I have started studying free groups but cannot get motivation behind it. I know that a free group $F_A$ over the set $A$ consisting of all combinations or expressions that can be built from the members of $A$.  But why would one want to do such a construction? What is the motivation behind it
In , one learns two definitions of free groups associating with sets.        Let $A$ be a set, the free group of $A$, $F(A)$ is the initial object in the category $\mathcal{C}$, where \begin{equation} \operatorname{Obj}(\mathcal{C})=\{A\xrightarrow{g}G\}, \end{equation}where the codomain $G$ are groups, and \begin{equation} \operatorname{Hom}(\mathcal{C})=\{\text{Commutative Diagrams } A\xrightarrow{g_1}G_1\xrightarrow{\phi}G_2\xleftarrow{g_2}A\}  \end{equation} where $\phi$ are group homomorphisms.   Also $F(A)$ has the concrete construction with elements being non-redundant words with alphabet $A$, and the group multiplication being juxtaposition and reduction.      But what good are free groups? Why are they useful?  says they are useful in topology, but does not explain why explicity.   Can someone give some examples that an undergraduate can understand? I am asking about examples that can show the usefulness of this abstract construction. So I guess the identification of $\mathbb{Z}=F({a})$ does not really count.
rsync over ssh "error in protocol data stream" (code 12). ssh works I tried:  rsync -v -v -e 'ssh -p YY' ./testfile me@XXXXX:/home   Error Message  opening connection using: ssh -p YY -l me 146.6.84.206 rsync -&gt;-server -vvve.s . /home  [sender] make_file(testfile,*,0) send_file_list done send_files starting server_recv(2) starting pid=17537 rsync: connection unexpectedly closed (9 bytes received so far) [sender] rsync error: error in rsync protocol data stream (code 12) at io.c(600)[sender=3.0.6] _exit_cleanup(code=12, file=io.c, line=600): about to call exit(12)   This works fine:  ssh -p YY me@XXXXX   Which suggests to me the problem is neither that sshd is not running nor that port YY is firewalled. I have checked anyway.  What other problems could there be?  EDIT: The problem seems to have been "self resolving." I could not replicate the following day. I started my local computer up. Perhaps notably I had a different IP address than last time. And now rsync magically works. I'd appreciate guesses as to what it could have been in light of it going away. FontAwesome font not found? Sampling with replacement or without replacement I'm writing a program in R that simulates bank losses on car loans.  Here is the questions I'm trying to solve:  You run a bank that has a history of identifying potential homeowners that can be trusted to make payments. In fact, historically, in a given year, only 2% of your customers default. You want to use stochastic models to get an idea of what interest rates you should charge to guarantee a profit this upcoming year.  A. Your bank gives out 1,000 loans this year. Create a sampling model and use the function sample() to simulate the number of foreclosure in a year with the information that 2% of customers default. Also suppose your bank loses $120,000 on each foreclosure. Run the simulation for one year and report your loss.  B. Note that the loss you will incur is a random variable. Use Monte Carlo simulation to estimate the distribution of this random variable. Use summaries and visualization to describe your potential losses to your board of trustees.   C. The 1,000 loans you gave out were for 180,000. The way your bank can give out loans and not lose money is by charging an interest rate. If you charge an interest rate of, say, 2% you would earn 3,600 for each loan that doesn't foreclose. At what percentage should you set the interest rate so that your expected profit totals 100,000. Hint: Create a sampling model with expected value 100 so that when multiplied by the 1,000 loans you get an expectation of 100,000. Corroborate your answer with a Monte Carlo simulation.  I'm confused about how to set up this simulation up from a high level point of view and have the following questions: 1. For part A, Should I create a pool of 1000 customers or should I create a larger pool of customers? 2. For part A, when sampling, do I sample with or without replacement? 3. For part B, I'm confused about how to set up the monte carlo simulation.  Am I varying the size of the customer pool? 4. For part C, I'm not sure how to set up a sampling model that involves the interest rate.  Any advice or guidance would be appreciated.  I'm also thinking that if I fully understood the high level concepts for parts A and B, part C might not be such a mystery. If I insert a flash drive and try to drag any files onto desktop - nothing happens.  Or from any other folder.  Drag 'n copy only works if you open desktop in nautilus ("show desktop in files").  I already ran the command:  gsettings set org.gnome.nautilus.preferences use-experimental-views false   It didn't help. How to fix this? We are going to change the IP address of our SharePoint Servers can any one point out the RISK and IMPACT Programmatically open new pages on Tabs How to turn off magic quotes on shared hosting? I want to turn off PHP's magic quotes.  I don't have access to php.ini.  When I tried to add php_flag magic_quotes_gpc off to my .htaccess file, I get a 500 internal server error. This is what my .htaccess file looks like:  AddType x-mapp-php5 .php php_flag magic_quotes_gpc off   Then I tried to use ini_set('magic_quotes_gpc', 'O'), but that had no effect.  How do I turn magic quotes off? Need to develop a tapered spacer for under a car lift on an angled concrete slab I need to fabricate/develop an angled spacer.  13 inch wide by 10 inch long, by 3/4 inch to 1 3/8 inch thick. This spacer is needed to go under two of the posts of a hobbyist car lift in my garage. The garage floor is angled by the door where two of the four corner posts will go. These two rear posts need to be on a flat surface and at the same level as the two front posts.  Each post pad will need to handle approximately 2000 pounds. I was thinking of building a 4-sided mold where the column pads would be on the angled concrete and pouring some type of epoxy, resin, or cement into the mold. Which the metal plate from car lift posts would sit on. enter image description here What does `mv ./*` without specifying destination do? I accidentally forgot to specify destination before hitting the Return key. Where does mv ./* without specifying destination move the files and directories under current directory to? Strategy for "Which is better" questions In questions that are asking:    "Which is better option A or option B?"  It seems that several close reasons immediately come to mind:   Primarily Opinion Based.  In many cases the choice between A and B is simply a personal preference, a "6 vs. half dozen", either approach will work just as well and answers to this type of question will be almost entirely opinion based. Too Broad.  In some cases A is better and in others B is better, to go through every possible use case would lead to too many possible answers or one answer that reads like a short text book. Off-topic, Minimal Understanding, include attempted solutions.  This is perhaps the the most important one. If the OP needs to know "Which is better", they should probably have tried both and decided which is better, based on a practical trial.   Yet another option would be to downvote and/or comment with a "Try it and find out".  So...  Are there cases when these questions are permissible?   Could we add some more specific guidance in the help center?  Or should we continue to handle these case by case and just downvote/close appropriately when they pop up?  Strongly related, but hopefully not a duplicate of:   "Which is faster" isn't the only case where these pop up. ...on Finnis's view all distinct instances of basic goods are incommensurable — none is of more, less, or equal value with any other. &mdash;    I expect to see more or less than any other, but then equal shows up, and is followed by with. But since with any other cannot replace than any other, do you think this sentence illegitimately omits something? I've never been quite able to figure out how groups work in Blender (at least not since 2.5/6x). Is it possible to group several objects and then manipulate them (scale, rotate, translate) as one? How to repair Ubuntu installation after interrupted dist-upgrade without losing any custom settings? Travelling as an unmarried Western couple in Indonesia Set the location in iPhone Simulator How can I set the location (as it's picked up in CoreLocation services) in the iPhone Simulator? Duty Free in the USA. Why do I have to be given my purchase as I board the plane? I am not one for buying things in airports but recently my wife asked me to get her an alcoholic beverage while flying from the USA to Ireland (JFK-DUB). I made the purchase in JFK and was surprised to be told I had to collect it when departing for my flight as I board the plane. The sales assistant's English wasn't their best so I may have misheard and my Yorkshire twang may have thrown her, but I was somewhat surprised as to why I couldn't carry the bag myself to the plane.  Is there a law that prevents me from carrying my purchase to the plane or is this an added service I was charged, which I was duped into paying?  When boarding the plane there were several others who had to pick up their sealed duty free bags (why sealed?) so it wasn't just me. No explanation was given. What are "schlieren" in English? The German word for these patterns is "Schliere, pl. Schlieren", and apparently this word has entered the English language as a loanword (cf. article  in Wikipedia or entry  in OED). But the German loanword is probably not something most native speakers of English understand without the help of a dictionary, so what would you call, or how would you describe, the following patterns?  Concrete:    Air:    Oil on water:    Coffee and milk:    Paint:    Smoke:    Rock:    Marbled paper:    And so on.    "Waves" is not applicable, because waves are vertical undulations of the surface, while these are a pattern on, or inside, the surface level, with no amplitude.  "Striæ" does not really fit either, because those refer more to the parallel layers, than to the curly forms. SQL MAX of multiple columns? Why isn't the symmetric twin paradox a paradox? Two twin sisters synchronize their watches and simultaneously (from the earth frame) depart earth in different directions. Following a predetermined flight plan, each sister accelerates identically to 99.9%c and then returns home at the same time (again in the earth frame). The observers on earth see each twin as having aged identically, as the symmetry of the problem dictates. Each twin however should see the other as having aged... What gives? Circuitikz symbols in TikZ circuit library Some time ago, I asked about . wh1t3 wrote: "However, as far as I know there is nothing the circuitikz package does that the TikZ circuit library can't."  So if it is true that the tikz library is more powerful, how do I get all the symbols described in the  using the TikZ circuit library?  Especially I am missing the symbols for ammeters and voltmeters.