Suppose $M$ is a closed $k-$manifold in $\mathbb R^n$ without boundary, can we always find a smooth function $f:\mathbb R^n\to\mathbb R^{n-k}$ such that $M$ is the level set where $f=0$?
It's a well-known theorem (Corollary 8.10 in Lee Smooth) that given a smooth map of manifolds  $\phi:M\rightarrow N$ and a regular value $p\in N$ of $\phi$, the level set $\phi^{-1}(p)\subset M$ is a closed embedded submanifold. Is the converse true? That is, given an embedded submanifold $S\subset M$, is there necessarily a manifold $N$, smooth map $\phi:M\rightarrow N$, and regular value $p\in N$ of $\phi$ such that $S=\phi^{-1}(p)$?   Prop. 8.12 in Lee Smooth shows that this is true locally; specifically,     Let $S$ be a subset of a smooth $n$-manifold $M$. Then $S$ is an embedded $k$-submanifold of $M$ if and only if every point $p\in S$ has a neighborhood $U\subset M$ such that $U\cap S$ is a level set of a submersion $\phi:U\rightarrow\mathbb{R}^{n-k}$.   (and any level set of a submersion is of course the level set of a regular value). I feel like this is the kind of question where, if there is a counterexample, it probably is very simple, but I wasn't able to come up with one.
Does Thanos's choice make any sense? So we know from Infinity War that Thanos was collecting all the Infinity Stones to basically solve the problems brought about by overpopulation on many planets.  But does it make any sense? He could've just easily increased the resources or changed the area or taken appropriate measure to handle that situation. As far as I know, collecting all the Infinity Stones gives the user limitless power and control over the universe in the MCU.  Is Thanos just a mass murderer or is there something I'm missing? How can I fix "unbalanced brackets" in my data tag? Often, when making a command block contraption, you need to set an entity's data tag. These tags can sometimes be very long, with many layers of nested curly and square brackets. For example, this is the command to summon a villager with a single custom trade:  /summon Villager ~ ~ ~ {Profession:3,Career:2,Offers:{Recipes:[{buy:{id:minecraft:diamond,Count:6b}},sell:{id:minecraft:diamond_hoe,tag:{ench:[{id:16s,lvl:10s}]}]}}   The single-line command block interface of Minecraft makes it incredibly difficult to find and fix errors, especially when it comes to mismatched brackets. In fact, for illustration purposes, I put an error in the above command.  What techniques can I use to find and fix unbalanced square or curly brackets when writing long and complicated data tags? If the center of a group $G$ is of index $n$, prove that every conjugacy class has at most $n$ elements. (This question is from Dummit and Foote, page 130, 3rd edition.)  Here is my attempt: we have  $|G| =|C_G (g_i)| | G : C_G (g_i)| $  $|C_G (g_i)| =|Z(G)| \cdot  |C_G (g_i):Z(G)| $.  Then   $|G|= |Z(G)|\cdot |C_G (g_i):Z(G)|\cdot | G : C_G (g_i)| $.  Then  $n = |C_G (g_i):Z(G)|\cdot  | G : C_G (g_i)| $.  But $|C_G (g_i):Z(G)|$ is positive integar as  $ Z(G)$ is   subgroup of  $C_G (g_i)$.  If $| G : C_G (g_i)|$ bigger than $n$, then $|C_G (g_i):Z(G)|\cdot | G : C_G (g_i)|$ is bigger than $n $.  But   $|C_G (g_i):Z(G)|\cdot | G : C_G (g_i)|=n$, contradiction.  Then $| G : C_G (g_i)|\le n$ and this completes the proof.   Is this proof right or not ??!!! Dear all I think I have a problem with the collation of makeidx.  following some suggestions found here I have got in the text something like this  \index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00855 @855A|qt}  \index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00868 @868f-869b|nn{\thefootnote}}   the first is an \index entry in the text, the second in a footnote. My problem is that even if I put the sort also on the second level of indexing it keeps keeping them separated.  I use this package information  \usepackage[splitindex]{imakeidx} \makeindex[name=n, title=Nomi di persone e luoghi]  \makeindex[name=p, title=Passi citati nel testo] \makeindex[name=t, title=Altri testi citati] \makeindex[name=pap, title=Papiri e manoscritti citati nel testo]  \newcommand{\nn}[2]{#2 \textit{n}#1} \newcommand{\qt}[1]{\textbf{#1}}   in the .idx file I get this  \indexentry{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00855 @855A|hyperindexformat{\qt}}{8}  \indexentry{Plutarco!demalignitate@\textit  {De Malignitate Herodoti}!00868 @868 b-c|hyperindexformat{\nn{33}}}{10}   can somebody advise me on what I am doing wrong here, please? I have tried to use \string after the first sort with no effect.    sure, sorry: this is the file I'm using  \documentclass[12pt,a4paper]{book} \usepackage[utf8x]{inputenx}  \usepackage{palatino}        \usepackage[LGR,T1]{fontenc} \usepackage[english,german,polutonikogreek,italian]{babel}  \usepackage[splitindex]{imakeidx} \makeindex[name=p, title=Passi citati nel testo,columns=1] \newcommand{\nn}[2]{#2 \textit{n}#1} \newcommand{\qt}[1]{\textbf{#1}} \usepackage[hidelinks]{hyperref} \begin{document}  (855A\index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00855 @855A|qt})% \footnote{DHM 868 B-C\index[p]{Plutarco!demalignitate@\textit{De Malignitate Herodoti}!00868 @868 b-c|nn{\thefootnote}}} \printindex[p] \end{document} Finding the sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$ Deteremine the sum of the series $$\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$$  So I first write down the $n^{th}$ term $a_n=\frac{\frac{n(n+1)}{2}}{n!}=\frac{n+1}{2(n-1)!}$.  So from there I can write the series as $$1+\frac{3}{2}+\frac{4}{2\times 2!}+\ldots +\frac{n+1}{2(n-1)!}+\ldots $$  I am quite sure I can do some sort of term by term integration or differentiation of some standard power series and crack this. Any leads? How to create a new partition from the existing partition? I wanna create a new partition from the existing root partition.My root partition is about 250GB.I wanna create a new partition say 150 GB taking space from this root partition.Can anyone help me with the procedure on doing this?  output: fdisk -l  karthick@karthick:~$ sudo fdisk -l [sudo] password for karthick:   Disk /dev/sda: 250.0 GB, 250000000000 bytes 255 heads, 63 sectors/track, 30394 cylinders Units = cylinders of 16065 * 512 = 8225280 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x28219744  Device Boot      Start         End      Blocks   Id  System /dev/sda1   *           1          61      487424   83  Linux Partition 1 does not end on cylinder boundary. /dev/sda2              61       30395   243650561    5  Extended /dev/sda5              61         124      498688   82  Linux swap / Solaris /dev/sda6             124         186      498688   82  Linux swap / Solaris /dev/sda7             186       30395   242651136   83  Linux Recover accidentally removed files on NTFS partition I accidentally removed some files on a NTFS partition  How can I recover them staying inside Linux without rebooting ? New global preferences: keyboard shortcuts and sticky topbar Showing that $\lnot Q \lor (\lnot Q \land R) = \lnot Q$ without a truth table I've done a truth table after reducing it to this and it seems to be equal to $\neg Q$:  $$\lnot Q \lor (\lnot Q \land R) = \lnot Q$$  But when I try to show it without a truth table (with just transformations), I end up in a loop between that and:  $$\lnot Q \land (\lnot Q \lor R)$$  Is there a way to show this is true without using a truth table? What am I missing?!  Thanks in advance! Is it possible to use the laptop's audio via headphone jack and Bluetooth speakers at the same time? Lolo Joe wants earphone jack to plug and use big speakers but at the same time he wants to partner the music coming out the earphone jack and his Bluetooth system and speakers. Is it even possible to use the earplug connection and the Bluetooth system and devices at the same time? How to check if a column exists in a SQL Server table? Word for someone that always has to be right Is there a word for someone that always has to be right? The person gets angry if they are not. I am asked to solve the following question using central limit theorem. In an election between two candidates, A and B, one million individuals cast theirvote. Among these, 2000 know candidate A from her election campaign and vote unanimously for her.The remaining 998000 voters are undecided and make their decision independently of each other byflipping a fair coin.Approximate the probability pA that candidate A wins up to 3 significant figures.  It's easy to solve directly. PA = $\frac{0.5(10000-2000)+2000}{10000}$ = 0.501. However, I am quite confused about how to solve this problem by central limit theorem. I'm having trouble solving a Fitch Style Proof and I was hoping someone would be able to help me.  Premises: p ∧ q  Goal: q ∨ r  Would be an assumption (p ∧ q => q ∨ r) in first step a correct one?   Then, q ∨ r Implication Elimination: 2, 1  Why is it not correct?  Thank you very much for any help! When I try to connect to Google Play, it would say "download Google play service". After I downloaded it, it says that it is "not supported by your device, please contact the manufacturer". Evaluating $\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx$ using complex analysis I'm taking a complex analysis course and doing some practice computing residues &amp; evaluating integrals. I pulled out an old book called     On page 196-197, there are some interesting integrals to evaluate. I'm at 5.4.3.10.: I was able to do question 1, but was stumped at how to even begin with question 2:     Evaluate the integral   $$\int_0^{2 \pi} e^{\cos x} \cos (nx - \sin x) \,dx ,$$   where $n$ is an natural number.   The answer is simply $\frac{2 \pi}{n}$. Any hints? If you willingly fall on your turn does it cost movement?  As a DM I'm thinking about how to handle things such as creatures who are immune to being prone, diving into the water from a height, or a flying creature diving off a cliff and taking a fall before taking flight. There are many circumstances where you can fall without taking damage or being knocked prone afterward, and I'm not sure how to determine movement following the dive.  For example, my character walks up to a ledge above a 10 foot drop to the ground below. After taking 5 feet of movement to step off the ledge, I allow myself to fall. When I hit the ground have I used 15 feet of movement or just 5? Each event has two possible outcomes: a and b.  Each event is independent.  Outcome a has $X\%$ chance of happening.  If we run $Y$ number of such events, what are the chances of getting $K$ outcomes a in a row, at least once?  To put it in practical terms imagine a series of coin tosses. Each coin toss has two possible outcomes: a = heads and b = tails. Outcome a = heads has a chance of $50\%$. If we run $Y = 100$ number of coin tosses, what are the chances of getting $K = 10$ outcomes a=heads in a row, at least once?   I'm using the practical example only so I can express the idea more clearly. I'm interested in finding out the general formulas and computations so that I can run the calculations for any $X$, $Y$, or $K$.  I would also be very interested in knowing how the probabilities would change if the events would not be independent. What if the more outcomes a we have in a row the bigger the chance of getting an outcome a in the next event? What if it's the other way around, and the more outcomes a in a row we have the lower the chance of getting another a? Run python script when a file has been added to a folder? Is it possible to run a Python script when a file has been added to a folder?  In my scenario I have a folder that contains movies and the Python script gets the details (name, year, plot, rating, etc) from OMDb and puts them into a file, which MySQL uses.   Every time I add a movie I have to run the Python script.  I want to know if it is possible to automatically run the Python script when a file has been added to the folder?  Thanks in advance. SF short story ID, written 40s—60s, primitive aliens who are much smarter than us I'm looking for a story that was probably written somewhere from the 1940s to the 1960s. I think I read it in an anthology, possibly in the 1970s.  A group of human explorers landed on a planet with primitive aliens, living in hunter-gatherer villages. They hoped to make a profit from trading our goods for—well, I'm not clear on what they wanted from the aliens.  The native people learned the human language before the explorers could learn theirs. Then the natives suddenly started producing technological innovations far beyond what they originally used. I remember that wooden sailing ships suddenly replaced their canoes. All the natives had to do was hear a story once from the explorers before copying the technology in the story.  Finally, the explorers realized that the natives were much, much smarter than us and that Earth faced a serious problem.  I'm fairly certain that it wasn't a Heinlein story, but I remember that it had some qualities similar to his work.