Why is the fundamental group of torus with finitely many points removed a bouquet of circles?
Fundamental group of a torus with points removed
Proof that any self-complementary graph has to have $4k$ or $4k+1$ vertex, for some $k \in \mathbb N$ What type of buildings offer what level of jobs? There are three tiers of workers (as is everything else); tier 1, 2 and 3. What I am having a hard time with is figuring out what I can build to target workers from each of the tiers. I gather what I am sure everyone else has, low tech industrial (regardless of density really) provides mostly tier 1 jobs.  But what are my other building options for tier 1 jobs? If I want to target tier 2 or 3 what types of buildings should I consider? Creating shapefile from current dataframe extents in layout view of ArcMap? Where is the ArcGIS 10 tool for creating a shapefile from current dataframe extents in layout view?  Have looked around and the closet thing i can find is Toolbox's Grid/Strip Map Index tools under Data Drive Pages.  I just want to be able to create a single polygon rectangle shp file based on the data frame (in layout view) for any given scale / page setup. Do I have to worry about copyright issues for code posted on Stack Overflow? I was wondering about the copyright of code snippets from answers. Let's say you ask a question and the solution consists of ten lines of code, which are given in an answer. Are there any restrictions about the use of that code?   It seems that the content of this page is under the . So does one have to attribute the author if code is used in an open source project? Can it be legally used in closed source projects at all?  I guess ten lines of code are not really problematic, but is there a critical number of lines? When is $\phi(n)$ a prime number? I was thinking about this problem today. If $p$ divides $n$, then $p-1$ must divide $\phi(n)$. But $p-1$ is even for all primes greater than $2$, so must $\phi(n) = 2$ - the only even prime? I feel like this argument is shaky and not very rigorous! How can I change what keys on my keyboard do? (How can I create custom keyboard commands/shortcuts?) Pre-1972 sci-fi short story or novel: alien(?) tunnel where people try new moves and get destroyed if they're not the correct ones What is the noun for someone who receives a referral? When a referrer (noun) gives a referral (noun) to another person, what is the term (noun) used for the recipient of the referral? "The referred" may work, but am I missing another term? "Referred" indicates past-tense, but I intend to use the term in future and present tense as well. Limit with inverse trignometry We have to find the value of $$\lim_{n\to \infty}\sum_{k=2}^n\arccos\left(\frac{1+\sqrt{(k-1)k(k+1)(k+2)}}{k(k+1)}\right)$$  I tried it a lot but could not find any start.   Please help me in this . We have a Barbarian on our team that very much likes to switch up his weapons; he goes from a Dwarven Hammer, to a pair of Hand-Axes, to occasionally just going bare-fisted. No issues there, he's fairly adept with all 3.  However, I have only just recently picked up on the way that he rolls his attacks for the two Hand-Axes: He rolls the first attack, adds his modifiers etc. Then he rolls the second attack, this time without modifiers, which is all well and good. But I'm not sure if this action uses his bonus action or not, or is considered a separate action (though, being a level 4 Barbarian, he only has one action per turn), or is merely part of the same action.  If we have a look at the Two-Weapon Fighting feature from the Fighter's fighting styles:     Two-Weapon Fighting   When you engage in two-weapon fighting, you can add your ability modifier to the damage of the second attack. (PHB5e p.72)   Similarly, the Dual Wielder feat also uses the Two-Weapon Fighting feat:     Dual Wielder   You master fighting with two weapons, gaining the following benefits:      ...         You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light. (PHB5e p.164, emphasis mine)      So; yes, this gives a bonus to the second weapon attack, but having asked a  to clarify which action this secondary weapon attack belongs to, it was determined that this uses your bonus action;     [...] the second attack requires a bonus action   So after that long winded explanation, here is the question: Without any additional feats, can you wield and use two weapons at once, and if so, how many actions does it require to make a full attack with both weapons in a turn? How do we know the brain flips images projected on the retina back around? Why do we turn images upside down again rather than dealing with them directly, still vertically rotated after passing through our lens?   I don't see how that would cause any problems, and how we'd ever be able to figure out if we are presented with flipped images after getting used to interacting based on visual input, whether flipped or not.   What am I missing? How to derive the conjugate prior of an exponential family distribution I am trying to derive the conjugate prior of the univariate Gaussian distribution over both the mean and the precision. I know that the prior I'm looking for is the normal-gamma distribution, but the idea is to derive this result.  It seems that you can write down the conjugate prior of an exponential family distribution immediately if you represent it in the canonical exponential family form.  suggests a way to do just that. So far I have written the Gaussian probability function in this form and equated the following, but I don't see how the form of the conjugate prior is immediately obvious.  $$p(\mu, \lambda | D) \propto p(\mu,\lambda)p(D | \mu, \lambda)$$ Foreign key referring to primary keys across multiple tables? I have to two tables namely employees_ce and employees_sn under the database employees.   They both have their respective unique primary key columns.   I have another table called deductions, whose foreign key column I want to reference to primary keys of employees_ce as well as employees_sn. Is this possible?  for example  employees_ce -------------- empid   name khce1   prince  employees_sn ---------------- empid   name khsn1   princess   so is this possible?  deductions -------------- id      name khce1   gold khsn1   silver Adding x and y values while creating a label QGIS I'm trying to create a label on a map in QGIS.  I can open the setting where there are an X and Y button and a few other properties, like opacity and distance.  I'm pretty sure the problem is that I need to enter the proper X and Y coordinate here, but I don't know how to get to where it lets you enter the coordinates. Tools for automating document compilation A lot of people write makefiles that say something like  paper.pdf: paper.tex         pdflatex paper         bibtex paper         pdflatex paper         pdflatex paper   To handle re-running TeX to get new/changed references and so forth. Is there a better way to do this? Having recently reviewed a few suggested edits, I've found that not being able to see the comments on the post is hindering my understanding of the context of some edits.  In particular, there have been a few posts where someone has made a substantial edit to a question or answer which significantly alters its meaning. Typically I'd reject that as "too radical", since it can't be known that the new meaning of the post was what the original author intended.  However, in some cases it is clear from the comments that the original author is aware of the intended change, and as such the editor is not in fact misrepresenting the author's intentions. In these cases, it would have been useful to see the comments when making the review decision.  As an example, a common cause of this is when an inexperienced user is asked to clarify their question, and ends up writing a comment that would be better off as an edit to the question (due to length, excess code content, etc), presumably due to lack of awareness that the question can be edited. Often, other users will helpfully edit the question to include the information the OP put in the comment (see ,  and  for examples). In this situation, it is clear that the edit reflects the OP's intentions, but this information is only present in the comments which are not visible in the review interface.  Following is the suggested edit screen showing the suggested edit. It seems indentation is increased and one line is moved a few lines upwards and some empty lines added to increase readability. But it look like an invalid edit since changing the related line may cause alteration on the runtime. So not looks lie a valid edit. Better open the answer and examine it        Checking the Q&amp;A itself reveals that a talk had taken place between the OP and the answerer and revision done by the OP with the permission of the answerer. So it was indeed a valid edit.        In short: is there any particular reason comments are not displayed in the suggested edit review interface? I think it would add useful contextual information when making review decisions, and presumably this would be trivial to implement.  Note: this is not a dupe of , as it does not concern comments posted after the suggested edit. I understood the very basic concept of how to use /etc/network/interfaces, but all I find online are examples, example after example, which I can copy-paste from. What I miss is an explanation of the syntax, an explanation of the meaning of the commands and which order the commands require. I want to understand, because most of the time copy-paste is not enough, because I'm not working on a fresh machine, so I can't just overwrite existing configurations because it would break a lot of stuff. man interfaces was not very helpful since it is written very complicated.  Example questions I have: what does inet in an iface line mean exactly (I could not even find it in the manpage), what does manual in an iface line mean exactly (many examples use it, but according to manpage it needs an extra config file then, which the examples don't present), when do I use or need them? When not? When I create a bridge, what exactly happens to the interfaces? I'm using GNU Emacs from  After Migration to OSX 10.11 "El Capitan", Auctex can't run the *tex utilities anymore:  Running `LaTeX' on `poisson' with ``pdflatex  -file-line-error   -interaction=nonstopmode "\input" poisson.tex'' /bin/sh: pdflatex: command not found   There is no problem running for example pdflatex on the shell:  #pdflatex This is pdfTeX, Version 3.14159265-2.6-1.40.16 (TeX Live 2015) (preloaded format=pdflatex)  restricted \write18 enabled.   but it seems that auctex calls /bin/sh pdflatex instead, and that does not work anymore:  /bin/sh pdflatex /Library/TeX/texbin/pdflatex: /Library/TeX/texbin/pdflatex: cannot execute binary file   maybe due to El Capitan's changes to Gatekeeper sandboxing.  How can I configure auctex to run pdflatex again? Existence of irreducible polynomials over finite field Let $F$ be a finite field. How do we prove that for each $n \in \mathbb{N}$ there is an irreducible polynomial of degree $n$?  One can assume that $F = \mathbb{F}_{p^m}$ where $p$ is prime. If $n \ge |F|$ then I can construct an irreducible polynomial, namely  $ p(x) = 1 + \prod_{j=1}^{|F|} ( x - a_j )$  where $a_j$ are all the field elements. It is clear that $p(x)$ has no roots in $F$.  This trick doesn't work for $n &lt; |F|$. A counter-example: Let $F = \mathbb{F}_3$ and $p(x) = 1 + (x-1)(x-2)$, then $p(1) = 1$, $p(2) = 1$, $p(0) = 1 + (-1)(-2) = 1 + 2 = 3 = 0 \pmod 3$.  I know there is a way to count them using the Möbius function $\mu(n)$ but I want a proof without it that just shows existence. Setting READ UNCOMMITTED when reading DMVs I've seen several people call SET TRANSACTION ISOLATION LEVEL READ UNCOMMITTED before reading system DMVs. Is there ever any reason to do this, assuming you aren't mixing calls to DMVs and tables in the same transaction?