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Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator.    I did in the following way. Are there other ways?  Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then,  $$e^{\pi}-{\pi}^e=e^{f(e)}-{e}^{f(\pi)}\tag1$$ Now, $$f'(x)=\frac{e\pi(1-\ln x)}{x^2},\quad f''(x)=\frac{e\pi (2\ln x-3)}{x^3},\quad f'''(x)=\frac{e\pi (11-6\ln x)}{x^4}.$$ Since $f'(x)\lt 0$ for $e\lt x\lt\pi$, one has $f(e)\gt f(\pi)$. By , there exists a point $c$ in $(e,\pi)$ such that  $$\begin{align}f(\pi)&amp;=f(e)+(\pi-e)f'(e)+\frac{(\pi-e)^2}{2}f''(c)\\&amp;\gt f(e)+(\pi-e)\cdot 0+\frac{(\pi-e)^2}{2}\cdot \frac{e\pi(2\ln e-3)}{e^3}\\&amp;=f(e)-\frac{\pi(\pi-e)^2}{2e^2}\tag2\end{align}$$ because $f'''(x)\gt 0\ (e\lt x\lt \pi)$ implies $f''(c)\gt f''(e)$.  By the  and $(2)$,  $$e^{f(e)}-e^{f(\pi)}\lt (f(e)-f(\pi))e^{f(e)}\lt \frac{\pi (\pi-e)^2}{2e^2}\cdot e^{\pi}=\frac{e\pi(\pi-e)^2}{2}\cdot e^{\pi-3}\tag3$$  Since $e^x\lt \frac{1}{1-x}\ (0\lt x\lt 1)$ and $0\lt \pi-3\lt 1$,  $$e^{\pi-3}\lt\frac{1}{4-\pi}\tag4$$  From $(1)(3)(4)$,  $$e^{\pi}-{\pi}^e\lt \frac{e\pi(\pi-e)^2}{2}\cdot\frac{1}{4-\pi}\lt\frac{3\times\frac{22}{7}\left(\frac{22}{7}-2.718\right)^2}{2}\cdot\frac{1}{4-\frac{22}{7}}\lt 0.993\lt 1$$ Using same modulus for RSA Problems with booting the system I have yesterday updated packages and today the system cannot boot properly. After seeing the Kubuntu logo there is a black screen which shows repeatedly.     Stopped User Manager for UID 121.      Removed slice User Slice of gdm.      Created slice User Slice of gdm.       Starting User Manager for UID 121...      Started Session c3 of user gdm.       Started User Manager for UID 121.   Please see the screenshot below urlEvent is undefined Why can't I connect to Pokemon Go via mobile data? When activating the evolution for a caught Pokemon, it seems to amplify the Combat Power (CP). Thus it seems most beneficial to evolve your Pokemon with the highest CP, if you have multiple of the same kind.   But what I am not sure about is whether it makes an efficiency difference if you spend candies and dust to boost CP before evolving, to gain more from the amplification? Or is it more efficient to just evolve without spending, and then using the leftover candies afterwards, since they now give a bigger CP boost per candy?   I only tested with a Rattata so far, which gained +8 CP per candy before evolving, and +15 CP afterwards. I collected no numbers on the amplification efficiency though, and now ran out of materials for further testing.  Or maybe it is a balanced calculation and makes no difference at all? I would love to have some insight on that.    Some data I collected thus far: (GPC = CP gain per candy)  79 CP Weedle (GPC 6) evolved to: 84 CP Kakuna (GPC 6), evolved at 90 CP to: 297 CP Beedrill    Note: The max-CP "half circle" in the profile stayed filled roughly at the same relative position throughout the evolutions. Bind variable with dotted name not working in dynamic SOQL As mentioned in , it's possible to use bind variables in dynamic SOQL queries.  This seems like a great way to avoid SOQL-injection, and to build a query string once but use it multiple times with different values in the variables.  I'm working on a method that returns a list of objects.  It accepts a class with multiple properties as its only parameter.  The properties of this class are used to filter the results to just the required records.  It will eventually be driven by a Visualforce component, which will be configured by querystring parameters in the URL.  I just ran into an issue where the query fails when I use dotted bind variables.  If I copy the constructed dynamic SOQL into a static query, it works fine.  This may be a bug, because in doing tests I found that :foo.bar would actually try to use local variable bar.  Here's one of my queries, which uses related custom objects.  I've stripped it down to just the part that's failing (wrapped for legibility):  result = Database.query('SELECT Name, Another_Object__r.Name,      Another_Object__r.Custom_Field__c      FROM Custom_Object__c       WHERE Another_Object__r.Custom_Field__c = :filter.deviceType');   The error reported is System.QueryException: Variable does not exist: filter.deviceType.  As mentioned above, the query works fine as static SOQL, just not dynamic:  result = [SELECT Name, Another_Object__r.Name, Another_Object__r.Custom_Field__c FROM Custom_Object__c WHERE Another_Object__r.Custom_Field__c = :filter.deviceType];   It also works in this case:  String deviceType = filter.deviceType; Database.query('SELECT Name, Another_Object__r.Name, Another_Object__r.Custom_Field__c FROM Custom_Object__c WHERE Another_Object__r.Custom_Field__c = :deviceType');   I haven't read about this being a known limitation.  I don't know if I should be using some other syntax/notation in the query -- the dotted notation appears correct.  Time to report a bug?  Thoughts on workarounds other than creating local variables to stand in for each member of the class? Where does gravity get its energy from? I would like to know where gravity gets its energy to attract physical bodies?  I know that the law of conservation states that total energy of an isolated system cannot change. So gravity has to be getting its energy from somewhere, or else things like hydropower plants wouldn't be able to turn the power of the falling water into a spinning rotor.  Just to be clear, Lets create an example:  Lets say we have two objects with equal mass close to each other. So gravity does its job and it pulls each other closer, this gets turned into kinetic energy. This is where I'm lost. According to the law of conservation energy can't be created or destroyed and the kinetic energy comes from the gravitational pull so where does the gravitational pull gets its energy.  If that energy isn't being recycled from some where else then that means you have just created energy, therefore breaking the law of conservation. Careers tag search should only search tags What's the limit on the no. of partitions I can have? I would like to know how many Primary and Extended Partitions I can create on a x86_64 PC with Linux running on it?     Update  : If there is a limit to the number of partitions, why is that the limit? Difference between StringBuilder and StringBuffer In this sentence:  "We would also accept an international transfer, but in that case we would ask you to pay your bank's charges at the time of the instruction to ensure that we received the full amount"  is it correct to use "received" or should it be in present tense? Redirect console output to textbox in separate program I'm developing an Windows Forms application that requires me to call a separate program to perform a task.  The program is a console application and I need to redirect standard output from the console to a TextBox in my program.    I have no problem executing the program from my application, but I don't know how to redirect the output to my application.  I need to capture output while the program is running using events.    The console program isn't meant to stop running until my application stops and the text changes constantly at random intervals.  What I'm attempting to do is simply hook output from the console to trigger an event handler which can then be used to update the TextBox.  I am using C# to code the program and using the .NET framework for development.  The original application is not a .NET program.  EDIT: Here's example code of what I'm trying to do.  In my final app, I'll replace Console.WriteLine with code to update the TextBox.  I tried to set a breakpoint in my event handler, and it isn't even reached.      void Method()     {         var p = new Process();         var path = @"C:\ConsoleApp.exe";          p.StartInfo.FileName = path;         p.StartInfo.UseShellExecute = false;         p.OutputDataReceived += p_OutputDataReceived;          p.Start();     }      static void p_OutputDataReceived(object sender, DataReceivedEventArgs e)     {         Console.WriteLine("&gt;&gt;&gt; {0}", e.Data);     } Can “due to” and “because of ” be used interchangeably? Is it fine to use due to in place of because of ? How about the other way around? Are any of these sentences ungrammatical?   He was lost because of the storm. He was lost due to the storm. He lost his way due to the storm. He  lost his way because of the storm. Display Animated GIF I process all my photos to black and white. Although I use a polariser where possible, I often find myself wanting to drop the luminance of the sky further which I do using a Black and White adjustment layer (in Photoshop). This usually results in a lighter halo following the line of the horizon, where the blues of the sky meet whatever the colour of the land is.  My current solution to this problem is to run a brush along this margin with a colour set to the same as the sky and the blend-mode set to Darken. This effectively paints in the lighter band with the sky colour while not laying down the colour on top of the darker horizon. Whilst this is effective, it is very time consuming.  Is there a better approach to dealing with these halos?  Here is a video that illustrated the Darken solution that I mention: Connecting multiple grounds As I'm relatively new to the electronics world, I was wondering if you can connect multiple  ground terminals (corresponding to several different voltage outputs) to the same ground. Or do they each need their own?  In my case I have 5 lines, one 12V, one 3.3V, two 5V and one variable from a 12V in using a  voltage regulator that already has its own ground. So I'm guessing that each would need its own ground, but it is never bad to ask. How to move a database from SQL Server 2012 to SQL Server 2005 What are my options if I need to move a database from SQL Server 2012(32bit) to SQL Server 2005(64bit)?  I know I cannot:   restore a backup of the database on SQL Server 2005 detach &amp; attach   I know I can:   use import data wizard, and I tried it on one database, but it only moves data, and even that was troublesome as I needed to do a lot of work creating temp tables to maintain identity columns, recreating all FKs, indexes etc.    Is there an easier option? Errors "*ERROR* UVD not responding, trying to reset the VCPU!!!" Why not make it mandatory to leave a reason or at least a remark of some sort whenever down voting?  You see more and more often questions being down voted while many times they are simply perfectly legitimate, as questions/as people being unable to achieve some desirable result. I am not talking about "Hey, please do this and that for me, thanks :D " posts.  I would like to know what reasons there are for not making it mandatory to leave a reason for a down vote.