, n is not composite, […]