I'd appreciate if anyone could help me figure out what font Munkres used in his famous "Topology" book. I am interested in the font used in the main body as well as the font used in his "theorem" environment.  I searched the Internet for similar questions with no success. I also tried "myfonts.com," but that did not yield any satisfactory results. Finally, I looked at PDF version of his book and by going to "documents properties," I checked the fonts that the pdf viewer was able to detect. However, the results (mainly variations of Courier, Helvetica, and Times) do not look quite similar to what it really is.  The reason I am asking this question is that Munkres strikes me as an incredibly easy book to read and I would like to format my papers in a similar manner. As a example, his "f" in a theorem environment looks differently than his "f" in a math mode, which makes it extremely easy to distinguish between the two.  Below you can a few screenshots from his book.  Thank you for all your help.  EDIT: I understand that some of you may think it is a duplicate of  , but I tried methods mentioned in that post and it did not solve my problem. However, thankfully some of you have already answered my question, which I am very grateful for.
Is it possible to identify the font used in a specific document/picture?  Answers to this question should identify:   Possible methods to do this (perhaps one answer per method) and adequately describe how to use it (as opposed to merely stating it); Ways of finding the identified fonts, if possible (free or not); and Any prerequisites associated with the method used, if required (for example, "In order to use method X, your document has to be in format Y").   This question is meant as an FAQ, based on an . Its aim is to facilitate the community with the general procedures involved in font identification. Similar cases are solved on a per-usage basis on 's  tag.
How do I convert between big-endian and little-endian values in C++? How do I convert between big-endian and little-endian values in C++? For clarity, I have to translate binary data (double-precision floating point values and 32-bit and 64-bit integers) from one CPU architecture to another.  This doesn't involve networking, so ntoh() and similar functions won't work here.  Note: The answer I accepted applies directly to compilers I'm targeting (which is why I chose it).  However, there are other very good, more portable answers here. Crash course on algebraic geometry with view to applications in physics Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?   What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references. Is there a GLM bible? Is there consensus in the field of statistics that one book is the absolute best source and completely covering every aspect of GLM - detailing everything from estimation to inference? It's not difficult to show that  $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$  On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$.  Squaring the first power series and comparing terms gives us  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}2^{-2n}=1$$  that is,  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$$  My question: is there a more direct, combinatorial proof of this identity?  I've been racking my brains trying to come up with one but I'm not having much success. understanding the commutator of dihedral group Let  $G=D_{2n}=⟨x,y|x^2=y^n=e,  $ $yx=xy^{n-1}⟩$  I need to find $G'$ [ the commutator of G]  now I understand that $G'$ is the subgroup generated from $ U=xyx^{-1}y^{-1} , $ $ \ \forall x,y \in G$ So, $U=&lt;y^2&gt;$  whats the strategy  here ? Iterating each character in a string using Python In C++, I can iterate over an std::string like this:  std::string str = "Hello World!";  for (int i = 0; i &lt; str.length(); ++i) {     std::cout &lt;&lt; str[i] &lt;&lt; std::endl; }   How do I iterate over a string in Python? Title of short story: God granting immortality to others to find a way to kill himself I am running a fresh install of Linux Mint Nadia (14). I am following the instructions on Vagrant  but have gotten stuck on the . It seems the Vagrant box cannot connect outside and so I can't install anything using either Chef or Puppet.  In the basic Vagrant resolve.conf contains nameserver 10.0.2.3. But with that set I can't ping us.archive.ubuntu.com.  If I change it to 8.8.8.8 then I can ping us.archive.ubuntu.com but it does not stay set, and after a reboot it changes back to 10.0.2.3 - so provisioning fails again.  Ideally I would like for 10.0.2.3 to work on my setup. Failing that I would like a way to permanently change resolv.conf so that I can do provisioning. Why is the d'Alembert's Principle formulated in terms of virtual displacements rather than real displacements in time? Is there a way to do custom /say commands? I've looked it up, and apparently there is not a way to. But, there's an NPC in my map where I have it "talking" with a /say command. So, is there a way to make it say something like [(NPC Name)] (Text) without mods? Could I have a hint for testing the convergence of the following series please?  $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$  Edit  The integral test does not work because $\int_1^n\frac{1}{(\ln x)^{\ln x}}dx$ has not an elementary primitive.  Thank You. Backup with duplicity on Google Cloud Storage I have already set a backup routine that backups all my files to Amazon S3 using duplicity. I would like to create a similar backup routine with Google Cloud Storage. Is it possible to get duplicity to work with GCS? Good books for a high schooler self-studying Abstract Algebra? Does anyone have any suggestions for abstract algebra books particularly suited to self-study?     Here is some background and motivation, if it's helpful.  I'm currently a junior in high school, but I have some familiarity with groups, rings, polynomials and fields, as I went through Fraleigh's book. I liked its writing style, but many of the problems were not helpful. I would like to get better at algebra with some other book, particularly one that's aimed at someone who already has a little familiarity with the objects, but is by no means completely comfortable.  One of my longer-term goals is to learn a bit of algebraic number theory, as I've only been studying elementary number theory. That's why I feel it  good to learn more algebra first. (I'd also be interested in suggestions of other good subjects to learn prior to tackling algebraic number theory.)  I was considering using one of Lang's books, since they seem pretty universal, but I've heard rumors that they are very terse and not good for self-learning. How do we know that $F = ma$, not $F = k \cdot ma$ It seems intuitive that $a\; \propto \frac{F}{m}$, as the greater the force that is applied on an object, the greater its acceleration will be. Inversely, the greater the mass of the object, the slower the acceleration will be.  However, when rewriting proportions as equations, you must introduce a constant of proportionality, and in this case of a direct proportion, if $a \propto \frac{F}{m}$ then when rewriting as an equation you will have $$a = k\cdot\frac{F}{m}$$  In order to get the standard formula $F = ma$ this constant must be $1$. However, how do we know that this is the case? How do we know that the constant isn't $2$ and the formula $F = \frac{1}{2}ma$, for instance? How to disable Alt + ` (backtick/grave) CSV text files loading error / crash in QGIS 2.18.8 I got a problem when I try to load tab-separated text files in the "Create a layer from the delimited text file" tab (that one with de comma symbol). After normally selecting the X and Y fields and clicking OK, the reference coordinate system (SRC) selector screen appears, then the program closes by itself. I had uploaded more than a hundred of this type of file by the same procedure in the previous version of QGIS (2.14 Essen) and have no problem, but since I updated to version 2.18.8 (Las Palmas) I can not load any files or .CSV type.   Do anyone knows what's happening? Am I doing anything wrong? Convert 12-hour date/time to 24-hour date/time I have a tab delimited file where each record has a timestamp field in 12-hour format:     mm/dd/yyyy hh:mm:ss [AM|PM].   I need to quickly convert these fields to 24-hour time:     mm/dd/yyyy HH:mm:ss.   What would be the best way to do this? I'm running on a Windows platform, but I have access to sed, awk, perl, python, and tcl in addition to the usual Windows tools. How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$? How can I compute $\mathrm{Aut}(\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z})$?  If $\varphi$ is one of this automorphism then $\varphi((1,0))=(1,0),(1,3),(3,3),(3,0)$ and $\varphi((0,1))=(0,1),(0,5),(2,1),(2,5)$.  So it seems to me that this groups contains 16 elements, am i right? And if I'm right there is a way to understand to which group of cardinality 16 this is isomorphic to?  EDIT: $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}\cong\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$. And $\mathbb{Z}/8\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ are characteristic (am I right?), so $\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z})\cong\mathrm{Aut}(\mathbb{Z}/8\mathbb{Z})\times\mathrm{Aut}(\mathbb{Z}/3\mathbb{Z})\cong\mathbb{Z}/8\mathbb{Z}^*\times\mathbb{Z}/3\mathbb{Z}^*$ and this group has cardinality 8, right? Where is my mistake? Computing the Lie bracket on the Lie group $GL(n, \mathbb{R})$ Consider the Lie group $GL(n, \mathbb{R})$. Since $GL(n, \mathbb{R})$ is an open subset of the space $M_{n,n}(\mathbb{R})$ of $n \times n$ matrices, we can identify the tangent space (Lie algebra) $T_{e}GL(n, \mathbb{R})$ with $M_{n,n}(\mathbb{R})$, where $e$ is the identity matrix. I'm trying to understand the following:  For all $x, y \in M_{n,n}(\mathbb{R}) = T_{Id}GL(n, \mathbb{R})$ we have $[x, y] = xy -yx$ where $[\cdot, \cdot]$ is the Lie algebra bracket.  According to the question , this seems to be the case for matrix Lie groups in general.  So far, I've worked out the following: Let $x \in M_{n,n}(\mathbb{R}) = T_{Id}GL(n, \mathbb{R})$ and let $X \in \Gamma^{inv}(TG)$ be the left invariant vector field corresponding to $x$, i.e. with $X(e) = x$. By openness, we can choose $\epsilon &gt; 0$ such that the curve $(-\epsilon, \epsilon) \ni t \mapsto e + tx$ through the identity is contained in $GL(n, \mathbb{R})$. For $g \in GL(n, \mathbb{R})$ let $L_g$ be the left multiplication by $g$ in $GL(n, \mathbb{R})$. Using left invariance of $X$ calculate: $$X(g) = ((L_g)_{*}X)(g) = D_eL_g(X(e)) = D_eL_g(x) = \frac{d}{dt}|_{t=0}g(e+tx) = gx$$ Now, the above implies for $f \in C^{\infty}(GL(n, \mathbb{R}))$ and $g \in GL(n, \mathbb{R})$: $$X(g)(f) = (D_gf)(gx)$$ where we identify the tangent vector $X(g)$ with the corresponding derivation at $g$.  Now how do I calculate $[x,y] \stackrel{def.}{=} [X, Y](e) \stackrel{\text{WHY?}}{=}xy-yx$ from this? Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i &lt; j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to express $\Sigma$ as a polynomial in the elementary symmetric functions, for $n=2,3,4$. Case $n=2$ is trivial: $\Sigma(X_1,X_2)= \sigma_1 = X_1+X_2$. Case $n=3$ is easy. I write $\Sigma$ as follows: \begin{align*} \Sigma(X_1,X_2,X_3) = (\sigma_1 - X_1)(\sigma_1 - X_2)(\sigma_1 - X_3), \end{align*} then apply  formally, obtaining \begin{align*} \Sigma(X_1,X_2,X_3) = \sigma_1^3 - \sigma_1 \sigma_1^2 + \sigma_2 \sigma_1 - \sigma_3 = \sigma_2 \sigma_1 - \sigma_3. \end{align*} The case $n=4$ seems harder to me, and I'm unable to work it out. I hope there is some trick to avoid unnecessary computations. My attempt is to write down $\Sigma$ completely: \begin{align*} \Sigma = (X_1+X_2)(X_3+X_4)(X_1+X_4)(X_2+X_3)(X_2+X_4)(X_1+X_3) \end{align*} and then multiply factors in groups of two, obtaining: \begin{align*} \Sigma = (\sigma_2 - X_1X_2-X_3X_4)(\sigma_2 - X_1X_4 - X_2X_3)(\sigma_2 - X_2X_4-X_1X_3), \end{align*} but it doesn't seem to help much. How can I solve the exercise?