For what reasons would or wouldn't one want 0 to be the start of the natural numbers as opposed to 1? Why would one want it to be 1, or why wouldn't one?
Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?  It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.
Why is fundamental group insufficient to classify manifolds? To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$ \numcases with \tag I would like to label some cases with specific tags, and was hoping for something like  \begin{numcases}{A = } \frac{r}{a} &amp; r &gt; a \tag{G} \\ \frac{a}{r} &amp; r &lt; a \tag{H} \end{numcases}   utilizing the numcasesenvironment from the cases package, and \tag from amsmath. However, the two don't seem to be compatible:     ! Package amsmath Error: \tag not allowed here.   Is there any alternatives that would do what I hope to accomplish? Log of products to sum of logs How do I update Android on my device? How to develop intuition for conditional probability? Usage of :- (colon dash) in bash Finding the asymptotic of an integral I came across the following exercise on asymptotic behavior of integrals: $$I(a) = \int_0^\infty\frac{\cos x}{x^a} \, dx, \text{ where } a\to0^+.$$ I have tried integrating by parts or replacing $\cos x$ with the first summands of its Taylor series, but I end up with something equal to infinity (independent of $a$ and that is not what we want).  I just thought about the substitution $x \leftrightarrow\frac{1}{1+x^2}$ and I get the  following result:  $$I(a) = \frac{\cos(1/2)}{2^{2-a}} + \int_0^1\frac{2x^3 \cos\left(\frac{1}{1+x^2}\right)}{(1+x^2)^{3-a}} \, dx - \int_0^1 \frac{2x^3\sin\left(\frac{1}{1+x^2}\right)}{(1+x^2)^{2-2a}}\,dx$$ If I could prove that the last summands are $o(I(a))$ then it would be OK, but I think, in general my strategy won't work. Apparently this is supposed to be an easy exercise, but I can't come up with anything right now. limsup of intersection of events as a subset of intersection of limsups Berlin Public Transport Ticket Validity I have purchased an AB zone Berlin 7 day transport ticket and validated it on Sunday 17th April so hopefully it will be valid until Sunday 24th.  I validated the ticket at about 10:00am so I guess 1000 is the time I validated? But what are the numbers 15 and 6 and letters 'So' correspond to? How (if checked by a guard) can they tell if the ticket is valid? And how can I as a customer tell when it is validated to? Why do I give my pets "food" but my livestock "feed"? When I feed my cat or my dog, the package tends to say "Dog Food" or "Cat Food."  In contrast, I give my chickens "chicken feed" or "poultry feed."  Likewise, a cow's silage is her "feed."  Why does this distinction exist?  Is there an historical or grammatical explanation, or is it merely idiom? How do I change the default wifi network that Ubuntu connects to on startup? There is an unsecured public wifi network in my range, as well as my own home WPA secured wifi that I want to use by default. However, on startup Ubuntu always connects to the unsecured one. Sometimes I forget to reconnect to the other one, which is very dangerous.   I am running a default setup of Ubuntu Destkop 11.10 with Gnome 3. I have looked under network settings but I can't find a 'default' option. Solve $y’ = \frac{2-xy^3}{3x^2y^2}$ using Bernoulli form Solve, by bringing the equation to Bernoulli form: $$ y’ = \frac{2-xy^3}{3x^2y^2} $$  Therefore we want to bring it to a form like: $$ y’ + p(x)y = q(x)y^n $$ So working with the equation i get: $$ y’ - \frac{2}{3x^2y^2} = -\frac{xy^3}{3x^2y^2} = -\frac{y}{3x} $$ I don’t see how to get to Bernoulli equation from here... Does magnetic monopole violate $U(1)$ gauge symmetry? Does a magnetic monopole violate $U(1)$ gauge symmetry? In what sense and why? Insofar as I know, there are at least two types of magnetic monopoles. One is the Dirac monopole while the other is the monopole in Grand Unified Theories (GUT), e.g., 't Hooft-Polyakov monopole.  In the latter case, some non-Abelian gauge theory breaks down to a (compact) $U(1)$ gauge theory wherein monopoles can be observed from a large distance. The monopole field is not from the $U(1)$ degrees of freedom and nothing is singular. On the contrary, Dirac monopole is singular (topological defect?) and we must introduce Dirac string or patches of vector potential $\vec{A}(x)$. Are these no more than artifacts arising when one tries to describe the monopole field within the $U(1)$ gauge theory by assuming $\vec{B}=\nabla\times\vec{A}$? It seems true since two Maxwell's equations that magnetic monopole might affect can be cast in the differential form $\mathrm{d}F=0$ and $F=\mathrm{d}A$ in Minkowski spacetime, from which $U(1)$ gauge invariance manifests for $\mathrm{d}(A+\mathrm{d}\chi)=\mathrm{d}A$.  However, I later noticed two seminal papers on Dirac monopole  &amp; , in which the authors claimed the following   electromagnetism without monopole $\rightarrow$ connection on a trivial $U(1)$ bundle electromagnetism with monopole $\rightarrow$ connection on a nontrivial $U(1)$ bundle And the wave function of an electron around a Dirac monopole should be regarded as a section which is free from discontinuities.   I've no idea about fibre bundle. Anyway I guess $U(1)$ theory does not necessarily expel Dirac monopoles. It has something to do with the topology of some manifold in the theory? Which manifold? Can anyone shed light on this intriguing issue? Thanks in advance. This is the first part of the first half of the a proof in the Axiom of Choice. The proof that the Axiom of Choice is equivalent to the choice function. REsources: Keith Devlin's Joy of Sets: pg. 58. F.R. Drake and Singh Intermediate Set Theory: pg. 110).   I am confused on proving disjointness and will ask further questions of proof once I get this first part v. clear for myself.   AC1: Let $\mathcal F$ be a family of pairwise disjoint, non-empty sets. Then there is a set $\mathcal M$ that consists of precised one element from each member of $\mathcal F$  AC2: For every family $\mathcal F$ of non-empty sets, there is a choice function $\mathcal f$. (I.e. $\mathcal f(X)$ is a member of X, for each X in $\mathcal F$.   Disjointness: Two sets are disjoin if Q $\cap$ R = $\emptyset$. Meaning that they have zero elements in common. These sets do not overlap.   Show: AC1 $\leftrightarrow$ AC2   (1) ($\rightarrow$) Let $\mathcal F$ be a set of nonempty sets.   (2)For each X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X}) = {(x, X) | x $\in$ X})  (3) Show Disjointness.  An outline of how book proves  disjointness:  (a) if u = {(X $\times$ {X}) | x $\in$ X} then u is of the form $\lt  z, t \gt$ where z = X and t=x, i.e. x $\in$ X.   (b) then if u $\in$ ({X} $\times$ x) $\cap$ ({Y} $\times$ Y) $\rightarrow$ u = $\lt z, t \gt$ with z = X and z=Y, i.e. X=Y. Which means ({X} $\times$ x) = ({Y} $\times$ y).  Questions: (1) Apparently we cannot prove disjointness in $\mathcal F$ but must prove in a different set. If so, why is that?  (2) why is the case that if u $\in$ ({X} $\times$ X) $\cap$ ({Y} $\times$ Y) then z =X and z=Y? I am not following that step at all.   (3) why does the axiom of choice have to do with ordered pairs? Why are we setting X $\in$ $\mathcal F$, let $X^*$ = (X $\times$ {X})?    Thank you! If you go to  and are a 4k+ user on Anime.SE, you will see the following:        There is one reopen vote on this question, but the reopen vote count is displayed on a different line than the word "reopen". This is slightly silly, and would perhaps benefit from converting reopen (1) to reopen&amp;nbsp;(1), and likewise for close and delete votes. (Or any other way of preventing a line break between the vote type and the vote count.)  A repro (the close link) from here on Meta: I'm really new in using biblatex, compile the sample placed in , but what I get is    1.1 Foo    Some text [B02x ].    1.2 Bar    Some text [B02y ].   Do not get the bibliography by chapters, or the global references.  Really not remains to be done   \documentclass{report}  \usepackage[style=alphabetic]{biblatex}  \addbibresource{references.bib}  \begin{document}  \chapter{First chapter}  \section{Foo}  Some text \cite{childs_temperature}.  \printbibliography  \chapter{Second chapter}  \section{Bar}  Some text \cite{hashemian}.  \printbibliography  \newpage  \printbibliography  \end{document}   And the references.bib    @BOOK{childs_temperature,   title = {Practical Temperature Measurement},   publisher = {Butterworth - Heinemann},   year = {2001},   author = {Childs, Peter R N},   address = {Great Britain},   edition = {1},   isbn = {0 7506 5080 X}   }    @PHDTHESIS{hashemian,   author = {Hashemian, Hashem Mehrdad},   title = {Measurements of dynamic temperatures and pressures in nuclear power plants},   school = {{The University of Western Ontario}},   year = {2011},   type = {PhD {T}hesis}   }   I need to obtain references by chapter and global cited references, but i don't know how compile. A program I am using spits out LaTeX tables in LaTeX files (outreg2 command from Stata), which I run using texify to make pdfs of the tables.  I crop these tables (using pdfcrop) and embed them elsewhere.  However, the pdfs include page numbers, which prevent pdfcrop from cropping directly around the table.  I need an automatic way to remove the page numbers before cropping.  I want to insert the command \pagenumbers{gobble} into the LaTeX code to automatically remove the page numbers.    I believe   texify --pdf --texinfo="\pagenumbering{gobble}" filename.tex   should work.  It does not. No exception pops up, but the page number remains. I cannot find documentation anywhere showing examples of using the --texinfo option to splice LaTeX code into the file before running the LaTeX file to create a pdf.   How should I properly use the texify command?  Or is there another way to splice in  \pagenumbering{gobble} before running texify? Does commutativity imply Associativity? What do I need to get photos with a unifom black background (not with post)? Often i see portraits such as the one below and like the way they look. I wanted to know can I get this look in my shots.  I am sure there are professional materials you can get to outfit a studio that will deliver such results. I have no clue what these items would be, can i get clued in?  Also, I was wondering if there is a DIY way to achieve this as well. Maybe with a black bed sheet or something?  I want to set something up in a room in my house for studio type shots. I am looking to get an idea of what items I need to get this done and how the setup should go.