Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance!
Let $f$ be defined on $\mathbb{R}$ by $f(x) = e^{-1/x^2}$ for $x$ not equal to $0$. and $f(0)= 0$. Prove that $f^{(n)}(0)=0$ for all $n = 1, 2,3$ ...  Do I need to use Taylor expansion from calculus class?  Any hint would be appreciated.
Why we can't implement AES 512 key size? How to view all the Symbolic links,Junction points,Hard links in a Folder using dir? The command dir /a displays a list of all the files,folders in a given location. But it displays the type  for Junction Points as well as Symbolic Links to Folders. Is there any command which will differentiate and tell me which of these  are Junction Points and which are Symbolic links as well as which of the files are hard links ? Get a "step-by-step" evaluation in Mathematica Is there Markdown to create tables? It seems like a lot of people try to display tables in Stack&nbsp;Overflow questions, but don't do a very good job. Is there Markdown syntax that supports creating a proper table? In thinking about Sturm-Liouville theory a bit I see I have no actual idea what is going on.  The first issue I have is that my book began with the statement that given  $$L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  the problem $L[y] \ = \ f$ can be re-cast in the form $L[y] \ = \ \lambda y$.  Now it could be a typo on their part but I see no justification for the way you can just do that!  More importantly though is the motivation for Sturm-Liouville theory in the first place. The story as I know it is as follows:  Given a linear second order ode   $$F(x,y,y',y'') = L[y] = a(x)y'' + b(x)y' + c(x)y = f(x)$$  it is an exact equation if it is derivable from a differential equation of one order lower, i.e.  $$F(x,y,y',y'') = \frac{d}{dx}g(x,y,y').$$  The equation is exact iff  $$a''(x) - b'(x) + c(x) = 0. $$  If $F$ is not exact it can be made exact on multiplication by a suitable integrating factor $\alpha(x)$.  This equation is exact iff  $$(\alpha(x)a(x))'' - (\alpha(x)b(x))' + \alpha(x)c(x) = 0 $$  If you expand this out you get the Adjoint operator  $$L^*[\alpha(x)] \ = \ (\alpha(x)a(x))'' \ - \ (\alpha(x)b(x))' \ + \ \alpha(x)c(x) \ = 0 $$  If you expand $L^*$ you see that we can satisfy $L \ = \ L^*$ if $a'(x) \ = \ b(x)$ &amp; $a''(x) \ = \ b'(x)$ which then turns $L[y]$ into something of the form  $$L[y] \ = \ \frac{d}{dx}[a(x)y'] \ + \ c(x)y \ = \ f(x).$$  Thus we seek an integratiing factor $\alpha(x)$ so that we can satisfy this &amp; the condition this will hold is that $\alpha(x) \ = \ \frac{1}{a(x)}e^{\int\frac{b(x)}{a(x)}dx}$  Then we're dealing with:  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \alpha(x)f(x)$$  But again, by what my book said they magically re-cast this problem as  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ \alpha(x)c(x)y \ = \ \lambda \alpha(x) y(x)$$  Then calling  $$\frac{d}{dx}[\alpha(x)a(x)y'] \ + \ ( \alpha(x)c(x)y \ - \ \lambda \alpha(x) )y(x) \ = \ 0$$  a Sturm-Liouville problem.  My question is, how can I make sense of everything I wrote above? How can I clean it up &amp; interpret it, like at one stage I thought we were turning our 2nd order ode into something so that it reduces to the derivative of a first order ode so we can easily find first integrals then the next moment we're pulling out eigenvalues &amp; finding full solutions - what's going on? I want to be able to look at $a(x)y'' \ + \ b(x)y' \ + \ c(x)y \ = \ f(x)$ &amp; know how &amp; why we're turning this into a Sturm-Liouville problem in a way that makes sense of exactness &amp; integrating factors, thanks for reading! How to filter features in a layer? I am trying to display certain features in a shapefile-based layer using QGIS. I have found the subset function in the properties panel, which does exactly what i want, but is too tedious given that I have to manually sift through a large number of features. Is there a quicker way to do this?  I am using both shapefiles and postGIS best,  dietmar Speeding up a NOT IN subquery Good evening,  I'm currently up to try to improve the performance of some of my queries. As far as I know, statements like "IN" or "NOT IN" are even faster with a large amount of values if a subquery on an indexed field without conditions is used.  SELECT * FROM table1 WHERE field1 NOT IN (SELECT index_field FROM table2)   When it comes to use conditions at table2, the query becomes realy slow on a large amount of data.  SELECT * FROM table1 WHERE     field1 NOT IN (         SELECT             index_field         FROM table2         WHERE             user_id = '2'     )   I was thinking about using LEFT Join but when I need to to the filtering for the user_id, it also becomes slow.  How may I solve this problem? Currently, I've no clue about that.  The original query looked sth. like this  SELECT     i.* FROM stream_item si LEFT JOIN user__publisher_item ui     ON ui.user_id = 2     &amp;&amp; ui.item_id = si.item_id INNER JOIN item i     ON i.id = si.item_id WHERE      si.stream_id IN (5,7,8,9,19,24,29,42,43,44)     &amp;&amp; ui.id IS NULL || (         ui.`read` = 0 || ui.saved = 1     ) GROUP BY     si.item_id ORDER BY     si.`found` DESC LIMIT 0,10     id      select_type     table   type        possible_keys                                                                           key                 key_len     ref                 rows        Extra 1       SIMPLE          si      index       PRIMARY,stream_2_item,stream_id_found                                                   stream_2_item       4           \N                  663236      Using temporary; Using filesort 1       SIMPLE          ui      eq_ref      user_id_item_id,user_2_item,user_id_read_saved_hidden,user_id_saved,user_id_hidden      user_id_item_id     8           const,si.item_id    1           Using where 1       SIMPLE          i       eq_ref      PRIMARY                                                                                 PRIMARY             4           si.item_id          1      I have a table which represents the n:m relationship between stream and items. An unique item can be accessible via multiple streams. Based upon this i have a state-table for user dependent states of an item. Currently, the user-state-entry will only be generated, if the item was read or saved the first tim. Otherwise, there is no entry in the user_item table. In the beginning, this query was fine but now, it takes a lot of time. Even if I leave out the "INNER JOIN" it takes 7 seconds on the current tables with around 500k rows each.  The table-structure in the background is as follows:   stream stream_item //n:m relation between stream and items item //unique items stream_group //user-defined groups of multiple streams stream_group_streams //_n:m relation between stream_groups and streams_ user user_item //state-table that holds the user-dependent states of an item - if there is no state, no row exists for an item   I already asked this question at  but added my 2 Edits with further information to late so that I haven't got an answer anymore. Proof for alternative definition of the derivative I read somewhere recently that you can define the derivative as follows:  $$f'(x) = \lim_{h, k \to 0^+} \frac{f(x +h) - f(x - k)}{h + k}$$   I have been trying to prove this for about 2 hours, and can't seem to get it done. How should I proceed?  Edit: Assume $f$ is differentiable for all $x \in (x - \delta, x + \delta)$ for some $\delta &gt; 0$. Why do I see a saddle in this picture of a computer screen? I am not entirely sure this is an appropriate question for PSE, however since many of you have such diverse backgrounds, I'll give it a shot.  I have noticed that when one takes a picture of a computer, TV screen with a phone, you can "see" what looks like a saddle phase portrait.     Why does this happen? Does it have any relevance to differential equations or other areas of math? How to call a function on a directive scope from the controller C# AutoComplete In a world without air, I understand they definitely would. However, with drag taken into account, I think they wouldn't. Since the drag force varies proportional to the square of speed (ignoring the change in coefficient of drag with speed), wouldn't the total impulse due to drag on a bullet fired from a gun from a specific height horizontal to ground be higher than the total impulse on an identical bullet dropped from the same height with no horizontal velocity?  Also, when bullet spin is taken into account, the bullet fired from a gun should resist change in orientation; therefore, it should maintain a small but non-zero angle of attack, and also have a lift force, correct? network connection over thunderbolt bridge between linux and mac grub2 not able to find the new kernel This is very strange issue, i have CentOS7 running on HP server and everything was working till date until i did "yum install kernel" after reboot it drop me in grub&gt; shell that means it won't able to read or find grub.conf file.   in grub&gt; i found grub.conf has still old files there is no new entries of new kernel   I did this, and found there is no new kernel entries but if i boot with old kernel then i can see latest grub.conf (with new kernel)  grub&gt;cat /grub2/grub.conf   Question: where grub2 getting that old file from? why it's not able to find new kernel ? No feedback when comment fails to post on Android 2.3.5 In the mobile theme on Android 2.3.5, comments that do not meet the minimum length will fail to post but no feedback message is displayed to the user.  The submit button changes to its active state, but nothing else happens. Im confused about fire. The way I see it :   Heat creates (kinetic) energy in mass and this creates stronger vibrations of atoms. When those vibrations are strong enough the electrons interact stronger due to electromagnetic forces. This causes the electrons to fly away. This is the creation of a plasma and the direction of most electrons is the direction of the flame. the shape and heat of the flame depends mainly on the amount of electrons and their velocity ( apart from the source of heat and the considered gas and solid material) the electrons move freely ( but not random ) in the flame. there is a chemical reaction going on that usually produces smoke or such. the entropy increases in all the above 6 steps and the process is repeated. the electromagnetic energy is converted into kinetic energy of the electron and conductivity in the plasma (flame). the moving electrons create an electromagnetic field.   That is more or less how I see it. What steps are ok ?  And the main question , why is there light involved ?  I mean why cant we just have kinetic energy of the electrons and a changing magnetic field ? Why are there photons ? If the electrons are in a high energy state why cant they just move faster instead of emitting a photon ???  Why is the light usually visible ? is that because of our structure of air ?  If the light is an electromagnetic wave why does the flame go up ? does that imply all electromagnetic waves go up ?? But electromagnetic waves are not influanced by gravity or buoyancy are they ??  What is the final destiny of the electrons and photons if the heat is gone ??  I am puzzled by this. Display error message if same record exist with combination of some fields? I have taken three fields from case object called AccountName, IntegrationTo__c and SfType__c (checkbox).  I have created one case record with a specific combination of three fields.  For the second time I create a record with that combination, then  display error message only if SfType__c is true in first record otherwise allow  for record creation.  trigger AvoidDuplicate on Case(before insert)  {          set&lt;string&gt; newNameSet = new set&lt;string&gt;();     newNameSet = new set&lt;string&gt;();      set&lt;string&gt; newEmailSet = new set&lt;string&gt;();      set&lt;string&gt; dbNameSet = new set&lt;string&gt;();      set&lt;string&gt; dbEmailSet = new set&lt;string&gt;();      for(Case newLead : trigger.new){          newNameSet.add(newLead.Account.Name);          newEmailSet.add(newLead.IntegrationTo__c);      }      for(Case dbLead : [select id, Case.Account.Name, IntegrationTo__c from Case where IntegrationTo__c IN: newEmailSet OR Case.Account.Name IN: newNameSet]){          dbNameSet.add(dbLead.Account.Name);          dbEmailSet.add(dbLead.IntegrationTo__c);      }      for(Case newLead : trigger.new){          if(dbNameSet.contains(newLead.Account.Name) &amp;&amp; dbEmailSet.Contains(newLead.IntegrationTo__c))              newLead.addError('You are inserting Duplicate Case');      }  }   i wrote this this not checking the account field name and integration__c values only with combination of account and integration__c shows error Word for "someone who talks too much" cp -r is meant to copy files recursively, and cp -R for copying directories recursively. But I've checked, and both appear to copy both files and directories, the same thing. So, what's the difference actually? Fantasy/scifi combo - band of travelers who must get from "sphere" or domain to domain - each geographic area is ruled by a magician - it turns out later these are people who are unwittingly tapping into the computer grid they all live on to alter their own reality?