The formula $\exists x[x=a]$, where $a$ is a constant, is always valid in first-order logic. See discussion and proof . The formula $\exists x[f(x)=a]$ is not valid, since the universal set, defined as the set that contains all elements, does not exist. Discussion . Now, I would like to know if the formula $\exists y[y=f(a)]$ is valid. Since $y$ belongs to the universal set, I would assume we can substitute it by $f(a)$ and since $f(a)=f(a)$, the formula is valid. If that's the case, I infer that the formula $\exists y[f^{-1}(y)=a]$ is valid as well.
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.
Can you enter the Schengen area with a UK visa? I'm a UK passport holder; my wife isn't. We don't live in UK. If she gets a UK visit visa, does this entitle her to travel to the Schengen area or do I need to get her a separate Schengen visa? @future method: Unsupported parameter type How do I use a particular font for a small section of text in my document? The standard font packages make global changes to the fonts in my document. What if I want to use a particular font for just a portion of my text?   How do I find the right name of the font? How do I select the font? How do I restrict the scope of the font change?     There are separate answers for different flavors of TeX (quick links): Do you need a comma before a subordinating conjunction (like if)? Are both of the following sentences correct?  a: You can call me, if you need me.  OR  b: You can call if you need me.  Note that in a:, the comma is placed before the "if" and is not present in case b.  From this link, I gather that it isn't necessary because it's a short sentence:   But can someone point out an "official" source on this usage? Here the lower box dimension and the upper box dimension are exactly what Falconer talks about in his book "Fractal Geometry". I already know that the upper box dimension is finitely stable since $N_\delta(A\cup B)\le N_\delta (A)+N_\delta(B)$, where $N_\delta(A)$ is defined as usual, and Falconer gives an example to show the box dimension may not be countably stable. (We know that, however, the Hausdorff dimension is countably stable).  But now I want to find the example to show the lower box dimension may not even be finitely stable.It seems that we can use von Koch curve to illustrate this, but what is the explicit explanation for this? Since the lower box dimension is monotonic, we need to find set A and B and show that   $\underline{\dim}_B(A\cup B)&gt;\max\{\underline{\dim}_B(A),\underline{\dim}_B(B)\}$. Pass php variables to js without drupal behaviour I stand on the following tricky situation:  I have a node with geofield coordinates attached. I am constructing an array of points. Something like:  Array ( [86] =&gt; Array     (         [0] =&gt; Array             (                 [lat] =&gt; 39.1641410477                 [lon] =&gt; -2.633972168             )          [1] =&gt; Array             (                 [lat] =&gt; 41.0420738489                 [lon] =&gt; -2.633972168             )          [2] =&gt; Array             (                 [lat] =&gt; 41.0420738489                 [lon] =&gt; 1.2524414063             )          [3] =&gt; Array             (                 [lat] =&gt; 39.1641410477                 [lon] =&gt; 1.2524414063             )         [4] =&gt; Array             (                 [lat] =&gt; 39.1641410477                 [lon] =&gt; -2.633972168             )     )  [85] =&gt; Array     (         [0] =&gt; Array             (                 [lat] =&gt; 39.1641410477                 [lon] =&gt; -2.633972168             )          [1] =&gt; Array             (                 [lat] =&gt; 41.0420738489                 [lon] =&gt; -2.633972168             )          [2] =&gt; Array             (                 [lat] =&gt; 41.0420738489                 [lon] =&gt; 1.2524414063             )          [3] =&gt; Array             (                 [lat] =&gt; 39.1641410477                 [lon] =&gt; 1.2524414063             )         [4] =&gt; Array             (                 [lat] =&gt; 39.1641410477                 [lon] =&gt; -2.633972168             )     ) )   I want to pass this to my-googlemaps.js in order to be used by google's init map Google map is defined as:  my-map:   header: true js:   js/my-googlemaps.js: { }   https://maps.googleapis.com/maps/api/js?key=my_key&amp;callback=initMap: {type: external, attributes: {defer: true,async: true}} dependencies:   - core/drupal   - core/drupalSettings  my-googlemaps.js:   console.log(Drupal.settings);   function initMap() {}   This way I get Drupal.settings undefined, which makes sense I guess. On the other hanve if I place initMap into a behavior It can not be seen as a callback as defined in googleapis.com before.  Has anyone stumbled with this before? Any idea how to attack the problem? $(\Bbb Z\oplus \Bbb Z)[X] \not\cong \Bbb Z[X]\oplus \Bbb Z[X]\,$? Well i tried to prove it but failed. They are both commutative and have the same set of units. So i can't see a property of one ring which the other ring does not possess. Thank you for any help. Java, Collection constructor What statistical test to use for discrete predictor and response variables? I'm doing a statistical analysis of a model for college. It's modelling a bear catching fish and I'm asking will the amount of time taken to digest the fish have an effect on the amount of fish caught? (stupid question I know!) So the predictor variable is the amount of fish the bear can eat before it needs to digest and the response variable is the amount of fish it catches. The predictors varies between 1 and 30 in integers and the response is also in integers. I repeated each level of the predictor 5 times creating a sample size of 150 altogether. I was thinking of just using a simple correlation test such as cor.test in R, but I've read online that this is for continuous data. Does that matter or would this test still be appropriate? Thanks a million, Eamon I read a novel when I was 12 or 13 (late 80's, early 90's) about two timelines combining on Earth.  It was a full length novel for adults that I swiped from my mom (she doesn't remember it at all.)  I've been trying to figure out what the title is for some time but my google-fu is unable to find the right search terms.    The book was published in the 80's or perhaps early 90's.  The story: It was set on Earth and featured two timelines running concurrently (in the 80's?)  Timeline 1:  The nuclear missile crisis had gone badly and a low level nuclear war had more or less destroyed the US, but small towns still survived but are slowly fading away.  Much of the story was about the scarcity of goods, and the protagonist in the main thread of the book started the story by finding a 6 pack of Coke buried in a ruined house.  Throughout the story people were finding things that had grown scarce (deer, bullets, human beings.)  One strange scene I recall involved mutant/radioactive sweet potatoes that had driven a man insane.  Eventually people in this timeline began having dreams of their lives in Timeline 2.  Some of the people were not having dreams, and it was eventually discovered that they had died in Timeline 2.    Timeline 2:  There is panic in a world recognizable as our 1980's as nuclear war draws near.  Since more and stronger missiles have been developed, the end of humanity is at hand.  Things start disappearing from that world as the war begins.  Timeline 2 was mostly seen as vignettes of panicked people running around.  Combination of Timelines: In a weird, incongruous, almost surely poorly written scene, Gaia (yes, Earth itself) is seen to be combining a dying timeline with a doomed timeline in order to save humanity.  At the end of the novel, kids from Timeline 2 are transported into Timeline 1 in order to repopulate.  The new kids drink all the Coke, in case you were wondering...  Any help finding the title would be welcome.  I recall enjoying the book a lot when I was a kid, but I suspect my mom traded it in at our local used book store because at some point it just disappeared. Hide processes from other users based on groups (under Linux)? Is it possible to configure process hiding for certain user groups under a linux system?  For example: Users from group X should not see processes owned by users from group Y in ps/top or under /proc.  Is it possible to configure such a setup with SELinux?  (I vaguely remember a similar feature in the funny grsecurity patch set - but IIRC, it was more generic - and besides, I want to configure a stock linux distro without having to maintain a custom kernel.)  Edit: For better illustration, . The example is not that generic, but one can configure that a user or some users can only see information of their own processes in ps etc. Is my US visa still valid after change in name? GeoServer cutting off symbols close to tile edges? I am looking for a book that goes through the mathematical aspects of neural networks, from simple forward passage of multilayer perceptron in matrix form or differentiation of activation functions, to back propagation in CNN or RNN (to mention some of the topics).  Do you know any book that goes in depth into this theory? I've had a look at a couple (such as Pattern Recognition and Machine Learning by Bishop or Deep Learning by Goodfellow, Bengio and Courville) but still have not found a rigorous one (with exercises would be a plus). Do you have any suggestions? There is no fruit jet pack. I can't get it. How do I get it? I looked at the jetpacks many times. There are none. I accidentally killed a noble, now every guard wants to kill me. Is there any way I can get rid of the bounty? I want to kill a whole process tree.  What is the best way to do this using any common scripting languages? I am looking for a simple solution. How do you take profit from stock trading while keeping capital invested? I'm looking to download and install the latest version of Ubuntu to replace the current version of Windows I had pre-installed on my machine when bought it.   I have some really neat features on the machine that I really am hoping will still work when I move over to Ubuntu.  Can someone please confirm if the below will still work and, if not, how I can get them to work:   The USB ports on my machine The ethernet or Network work port on the machine The HDMI output on the machine and finally, the built in webcam   Thanks Sam where does lspci gather its information from? In  I found "[The   latest   version    of   this   document   ". There I says: "… lspci is a synonym for cat /proc/pci".  Which isn't the case here, on a Crunchbang 10 system (Debian based). No such directory. I do get the basic idea and, as far as I  know, the content of proc is (mainly) created during runtime. (Wrong?). This made me curious:  Question: Where does lspci gather it's information from? And where is this documented? (Where did I miss something?)  &nbsp; Another difference I found: In kernel.org's documentation, under "Table 1-5: Kernel info in /proc"      pci &nbsp;&nbsp;&nbsp; Deprecated info of PCI bus (new way -> /proc/bus/pci/, decoupled by `lspci)