Why isn't math completely solved by expert systems? Everything in math can be perfectly defined or formalized. And we could derive logical conclusions from that. However, mathematicians still use human natural language to solve their theorems.  Why isn't automatic proof checking or automated theorem solving the kernel of math?
Why are mathematical proofs that rely on computers controversial? There are many theorems in mathematics that have been proved with the assistance of computers, take the famous  for example. Such proofs are often controversial among some mathematicians. Why is it so?  I my opinion, shifting from manual proofs to computer-assisted proofs is a giant leap forward for mathematics. Other fields of science rely on it heavily. Physics experiments are simulated in computers. Chemical reactions are simulated in supercomputers. Even evolution can be simulated in an advanced enough computer. All of this can help us understand these phenomena better.   But why are mathematicians so reluctant?
Does anyone know why typedefs of class names don't work like class names for the friend declaration?  class A { public: };  class B : public A { public:    typedef A SUPERCLASS; };  typedef A X;  class C { public:    friend class A;             // OK    friend class X;             // fails    friend class B::SUPERCLASS; // fails }; Which is faster in Python: x**.5 or math.sqrt(x)? I've been wondering this for some time. As the title say, which is faster, the actual function or simply raising to the half power?  UPDATE  This is not a matter of premature optimization. This is simply a question of how the underlying code actually works. What is the theory of how Python code works?  I sent Guido van Rossum an email cause I really wanted to know the differences in these methods.  My email:     There are at least 3 ways to do a square root in Python: math.sqrt, the   '**' operator and pow(x,.5).  I'm just curious as to the differences in   the implementation of each of these.  When it comes to efficiency which   is better?   His response:     pow and ** are equivalent; math.sqrt doesn't work for complex numbers,   and links to the C  sqrt() function. As to which one is   faster, I have no idea... Intuitive explanation for how could there be "more" irrational numbers than rational? Why does nothing show up in the Compositing tab? I am trying to follow Blender Guru's beginner tutorial. He went to the Compositor tab and was using it to Denoise an image but when I got there, no nodes show up. Can you guys help? How can I add a new user as sudoer using the command line? Why is Tony Stark in the Iron Man suit unaffected by G-forces, inertia and blunt force trauma? Tony's suit is an engineering marvel, and they explain many facets of the technology to help give legitimacy to the special powers, abilities and weapons it has.  I never read the Iron Man comics, but in the movie they really don't have a technological explanation for how he miraculously seems to be unaffected by crushing G-forces of sudden acceleration, the inertia of sudden stops and hair-pin turns, and especially the blunt force trauma from getting smashed in the face with Thor's hammer.  Certainly it appears the suit takes punishment, but Tony always remains unscatched, never a collapsed lung, broken bone or even a bloody nose.  In my infantile understanding of theoretical physics, the same technology that would be used to counter G-forces and sudden blunt force trauma to the body inside would be the very same technology used for  which in my honest opinion would be a much more stunning technological achievement than his heralded Arc Reactor.  Was their ever an explanation proferred for this and if so was it indeed attributed to Anti-Gravity? parsing XML file gets UnicodeEncodeError (ElementTree) / ValueError (lxml) Convergence of Ratio Test implies Convergence of the Root Test In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise:     Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that $$\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}=L,$$ then $$\lim_{n\to\infty}|a_n|^{1/n}=L.$$    I've been trying to prove this with no luck. The only thing I've thought of doing is $$\lim_{n\to\infty}\left(\frac{|a_{n+1}|^n}{|a_n|^n}\right)^{1/n},$$but this hasn't lead me anywhere except dead ends. Will someone provide a hint for me about how to proceed? Thanks!  Minor update: I don't know if it's helpful yet, but I know we can write the limit as $$\lim_{n\to\infty}\left(\frac{|a_{n+1}a_n\cdots a_0|}{|a_n\cdots a_0|}\cdot\frac{1}{|a_n|}\right).$$This reminds me a lot of the geometric mean, which even has the exponents I'm trying to get... In "Settings" > "SD card &amp; phone storage settings" under "Internal Phone Storage" it lists "Available Space" as 13.78MB.  I'm trying to install an app that is 2.38MB.  Why do I get the "Insufficient storage available" error message?  I clearly have enough space.  If it's relevant, my phone is an LG Optimus V. What information does the trace of a matrix give? What graduation gown should a PhD-holder wear when later receiving a Master's in another field? Access nested dictionary items via a list of keys? No Wifi, HP Pavillion 15 Ubuntu 14.04 Can I mill my opponent into defeat with Mindcrank and Duskmantle Guildmage? Okay the scenario I'm thinking of is as follows:   I control a ,  and 3 untapped lands (2 swamps, 1 Island to be perdantic). In my opponents turn he casts a burn spell to kill my duskmantle.   I activate his first ability in response.   The way I am hoping the following interactions will go is:   Duskmantles ability will hit the stack at the top and resolve The burn spell resolves dealing its damage to Duskmantle The burn spell card hits the graveyard and triggers the infinite loop for duskmantle and mindcrank killing my opponent Duskmantle perishes   Is this the way that the stack would pan out? Do I win in this situation regardless of the stack being correct or not above? (I am not fully familiar with how the stack really works in magic) You can see the cards from my deck . I want to completely replace Ubuntu with Lubuntu I accidentally issued some commands I thought were safe in the terminal, and I ended up with no System Settings and lots of other missing functionality, but the OS still boots normally. I've decided I want to install Lubuntu instead of fixing Ubuntu for a fresh start and since my laptop is running on a 1GHz dual core AMD CPU.  I created a bootable USB as I usually do and placed Lubuntu 17.10 amd64 using Rufus 2.14.1086. I have GRUB that was installed with Ubuntu upon first install I still have Win8.1 which I still want to keep.  PS: I also tried creating a startup disk from Ubuntu  The question is: How can I boot from USB with GRUB2 so I can overwrite Ubuntu? This has been puzzling me for a few days now, and I asked a graphic designer I knew before trying here and haven't been able to solve it yet. It's quite weird so please bear with me for a bit.  I am using Adobe Illustrator to create a logo, and I have something that can potentially work. I have used only the pen tool with strokes (no fill), but the issue is, when I view the logo at 100%, it looks pixellated! I know vectors should not even be able to appear pixellated, but they are. Here is the screenshot from AI, without exporting to anything:     That was just done using the snipping tool within Windows, and it's at 100% zoom.  I have anti-aliasing turned on from Edit > Preferences > General, and the document resolution is at 300 ppi (I get the same result with 72 ppi).   Could anyone help me with this?   If relevant: Using CS6, and have a 24 inch Dell U2412M at 1920x1200 resolution. I do have a 21.6" Samsung at 1680x1050 and I get the same result there as well. Let me know if you need any other settings, or if you would like the file, I'd be happy to provide it to be able to solve this problem. I left my cast iron pot on the gas stove, on high flame, and then forgot about it for 3 hours. It was empty and I was drying it before I oiled it to store it. After I discovered it and let it cool, I oiled it with a paper towel and all this rusty stuff came off and continues to come off. Is my pan ruined? What should I do? I am working on the following problem from group theory:     If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd.   That is, for some integer $k$, there are $2k+1$ elements $a$ such that $a \in G,\;\; a*a = e$, where $e$ is the identity element of $G$. I am testing the new  &lt;lightning:datatable&gt;, it is really good, and is a good replacement for most of my use cases which today I use a custom list component that I have developed.  I am getting data from my server through apex method, to populate it into the &lt;lightning:datatable&gt; component. I can bind the returned list directly to the data attribute and if the column names are same as the field names, all data is bound correctly.  The problem occurs when I get data from a parent record. For example - I want the Account name of my contact.  This example can show the problem:  Component Markup    &lt;aura:handler name="init" value="{!this}" action="{!c.doInit}"/&gt;  &lt;aura:attribute name="mydata" type="Object"/&gt; &lt;aura:attribute name="mycolumns" type="List"/&gt; &lt;aura:attribute name="sortedBy" type="String"/&gt; &lt;aura:attribute name="sortedDirection" type="String"/&gt;  &lt;lightning:datatable data="{! v.mydata }"                      columns="{! v.mycolumns }"                      keyField="Id"                      onrowselection="{! c.getSelectedName }"                      onsort="{!c.updateColumnSorting}"                      sortedBy="{!v.sortedBy}"                      sortedDirection="{!v.sortedDirection}"                      hideCheckboxColumn="{!false}" /&gt;        Component Controller  ({     doInit: function (component, event, helper) {         component.set('v.mycolumns', [                     {label: 'Id', fieldName: 'Id', type: 'id', sortable:"true"},                     {label: 'First Name', fieldName: 'FirstName', type: 'text', sortable:"true"},                     {label: 'Last Name', fieldName: 'LastName', type: 'text', sortable:"true"},                     {label: 'First Name Hebrew', fieldName: 'First_name_Hebrew__c', type: 'text', sortable:"true"},                     {label: 'Last Name Hebrew', fieldName: 'Last_name_Hebrew__c', type: 'text', sortable:"true"},                     {label: 'Account', fieldName: 'Account.Name', type: 'text', sortable:"true"},                     {label: 'Email', fieldName: 'Email', type: 'email', sortable:"true"},                     {label: 'Phone', fieldName: 'Phone', type: 'phone', sortable:"true"}                 ]);          var action = component.get("c.getRecordsFromDb");         action.setParams({             sObjectName: "Contact",             commaDelimitedFieldNames: "Id,FirstName,LastName,First_name_Hebrew__c,Last_name_Hebrew__c,Account.Name,Email,Phone",             filter: ""         });         action.setCallback(this, function(response){             var state = response.getState();             if (component.isValid() &amp;&amp; state === "SUCCESS") {                 var data = response.getReturnValue();                  component.set("v.mydata", data);             }         });         $A.enqueueAction(action);     } })   APEX Aura Enabled Method  @AuraEnabled public static List&lt;sObject&gt; getRecordsFromDb(String sObjectName,         String commaDelimitedFieldNames,         String filter) {      system.debug('\n\n\n\ngetRecordsFromDb sObjectName: ' + sObjectName + ', commaDelimitedFieldNames: ' + commaDelimitedFieldNames + ', filter: ' + filter + '\n\n\n\n');      // create a dynamic soql to get the value of the field     String sql = 'SELECT ' + commaDelimitedFieldNames + ' FROM ' + sObjectName;      if (!String.isBlank(filter))         sql += ' WHERE ' + filter;      system.debug('\n\n\n\ngetRecordsFromDb sql: ' + sql + '\n\n\n\n');     List&lt;sObject&gt; records = Database.query(sql);     system.debug('\n\n\n\ngetRecordsFromDb: ' + records + '\n\n\n\n');      return records; }   The problem is with the Account.Name field - the &lt;lightning:datatable&gt; component does not display the column data.   I know that the data is returned ok, and I know that in the object that is returned, the Account is contained like an object (probably this is the problem):    Is there a good way to show parent record data with this component? Is there a Linux equivalent of Windows' "resource files"?