I've been wondering whether in Harry Potter canon there is any hint on whether there is energy expenditure involved in doing magic. Hints would be things like increased tiredness, fatigue after doing several spells, the need to eat more, etc.
Often in fantasies involving magic casting spells requires a certain physical or mental toll on the caster. I don't recall any mention of spells being physically or mentally taxing in the Harry Potter universe.  Does this mean they can cast an unlimited number of spells in a given day without any negative effect to themselves?
Why is there a big delay after entering a wrong password? Can you partition an infinite set, into an infinite number of infinite sets? How can I utilize the whole terminal, like htop or emacs do? I want to write a program that can "draw" in the terminal the same way as htop or emacs. By that I mean "get rid of the prompt and make the output dependent on the current size of the terminal".   To make this a bit more clear:   I want to use the terminal this way:     in stead of this way: When "Questions that may already have your answer" (similar questions) appear as a right column, it overflows and overlaps the footer.  (shown in the figure)        This happens on all SE network sites on which I have asked questions.  Also this happens to other columns also like "How to Format". When there is less than 5% battery left on a mac, the battery indicator goes red and system performance significantly drops - presumably in an attempt to make that remaining 5% last a little longer.   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I have: $$  \forall x \exists y \forall z(x^2-y+z=0) $$ such that $$ x,y,z \in \Bbb Z^+$$   My first question, can it be read like this:  $$  \forall x \forall z \exists y(x^2-y+z=0)  $$  The way I did it, is I started off with $x=1, z=1 $   $$ 2-y = 0 $$ $$ y =2 $$  Is this a good approach? Periodic function implies periodic primitive? If $f$ is a periodic function with $T = \pi$, does that mean that $F$ (a primitive of $f$) has the same period ? What about the inverse implication (periodic primitive implies periodic function) ? Using subcaption package but subfigures end up stacked rather than side by side Bolzano-Weierstrass Theorem Proof Abbott (Ax. of Choice) In Abbott's Proof of the Bolzano-Weierstrass Theorem, does Abbott use any form of the Axiom of Choice ?   I think he is since he chooses an $a_{n_k} \in I_k$ where there are multiple such $a_{n_k}$. How did Newton discover the universal law of gravitation? 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The maximal atlas is constructed by taking an atlas and appending all coordinate charts that are compatible with the atlas and using this he shows that any atlas on a locally Euclidean space is contained in a unique maximal atlas.  He also later shows that for any chart on a manifold, the coordinate map is a diffeomorphism onto it's image.   My question arises when Tu mentions in an aside that every compact topological manifold in dimension four or higher has a finite number of differentiable structures. What I don't understand is how it is possible for a smooth manifold to have more than one differentiable structure. Doesn't the fact that for any chart the coordinate map being a diffeomorphism coupled with the fact that composition of smooth functions is smooth mean that any two coordinate charts on a manifold are compatible and thus they must all belong to one maximal atlas? How is it possible to have another? Am I not understanding the word "maximal" or atlas or some other concept? 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