Convergence of $\int^\infty_0\frac{1}{1+x^2\sin^2(5x)}\,dx$
Does $\int_{0}^{\infty}\frac{dx}{1+(x\sin5x)^2}$ converge?
How do I study the "correlation" between a continuous variable and a categorical variable? Prepend 0 to sublists I found this confusing when I use the neural network toolbox in Matlab. It divided the raw data set into three parts:   training set validation set test set   I notice in many training or learning algorithm, the data is often divided into 2 parts, the training set and the test set.  My questions are:   what is the difference between validation set and test set?  Is the validation set really specific to neural network? Or it is optional. To go further, is there a difference between validation and testing in context of machine learning? How can I find posts related to lost reputation due to a "user was removed" message? Today I noticed that my reputation got truncated by 64 points, and when I looked at my reputation history, I saw  notifying me of points removal along to a "user was removed" message. Each has a slightly different timestamp, ranging from 12:39:04 to 12:39:43.    I can't find the questions/answers concerned by this (and I don't remember having won as much as 65 points once, making this even stranger). I would like to find out how I could have lost 65 points while I never won as much in a single question/answer.  Is there a way I can check which of my posts were impacted? How does DC work with natural 20? Requirements for a clan war? Schengen refusal formulae can be difficult to understand sometimes.    When an applicant receives a refusal with "Justification for the purpose and conditions of the intended stay was not provided", what does it mean? I have a issue with my Drupal installation, for example:   when I enable or disable the modules, It redirects me to a blank page, when I create a new content type and save it, redirects me to a blank page, when I add a new view and save it, it redirects me to a blank page, when I try to clear the cache, it redirects me back to a blank page, or in similar cases.   Basically all the confirmation pages redirect me to a white screen. When I refresh it again, it shows me the page. I tried to increase the PHP memory value but it doesn't help.  Are there any other solutions for this?  The error which I'm having:      Fatal error: Allowed memory size of 100663296 bytes exhausted (tried to allocate 8192 bytes) in sites/all/modules/views/plugins/views_plugin_localization_none.inc on line 1 Is $\exists x(x=a)$ a valid formula in first order logic with equality? Under , is $\exists x(x=a)$ true in every interpretation in first order logic (FOL) with equality? If so, could $\exists x(x=a)$ be considered a tautology? This question was motivated by  which was closed for lacking clarity. Here I have attempted to add that clarity myself so that the same question can be useful to others who are wondering a similar thing. This is a self answer question. Windows 7 File Explorer search not working Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them Prove if $n$ has a primitive root, then it has exactly $\phi(\phi(n))$ of them.   Let $a$ be the primitive root then I know other primitive roots will be among $\{a,a^2,a^3 \cdots\cdots a^{\phi(n)} \}$ because any other number will be congruent modulo $n$ to one of these. Then I figured as the answer is $\phi(\phi(n))$ so for $a^k$ to be primitive root, $k$ must be coprime with $\phi(n)$.But I don't know the reason.Am I missing any silly point? I don't know how to disconnect a face from the rest of the mesh in edit mode.  This isn't like pressing P to separate the selection into a completely different object. I just want to select a face and disconnect/rip it away from the rest of the mesh but keep it as a part of the original object.  One way to do it would be to duplicate the face with ShiftD and then delete the original face. But I suspect there is a "native" command for disconnecting selected faces and I would like to learn it. How should I deal with becoming discouraged as a graduate student? I am a third year (starting fourth year in the fall) PhD student in mathematics. I've passed all qualifying exams and am currently doing research. As far as I can tell, I am not doing poorly. I have the good fortune of having a great advisor, being in a very supportive department, and having friends and family who genuinely care about my success.   The fact is research is hard. It appears to consist primarily of staring at a problem for days and days and days without getting anywhere. Sometimes, rarely, I do figure something out and that feels wonderful, but the overwhelming majority of my time appears to be spent banging my head against a mostly figurative wall. I am not complaining about the material being hard, and I am not averse to putting in hard work, but I get frequently discouraged when I realize the vast volumes of mathematics that I yet know nothing about (and probably never will). It's very hard to quantify progress - in particular, there are too few tangible returns after too many hours worked. I find myself thinking along the lines of "Oh, if only someone actually smart were thinking about this problem they would have solved it in moments" and so on.   I've talked about this to some number of people; here is some advice I have received:   Take a day off. Putting in hours upon hours of trying things doesn't magically lead to a solution, particularly if the brain is tired and just wants to sleep. Have a hobby. Since math research doesn't exactly provide instant gratification, a hobby might provide some instead.  Talk to other graduate students. Realize that many graduate students go through this.   I'm interested to know how other folks have dealt with being discouraged as a graduate student. Does it get better with time and experience? Is this a sign that research is not for me and that I should seriously consider a life outside academia? E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable) I'm trying to run this command in the terminal:  sudo apt install software-properties-common   This is the error message I get:  E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable)   E: Unable to acquire the dpkg frontend lock (/var/lib/dpkg/lock-frontend),     is another process using it? If a thin metal plate is placed between two charges $+q$ and $+q$, will this cause a change in the electrostatic force acting on one charge due to another? What is the concept behind this? What will happen if the metal plate is thick? Is it posible to change the locale of my device? I have the following diagram:    However, this looks a bit ugly, I would like a version that looks a little bit better. For example, size the last arrow such that the T_\phi is moved to the right.  This is the code which I am using to do this:  \begin{align*}   \xi \colon &amp;\lb \to \mathcal{L}(\hh)\\   &amp;\phi \mapsto T_\phi. \end{align*}   Where \lb and \hh just give the names for those spaces. Does anybody have a suggestion how I would improve the look of this function definition? What exactly is "pressure", and what's its relation to force? Guid vs INT - Which is better as a primary key? getResourceAsStream() is always returning null