Trying to understand the very basics of category theory... While trying to solve a simple exercise on the subject I suddenly realized that I don't even know what morphism equality is! (The exercise asked to prove that such-and-such morphism composition in such-and-such category is well-defined - but to prove this I had to prove that it is associative, which requires me to prove equality of the sort $f;(g;h)=(f;g);h$ - which gave raise to this question)  On the one hand, many proofs in category theory consist of diagrams of some sort where it is shown that $f;g=h;j$ because a square whose edges are morphisms $f:A\rightarrow B$, $g:B\rightarrow D$, $h:A\rightarrow C$ and $j:C\rightarrow D$ 'commutes':    So it would seem that two morphisms are equal iff they lead from the same source to the same destination?  On the other hand, this would mean that there may be at most one morphism in one direction between two given objects - but this seems not the case.  I looked at Wikipedia article but could not find morphism equality defined.  Sorry if this is an extremely basic question - but when are two morphisms equal?
I am learning basics of the category theory (CT).  I do understand that CT is a modern powerful framework to describe various branches of mathematics in a unified way.  I do admit that category's definition tells a lot about the whole thing and focuses on the most important properties, shadowing unnecessary details. However, the more I go forward, the more it feels like it should explicitly require a notion of morphisms equality (like it does with morphism composition). It feels so because equality or inequality of morphisms arises everywhere, since the very beginning: any commutative diagram eventually reduces to an equality of two different paths and many other things involve equality as well. Even composition is defined grounding upon an (implicit) equality!  I'll try to give an example of my concern. Consider $\mathbb{SET}$. Its objects are sets and morphisms are functions. Now, let's recall that functions are sets as well - to be more precise, a function $f : A \mapsto B$ is a subset of a $A \times B$ obeying certain properties, which I omit. Thus, one could argue that $f$ is both a morphism between objects $A, B \in Obj(\mathbb{SET})$ and an object defined as a set of pairs $\{ (a_1 \in A, b_1 \in B), ..., (a_n \in A, b_n \in B) \}$ and those are equal because both are essentially exactly the same thing.  I hope my concern is clear and I need some kind of "conceptual insight" from knowledgeable people to move forward. Please give me the right perspective.
Google analytics: Tracking Not Installed What is the best way to include R code in the Latex paragraph I am a newbie in Latex. I have two simple questions regarding including R code in Latex.  Q1: For a block of R code, I use 'listings' package, as follows.  \documentclass[12pt]{report} \usepackage{listings} \begin{document} \begin{lstlisting}[language=R] &gt; dpareto(4, 2, 3) [1] 0.09375 &gt; dpareto(4, -2, 3) [1] NaN &gt; dpareto(seq(-1, 4, 1), 2, 3) [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.2962963 0.0937500 \end{lstlisting} \end{document}   I feel it's not professional. What is the best way to present the block of R code?  Q2: For code pieces in the paragraph, for example I wish to include a R function, or variable, or package name in my Latex context, what is the best way to do that? So to help prevent brute force attacks against my phpmyadmin install I've configured nginx to require HTTP Basic Auth (extra username and password) and then the requests need to come from a whitelisted IP Address. However the IP Address whitelist works fine if I run  however without adding the extra .php block I could load  without my IP being whitelisted. This issue doesn't affect the HTTP Basic Auth. Is there a reason this happens? Anyway to get around having to add extra .php blocks? A copy of the config area for phpmyadmin is below.  # Setup and secure phpMyAdmin location /phpmyadmin/ {   allow 1.2.3.4;   deny all;   auth_basic "phpMyAdmin - HTTP Basic Login";       auth_basic_user_file /etc/nginx/pma_pass;     location ~ \.php$ {     include snippets/fastcgi-php.conf;             fastcgi_pass unix:/run/php/php7.0-fpm.sock;     allow 1.2.3.4;     deny all;     } } Does priority matter in Magic? Find $P(n+1)$ for a polynomial P Fix a nonnegative integer $n$. Let $P(x)$ be a polynomial of degree $n$ (over the real numbers) such that for all $k\in\left\{0,1,\ldots,n\right\}$, we have $P(k)=\dfrac{k}{k+1}$. Find $P(n+1)$.   My attempt:  I consider a new polynomial $Q(x)=(x+1)P(x)-x$. This $Q$ is a polynomial of degree $\leq n+1$ satisfying: $Q(k)=0\quad\forall k\in\{0,1,2,\ldots,n\}$. Therefore $Q(x)=\lambda x(x-1)\cdots(x-n)$ for a certain real $\lambda$. Hence, $Q(-1)=\lambda \cdot (-1)^{n+1}(n+1)!$, but also $Q(-1)=(-1+1)P(-1)+1=1$. Comparing these, we get $\lambda \cdot (-1)^{n+1} (n+1)! = 1$, thus $\lambda=\dfrac{(-1)^{n+1}}{(n+1)!}$. Since $Q(n+1)=\lambda(n+1)!$ then $Q(n+1)=(-1)^{n+1} $, and from $P(x) = \dfrac{Q(x)+x}{x+1}$ we deduce: $$P(n+1)=\dfrac{(-1)^{n+1}+n+1}{n+2}.$$ I doubted that the answer were $P(n+1)=\frac{n+1}{n+2}$ but the term $(-1)^{n+1}$ wasn’t expected. Is there any mistake here? Does someone have an alternative proof? Problem is to find sum  $$\frac{1}{3!}+\frac{4}{4!}+\frac{9}{5!}\cdots$$  What I knew doesn't apply on this problem  Some series are telescoping, some types are solvable using binomial , both look useless here Binomial gives  $$n (n-1) (n-2)x^3=1$$  $$n (n-1)( n-2)(n-3)x^4=4$$  What approach to use here? Tools to get a pictorial function call graph of code I have a large work space which has many source files of C code. Although I can see the functions called from a function in MS VS2005 using the Object browser, and in MSVC 6.0 also, this only shows functions called from a particular function in a non-graphical kind of display. Additionally, it does not show the function called starting from say main(), and then the functions called from it, and so on, deeper inside to the leaf level function.  I need a tool which will give me a function call graph pictorially with functions callee and caller connected by arrows or something like that, starting from main() to the last level of function, or at least showing a call graph of all functions in one C source file pictorially. It would be great if I could print this graph.  Any good tools to do that (need not be free tools)? Are these intended activities legal to do in the USA under the VWP? I'm a British citizen traveling to the USA next month with an ESTA to see my fiancee. I'm traveling there to plan our wedding, talk about a location to get married, and probably finalise a date and a booking with my fiancee (US citizen). This is the primary reason for my trip to the USA. Am I allowed to do this as a tourist? I would like to know for when the CBP officer asks me if this thing is allowed.  Am I allowed to enter the USA, on a separate trip (probably later this year), for the sole purpose of marrying my fiancee, provided I can persuade the CBP officer I intend to leave at the end of my trip? My fiancee and I are moving to the UK next year. I am asked to solve this ODE: $$yy''-y'y^2-(y')^2=0.$$ Can I get some guidelines on how I can solve it? I am not sure how to approach without initial conditions. Too much memory getting used I have been feeling this way for a while, looking at task manager mentally ballparking how much physical memory should be in use, and noticing a large disparity. I finally got around to measuring it. Taking the results of tasklist and adding, gave about 3.87 GB. Task manager told me I was using 4.6 GB of memory. This seems like a large disparity. Where did it go?  EDIT: So its clear, it is not going to prefetch. It has also been suggested that its part of the OS, but not part of any of the system processes. If this is true, please find a way to get the OS to tell me how much memory is getting used. Since I want to find out where the memory went, asking me to look at the difference does nothing for me.  EDIT 2: It is not the following:   The kernel The cache Hardware reserved memory   Picture of resources taken up. Not quite sure what it will do but... Until I went to army, I (had been / used to be / was ) a introverted person. - which one is right? Until I went into the army, I (had been / used to be / was ) an introverted person. And after I I finished army service, I became more confident.   I think all of these are possible.  As for had been, I'm referring to the state of me being introverted before the accident that happened in the past. So, I think I can use had been.  And at the same time I feel I can use used to be and was.. If I don't want to emphasize the time gap between two accident, I feel I don't need to use past perfect. So I think this problem depends on how I'm going to say something.   which one is preferable? and what is the difference? Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even I am an Indian student in France on a 6 month visa for an Internship. Currently I don't have my passport with me as I have applied for UK visa and am not hoping to receive it back in next 10 days. I was wondering if I can travel to Germany without my passport in hand.   The documents I can carry are a photocopy of my passport's identity page, Residence proof here, proof of Internship in France,Indian national identity card, receipt from UK embassy stating my visa application.   Please note that I do NOT have a photocopy of French visa page. I've recently read a  from a local paper that included the following:     The pair was drinking prior to the shooting.   To me, this appears wrong and I would say that the proper way to make the statement would be:     The pair were drinking prior to the shooting.   Is the original article correct? Is there a general rule for determining the direction of tension force? Tension, for me, is a tricky thing.  After finishing a related chapter of my book and watching a video, I still can't get a hang of it.  Here is a situation:   My knowledge is that tension, just like normal force, happens in  just the same way, but with the difference of a string attached.  It is understandable that in the picture, the string attached to the hanging mass has a tension directed upward, because it's weight is directed downward. (Action and reaction)  But what intrigues me is that applying the same logic, the horizontal mass is being pulled to the right, it would have been understandable if the tension is directed to the left.   also Can I think of tension as the reaction force when you pull, and normal force when you push?  update If I was to think tension as a reaction force in the first picture for the horizontal mass, then can the frictional force be the action force? (Ehh, I don't think that makes sentence. Friction is always reaction) "Images with street signs" CAPTCHA impossible to solve I just got challenged (over on ) by the new "Mark all images with street signs" CAPTCHA.  I must have tried a dozen times to solve it by checking all images that had street signs, but I never succeeded.  Of note is a text input field without instructions of what to enter there. I suspect that is the problem: It is expecting input, but doesn't tell me what to input.  It's either broken or I am missing something.  I am using Chrome on an iPhone in "full site" mode. find() function in python2.7.5 How to pipe the result of 'date' command to 'mv' command I want to rename my file to the current timestamp.I know how to get the current timestamp("date +%s") but how can I pipe it to 'mv' command? How do you interpret a significant but weak correlation? Why does Ubuntu not allow the GUI users to login as root at the time of system start up like administrator in Windows?  Why they are restricting us to login only through the terminal?