Limit of $\lim\limits_{x \to \infty}\frac{1}{\cosh(x)}+\log\left(\frac{\cosh(x)}{1+\cosh(x)}\right)$ without l'hopital
Limit of function of hyperbolic
I've come across quite a few answers that request the asker to, for example: "send the user an email", after which the answerer will provide them with the code or other resources they're looking for.  A few examples:          Should we take action against these questions?  I feel like these answers aren't appropriate (see my answer below).  Update:  Perhaps there are some cases where this is acceptable. I've seen a few questions regarding a particular piece of commercial software where the author/developer of the software answers, telling the asker to contact either them or their organization's in-house support.  I'm on the fence about this case. If I came across a question here describing a similar problem I was having, it would be useful to me to have the support information and know that I could contact it for assistance.  I'm interested in others' opinions on this situation in particular. Can someone please verify my proof?     Is the collection $\tau_\infty = \{U:X-U$ is infinite or empty or all of $X\}$ a topology on $X$?   No. Let $X = \mathbb{R}$.  Clearly, $\{x\} \in \tau_\infty$ for all $x \in X$.  Set $\displaystyle{B = \cup_{x \neq 0} \{x\}}$. Clearly, $X - B = \{0\}$, but $\{0\} \notin \tau_\infty$. Since $\tau_\infty$ is not closed under arbitrary unions, it does not define a topology on $X$. How do you show monotonicity of the $\ell^p$ norms? I can't seem to work out the inequality $(\sum |x_n|^q)^{1/q} \leq (\sum |x_n|^p)^{1/p}$ for $p \leq q$ (which I'm assuming is the way to go about it). How to read ext4 partitions in Windows? I have two questions on ext4:   Is there currently a way to mount an ext4 partition in Windows (XP/Vista/7) to get basic read support? If not, are there any plans to support this someday? I am looking for an example such that in a pre-Hilbert space $H$ we have for a subspace $U$ that  (i) $\bar{U} \oplus U^\perp \neq H$  (ii) $ \bar{U} \neq U^{\perp \perp}$  Since finite and closed subspaces will probably not do it, I am not good at inventing nice examples. Does anybody here have a few at hand? Journals selection - alternative criteria to impact factor While trying to select the journals I can submit my present work and future work, I realize the time between the date the authors send the article to the journal and when the revised article is submitted. Usually this is something between 6 months and more than 12 months.  Althought I am focusing on the Impact Factor (IF) to select journals, I think that 12 months to have a feedback that might include "add more variables analysis","this article will be more valuable if you test also model x,y and z", can have a huge impact on you work. It will make you stop what you are currently doing and possible spend up to 2 months making the changes. This might make a big mess on your work if you have deadlines to respect, however this happens to everyone in academia.  My question is, besides the IF journal based selection, how can we select a journal with enought quality with lower IF? What alternative criteria should one use to select the journal? Should we choose newly created Journals? How can I serialize an object to JSON without properties that are null? To communicate with a REST interface on a private system, I need to send a JSON message from a trigger. I have a message object that has some fields, lists and a child object:  public class MerchantConfig {  public class MIDConfig {     public String mid;     public String currencyCode;     public String cardTemplateName;     public String paymentPanelRestriction;     public List&lt;String&gt; restrictedCountries;      public String paymentPanelRestrictionForCountries; }  public String merchantName; public String sopgTemplateName; public Integer dispositionTimeWindowInMinutes; public List&lt;String&gt; features; public Boolean activateImmediately; public List&lt;MIDConfig&gt; midConfiguration; }   I can serialize it to JSON using the built-in serializer:  String body = JSON.serializePretty([MerchantConfig instance]);   The result:  {   "sopgTemplateName" : null,   "midConfiguration" : [ {     "restrictionCountries" : [ "DE", "AT" ],     "paymentPanelRestrictionForCountries" : "MANDATORY",     "paymentPanelRestriction" : "OPTIONAL",     "mid" : "1000000001",     "currencyCode" : "EUR",     "cardTemplateName" : null   }, {     "restrictionCountries" : [ ],     "paymentPanelRestrictionForCountries" : "",     "paymentPanelRestriction" : "OPTIONAL",     "mid" : "1000000002",     "currencyCode" : "USD",     "cardTemplateName" : null   }, {     "restrictionCountries" : [ ],     "paymentPanelRestrictionForCountries" : "",     "paymentPanelRestriction" : "PROHIBITED",     "mid" : "1000000003",     "currencyCode" : "AUD",     "cardTemplateName" : null   } ],   "merchantName" : "Test Merchant",   "features" : [ XYZ ],   "dispositionTimeWindowInMinutes" : 10,   "activateImmediately" : true }   How can I serialize this object so that all properties that are null don't show up in the JSON string?   In this example, "sopgTemplateName" and all "cardTemplateName" properties are null and should not appear in the JSON string. The empty list "restrictionCountries" should still appear as an empty list, though. I have two tensors that i must calculate double dot product. matrix A is rank 2  and matrix B is rank 4. I want to multiply them with Matlab and I know in Matlab it becomes: A : B = trace (A*B) but it has one error and it says: Inner matrix dimensions must agree So how can I solve this problem? anybody help me? Thanks Cannot remove JDK – /usr/sbin/alternatives: No such file or directory When I try to remove Java with sudo apt-get remove jdk1.8.0-40 I get this:    Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be REMOVED:   jdk1.8.0-40 0 upgraded, 0 newly installed, 1 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 245 MB disk space will be freed. Do you want to continue? [Y/n] y (Reading database ... 231488 files and directories currently installed.) Removing jdk1.8.0-40 (1.8.040-1) ... find: `/usr/java/*': No such file or directory /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 586: /usr/sbin/alternatives: No such file or directory /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 598: /usr/sbin/alternatives: No such file or directory dpkg: error processing package jdk1.8.0-40 (--remove):  subprocess installed post-removal script returned error exit status 127 Errors were encountered while processing:  jdk1.8.0-40 E: Sub-process /usr/bin/dpkg returned an error code (1)   Output of  ls -1 /var/cache/apt/archives/*jdk*:  openjdk-7-jre_7u75-2.5.4-1~trusty1_amd64.deb openjdk-7-jre-headless_7u75-2.5.4-1~trusty1_amd64.deb   Output of sudo dpkg --purge --force-all jdk1.8.0-40:  (Reading database ... 231488 files and directories currently installed.) removing jdk1.8.0-40 (1.8.040-1) ... /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 586: /usr/sbin/alternatives: No such file or directory /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 598: /usr/sbin/alternatives: No such file or directory dpkg: error processing package jdk1.8.0-40 (--purge): subprocess installed post-removal script returned error exit status 127 Errors were encountered while processing:  jdk1.8.0-40   When trying the edit I get this:  (Reading database ... 231488 files and directories currently installed.) Removing jdk1.8.0-40 (1.8.040-1) ... /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 586: /usr/sbin/alternatives: Permission denied /var/lib/dpkg/info/jdk1.8.0-40.postrm: line 598: /usr/sbin/alternatives: Permission denied dpkg: error processing package jdk1.8.0-40 (--remove):  subprocess installed post-removal script returned error exit status 126 Errors were encountered while processing:  jdk1.8.0-40 E: Sub-process /usr/bin/dpkg returned an error code (1) $$  \lim_{n\to\infty} n^{1/2} \int_{0}^{1} \frac{1}{(1+x^2)^n}\mathrm{d}x=0 $$  Is my answer correct? But I am not sure of method by which I have done. Problem regarding radius of convergence of $\sum n!x^{n^2}$. $$\sum_{n=0}^\infty n!x^{n^2}$$   Can anyone tell me what will be the radius of convergence for this power series. According to me it is $0$.   If you can please show me how to go about it if I am wrong. Ubuntu GNOME 17.04: wi-fi not working -- mac address keeps changing? My Panda USB wi-fi adapter works just fine on 16.10, but when I try to connect to my wi-fi router in 17.04, GNOME network manager reports "Connection failed." I did some tinkering, and noticed that my MAC address for my wifi adapter, according to GNOME, is DIFFERENT every time I make it forget my wifi settings and try to reconnect. Any leads on a possible fix or work-around?   I'm running Ubuntu GNOME 17.04, kernel 4.10.0-19-generic, GNOME 3.24.0. I have my A* pathfinding algorithm right now working such that given a graph it can find the route from one starting node to one goal node.   This works well for many problems, but sometimes I find that multiple goal nodes are valid.  For example, in a game, an NPC might have to go to a switch to shut a door. If there are multiple switches that shut the door in question, it doesn't matter which switch it goes to - it just needs to go to one of them.  How can I augment my pathfinding algorithm to find the path to the best goal (the shortest path)? As in, the algorithm searches for multiple goals and when it finds just one, it returns the path to that one.  One solution would be to run the algorithm as many times as there are valid goals, but you can guess why that might be inefficient. We know that the needle that is used in a compass is a permanently magnetized ferromagnetic material and commonly steel is used.  If we used an unmagnetized iron needle instead, would it still align with Earth's magnetic field lines? If yes, how? And now, for something completely different: The mushrooms are among us I'm talking about at least two juvenile sci-fi novels, read between second and fourth  public grades (U.S.A.) which for me places them around 1956-9.  The protagonists are a brother and sister, presumably my (then) age. They are friends  with a single man, a scientist who has built a rocket ship. The scientist is little fey,  physically; a (relatively) large, round head, balding, short in stature and striking long,  thin hands and fingers.  The rocket ship's final preparation is to slop a viscous, clear gloop over EVERYTHING, inside and out from a pail with a paintbrush. In flight, there is an encounter with the moon, and a very small planet with (non-threatening) inhabitants not physically unlike the fey rocket scientist. The planet orbits either the moon or the earth, not the sun, and strangely, some kind of mushroom(!) theme runs through the books. Really.  That's it. That's all I got (I was nine at the most, fer chrissake). My recollection tells me there was concealment from parents, a particular name for the planet and people  and flight through a spatial fog. I mention these things separately because I am less sure  of them than the previous two paragraphs. The author was blissfully unrepentant of the laws  of physics:  Earth gravity on a micro-planet, fog in space and super-gloop to ready the  ricketiest rocket. Worked for me.  I'm guessing even Richard, my previous mentor/savior might not be familiar with these.  Anybody? Author, titles? Or do you just recognize one of these? Why won't Mathematica display an answer? It happens sometimes to receive or see a very good answer that arrives very late and that would deserve more upvotes. What can be done to give a boost to them? I would happily spend some of my reputation points to be able to give them such boost. This is not merely a question of reputation, but the idea is to let the answer &quot;bubble up&quot; faster so that it can be seen easily by other people to look into question. I hope I explained my question so that is understandable. Most of you got the point and there are some interesting proposals like the one that proposes to put randomly an answer at top. But I do have not enough experience to say if there is one that could solve the issue or not. I would be curious to know what Jeff thinks about it, if it feels there is something that can be improved or not. Pointer to class data member "::*" I came across this strange code snippet which compiles fine:  class Car {     public:     int speed; };  int main() {     int Car::*pSpeed = &amp;Car::speed;     return 0; }   Why does C++ have this pointer to a non-static data member of a class? What is the use of this strange pointer in real code? Assume you are given two objects whose exact locations are unknown, but are distributed according to normal distributions with known parameters (e.g. $a \sim N(m, s)$ and $b \sim N(v, t))$. We can assume these are both bivariate normals, such that the positions are described by a distribution over $(x,y)$ coordinates (i.e. $m$ and $v$ are vectors containing the expected $(x,y)$ coordinates for $a$ and $b$ respectively). We will also assume the objects are independent.  Does anyone know if the distribution of the squared Euclidean distance between these two objects is a known parametric distribution? Or how to derive the PDF / CDF for this function analytically? Related post   In Peskin and Schroeder's QFT p28, the authors tried to show causality is preserved in scalar field theory.  Consider commutator $$ [ \phi(x), \phi(y) ] = D(x-y) - D(y-x) \tag{2.53} $$ where $D(x-y)$ is the two-point correlation function,  $$D(x-y):= \langle 0 | \phi(x) \phi(y) | 0 \rangle  = \int \frac{d^3 p}{ (2\pi)^3} \frac{1}{ 2E_{\mathbf{p}}} e^{-ip(x-y)}\tag{2.50}$$  P&amp;S argued that each term in the right-hand-side of (2.53) is Lorentz invariant, since $$\int \frac{d^3p }{ (2\pi)^3} \frac{1}{2E_{\mathbf{p}}} = \int \frac{ d^4 p }{ (2\pi)^4} (2\pi) \delta(p^2-m^2)|_{p^0&gt;0} \tag{2.40}$$ is Lorentz invariant.  Since there exists a continuous Lorentz transformation in the spacelike interval $(x-y)^2&lt;0 $ such that $(x-y) \rightarrow - (x-y) $ and $D(y-x)=D(x-y)$, (2.53) equals zero in the spacelike interval. In timelike interval, since such continuous Lorentz transformation does not exist, (2.53) is non-zero in general.  My question is, consider a non-continuous Lorentz transmation in the timelike interval, $PT$, namely time reversal times parity transformation. I can also let  $(x-y) \rightarrow - (x-y) $. Why (2.53) in the timelike interval is non-zero?  I guess $PT$ will let (2.40) go to $p^0&lt;0$ branch. But I am not sure if it breaks the Lorentz invariant of (2.40) and (2.50).