FindMinimum, NMinimize, etc. with external process
What are the most common pitfalls awaiting new users?
Help Understanding How Long It Will Take To Heat My Wire How can one prove that $\lim_{n \to \infty}a^{1/n}=1$ for every $a>0$? Prove that $\lim_{n \to \infty} a^{\frac{1}{n}} = 1$ if $a &gt;0$.  In my textbook, we are given a suggestion to let $a^{\frac{1}{n}} = (1+h_n)$ and then show that the $h_n$ term goes to zero using a Theorem that states the following conditions:  $\lim_{n \to \infty} = \infty,$ if $a &gt;1, 1,$ if $a = 1,$ and 0, if $\vert a \vert &lt; 1$. I apologize for not formatting the above cases appropriately; I could not figure out how to use a giant left brace to group them altogether.  If I rewrite it as $a^{n^{-1}}$, then I thought that this might help, but I don't think the Binomial Theorem would then apply since the exponent is negative. Furthermore, the problem is only concerned with the positive, real $n$-th roots. I am quite stuck on this problem and any suggestions or advice would be greatly appreciated.   I am using the textbook Introduction to Analysis by Arthur Mattuck. What should I do if two libraries provide a function with the same name generating a conflict? What should I do if I have two libraries that provide functions with equivalent names? Tikzexternalize not compatible with Miktex 2.9 abntex2 package I'm working on this problem from Elementary Analysis by Ross which is intuitive when sketched but keeps stymieing me when I try to write it out.  Let $f$ be a continuous function on $[a,b] \subset \mathbb{R}$. Define $f^\star (x)$ as:  $$ f^\star(x) = \sup \{f(y)\mid a \leq y \leq x \} $$ Prove that $f^\star$ is a continuous increasing function on $[a,b]$.  Things I've figured out: (these are mostly trivial)   $f$ is uniformly continuous on $[a,b]$. Since $[a,b]$ is closed and bounded there exists some $c \in [a,b]$ such that $f(c) \geq f(x)\ \forall x \in [a,b]$. (In words: the supremum is actually attained.) If $c \in [a,b]$ is as above, then $f^\star$ is constant (and hence continuous) on $[c,b]$ (which is possibly a singleton).   This seems completely obvious when you actually draw a continuous function but translating that to a formal proof eludes me... In PHP, is there an easy way to convert a number to a word? For instance, 27 to twenty-seven. How can I read input from the console using the Scanner class in Java? Isomorphism involving Eilenberg-Maclane space, Tors. Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. Does there exist an isomorphism between $H_*(K(\pi, 1); A)$ and $\text{Tor}_*^{\mathbb{Z}[\pi]}(A, \mathbb{Z})$? Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$ Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1?  Thanks “It can be very cold…” vs “It may be very cold…” It can be very cold in winters.   It may be very cold in winters.   What is the difference between these two sentences?  People say when may is used then it means that it may be cold in winters or it may not be cold in winters. In other words, may means 'not sure'.  So does that mean if can is used then it is 100% sure that it will be cold in winters? My neighbor has asked me help with his laptop, an HP with lost power-on password. I tried inserting an Ubuntu cd to format the partition, but well... it doesn't even get there. Any ideas? In a modern SPFx Web Part, how do I check if a URL exists? C++ warning: deprecated conversion from string constant to ‘char*’ [-Wwrite-strings] What are the most important things I need to do to encourage Google Sitelinks? My website used to have sitelinks and now it doesn't. It's very possible that it's due to changing the website to a sidebar design instead of having an "interstitial" type landing page which limited the number of choices, but I'm not sure.  Here is how sitelinks might look for a site:    What are some things that I can do to improve my chances of getting sitelinks? Does non-mass-energy generate a gravitational field? At a very basic level I know that gravity isn't generated by mass but rather the  and when I wave my hands a lot it seems like that implies that energy in $E^2 = (pc)^2 + (mc^2)^2$ is the source of gravity.  If the total energy of a particle contributes to the generation of a gravitational field, that seems to imply some strange things.  For example, you could accelerate a particle like an electron to such a high velocity that it has enough momentum / kinetic energy to collapse into a black hole.  An observer traveling with the electron would see an electron and stationary observers would see an arbitrarily massive black hole flying by.  These observations seem to be contradictory.  The same would be true of an extremely high energy photon.  Another consequence is that a stationary particle could appear to have so much mass that it could attract / deflect the trajectory of a rapidly moving observer.  It seems odd that at low velocities you wouldn't even notice a stationary electron as you pass it by but at high velocities the gravitational field you observe could be significant enough to alter your path.  Am I correct that non-mass-energy generates a gravitational field?  If it does, how are these strange observations like a particle looking like a black hole in one reference frame but not another reconciled? Structure of ideals in the product of two rings $R$ and $S$ are two rings. Let $J$ be an ideal in $R\times S$. Then there are $I_{1}$, ideal of $R$, and $I_{2}$, ideal of $S$ such that $J=I_{1}\times I_{2}$.  For me it's obvious why $\left\{ r\in R\mid \left(r,s\right)\in J\text{ for some } s\in S\right\}$ is an ideal of $R$ (and the same for $S$) so I can prove $J$ is a subset of the product of two ideals and also that the product of these two ideals is an ideal. But I can't see how to prove equality without assuming existence of unity or commutativity. Trying basically to show that if $\left(r,s\right)\in J$ then also $\left(r,0\right),\left(0,s\right)\in J$. Thank you for your help! Python: list items are not changed in for loop What is the definition of $2.5!$? (2.5 factorial) I was messing around with my TI-84 Plus Silver Edition calculator and discovered that it will actually give me values when taking the factorial of any number $n/2$ where $n$ is any integer greater than $-2$. Why does this happen? I thought factorials were only defined for positive integers and $0$, so what is my calculator doing to get the answer $3.32335097$ when I enter $2.5!$? Is there actually a definition of $2.5!$ or is my calculator just being weird? How is the factorial function implemented?  I understand the binary implications of $2.5$, so that could possibly have something to do with it. I get a domain error when trying to take the factorial of $-1, 2.3, e$, and any number that is not of the form $n/2$ where $n$ is any integer greater than $-2$. How do I generate a smooth random horizontal 2D tunnel? I'd like to create a smoother version of the navigable and quite natural-looking random tunnel found in .  It should ideally be...   infinite, so more can be generated as the player progresses; parametrised, allowing control of its thickness over time; made of smooth curves, not rectangles as in the above game; precomputable, as knowing its bounds in advance allows collision detection for e.g. positioning powerups inside the tunnel.   I'm looking for a generic method I could implement myself. Further parameters and optimizations are welcome.  Asking here was suggested . I think this fits in both places, as it's as much about the algorithm as about gamedev. What are these fuse-looking cartridge case components on small, battery operated RF tranceivers? I have recently opened up 3 dead, battery-operated RF transceivers lying around, both wearable and key-chains. I've noticed that all three had components that looked like fuses (cartridge case), soldered down, that are quite large compared to the size of the circuit. See the pictures below. I'm not posting full circuit pictures because I don't know if I should, I can update if they are necessary.  Device 1:    Device 2:    Device 3:    I can't see why such a device will require a fuse, and I couldn't find any other common components manufactured in cartridge cases, maybe I'm not using the correct name. I don't see them on any applications notes or reference designs for the ICs used either.  I've tried to figure out how they are connected to the circuit, I don't have schematics and PCBs are more than 2 layers so they may not be complete, but see the schematics I've figured below.  Device 1:    Device 2:    Device 3:    Also, when I do a continuity test between the terminals of the components with a multimeter, resistance starts in ~50 Ohms range and goes down to short as I keep the probes on its terminals so my best guess is something for ESD protection, maybe Varistors or Thermistors?   Does anyone know what these components are and why they are there?