Proving that if $0<a<b$, then $a<\sqrt{ab}<\frac{a+b}{2}<b$
If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$
At 12 o'clock, the hour hand and minute hand of the clock can be swapped, and the clock still gives the same time, but at 6 o'clock, it can not be swapped. So in what cases when we swap the hour and the minute hand position does a clock still give a valid time? Google Chrome PPA upgrade invalid signature Negation in English Issue with using ContactID as subscriber key in Marketing Cloud We have set up our Sales Cloud - Marketing Cloud integration where Contact ID is used as the subscriber key.  What we've noticed is when we have contact ID as the subscriber relationship on our sendable data extension, Marketing Cloud will reference contact ID on All Subscribers list and if data matches, it will send to the email address listed on All Subscribers and not to the one on the data extension.  This has caused us headaches for a number of reasons:   Email addresses always change so the one on All Subscribers is out-of-date. Whilst preference center can update a recipients address and write this back to All Subscribers that not how most addresses are updated by the user Email address is typically updated in Sales Cloud instance and migrated to Marketing Cloud either through imports or via synchronized data extensions.  It does not map directly to All Subscribers. Marketing Cloud users assume (and quite rightly) the address listed on the data extension is what will be used in the send. Not the one on All subscribers.   Anyone else experienced this? Any solution to resolve this issue? What causes Nova's Triple Tap to go on a short cooldown? This isn't about the level 20 ultimate talent.  There is a brief "arming phase" where Nova locks onto her target before shooting three bullets.  When something happens to her target during that phase (allied Devouring Maw, Divine Shield, etc.), sometimes Triple Tap is placed on the full 100 second cooldown, other times it is placed on a much shorter 10 second cooldown.  What triggers this 10 second cooldown?   Is this a timing issue?  I believe if she is interrupted after the shots start going out, Triple Tap is always put on the full cooldown.  Maybe I'm just seeing the ability get blocked right as the first shot goes out? Is this ability-specific?  Does stunning Nova have a different effect than a Devouring Maw or Void Prison on her target?  And is that different from a Divine Shield/Sanctification on the target? Is this determined by whether the line to the target was interrupted by an ally or an enemy hero?  For example, is there a difference between an allied Void Prison and an enemy Void Prison on the target? Can you raise a number to an irrational exponent? The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. $\sqrt[y]{a^x}$. So what happens when you raise a number to an irrational number? Obviously it is not so simple to break it down like above. Does an irrational exponent still have a well formed meaning?  The only example that comes to mind is Euler's identity, but that seems likes a pretty exceptional case. What about in general? Does and has both are used with singular pronouns (He has the bottle , He does play cricket , etc) whereas Do and have are used with plural pronouns ( They have the bottle , Do they like cricket? , etc)  But still we use Does with have ( She does have a car ). Why? Shouldn't it be     She does has a car. Clock time is off on dual boot proper use google webmaster tools URL Parameters There are some urls like http://domain.com/user/:username and http://domain.com/post/:id on my website, the username and id is the parameter from database. I try to let google crawler more efficiently , can I use google webmaster tools &gt; URL Parameters &gt; add parameter ? I read the document but still don't get it how to set up? the column Parameter and the rest what should I filled? or this tools only for get method parameter? How to identify the process that's sending error messages to terminal? Is there a way to show that $\sqrt{p_{n}} < n$? Is there a way to show that $\sqrt{p_{n}} &lt; n$?  In , I show that $f_{2}(x)=\frac{x}{ln(x)} - \sqrt{x}$ is ascending, for $\forall x\geq e^{2}$. As a result, $\forall n \geq 3$ $$\frac{p_{n}}{ln(p_{n})} - \sqrt{p_{n}}\leq \frac{p_{n+1}}{ln(p_{n+1})} - \sqrt{p_{n+1}}$$ Also (and as a result), $\forall n \geq 3$ $$ \frac{p_{n}}{ln(p_{n})} - \sqrt{p_{n}} &gt; 0$$ Or $$ \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; \frac{\pi (p_{n})}{\sqrt{p_{n}}}$$  According to  $$\displaystyle\smash{\lim_{n \to \infty }}\frac{\pi (p_{n})}{p_{n}/ln(p_{n})}=1$$ Or, $\forall \varepsilon &gt;0$, $\exists N(\varepsilon )$: $\forall n&gt;N(\varepsilon )$ $$1- \varepsilon &lt; \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; 1+ \varepsilon$$ Or $$1- \varepsilon &lt; \frac{\pi (p_{n})}{p_{n}/ln(p_{n})} &lt; \frac{\pi (p_{n})}{\sqrt{p_{n}}}$$ As a result $\forall \varepsilon &gt;0$, $\exists N(\varepsilon )$: $\forall n&gt;N(\varepsilon )$ $$(1 - \varepsilon ) \cdot  \sqrt{p_{n}} &lt; \pi (p_{n}) = n$$  But this is not enough.  Interestingly,  is true iff function $f_{4}(x)=\pi (x) - \sqrt{x}$ is strictly ascending ($x &lt; y \Rightarrow f(x) &lt; f(y)$) for prime arguments.  If $f_{4}(p_{n}) &lt; f_{4}(p_{n+1})$ then $$\pi (p_{n}) - \sqrt{p_{n}} &lt; \pi (p_{n+1}) - \sqrt{p_{n+1}}$$ Or $$\sqrt{p_{n+1}} - \sqrt{p_{n}} &lt; \pi (p_{n+1}) - \pi (p_{n}) =1$$  And vice-versa, if $$\sqrt{p_{n+1}} - \sqrt{p_{n}} &lt; 1$$ Then $$-\sqrt{p_{n}} &lt; -\sqrt{p_{n+1}} + 1$$ Or $$\pi (p_{n})-\sqrt{p_{n}} &lt; \pi (p_{n}) + 1 -\sqrt{p_{n+1}} = \pi (p_{n+1}) -\sqrt{p_{n+1}}$$  So, if Andrica's conjecture is true then $\forall n \geq 3$ $$\pi (p_{n})-\sqrt{p_{n}} &gt; 0$$ Or $$\sqrt{p_{n}} &lt; \pi (p_{n})= n$$ Flex-box: Align last row to grid Here is the situation.  I have hexagonal board,and a unit on it,with speed or move value 4.Diffrent terrain has a diffrent cost.When i click on the unit,game should show me a move range.  My solution was to check each  hex in range of 4,with A* pathfinding,and if path cost was less than 4 then this hex was in range.Finally game nicely show me range of that unit.  My question is:   Is there other solution to search for range on hex grids or square grid,because even if i am really proud of what i did in my solution,i think,it is a little to exaggerated?:))  What make me ask this question?I noticed that when unit speed is 4 or 6 or even 8,time to computing range for my computer was really good,but when speed was 10 and more i noticed that i needed to wait few second to compute.Well in real games i rather dont see something like this and my A* pathfinding is rather well optimized,so im thinking that my solution is wrong.  Thanks for any replies. Unable to install Ubuntu on Lenovo Y500 Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$ Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$  So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called $a_{n}$ and the denominator be $b_{n}$ Is there a way to use this statement so that I could force the original sequence into the form of $1/\left(1+\frac{1}{n}\right)^n$ How do you judge the quality of a journal? Since I can't find any alternative to Linux'  elevation command, I have the following question:  How to define a PowerShell function which requires elevation? I mean UAC prompt.  Say, such function follows:  function system-check {     SFC /ScanNow }   System:  Windows 8.1 Pro 64-bit  PowerShell:   Major  Minor  Build  Revision -----  -----  -----  -------- 5      0      10586  117    EDIT1:  To be 100% understandable let me rephrase:   I run PowerShell as user I run the aforementioned function system-check I want the function to elevate in order to be able to execute the command; note, that I want the UAC prompt to appear Why is find not accepting '-exec cp {} dir +'? I have a directory, dir1 which contains many files whose names end in either .jpg or .png. I want to copy all the .png files to dir2 which is empty.  This command works:  find dir1 -name '*.png' -exec cp {} dir2 \;   but this command doesn't:  find dir1 -name '*.png' -exec cp {} dir2 + find: missing argument to `-exec'   I also tried:  find dir1 -name '*.png' -exec cp {} -t dir2 + find: missing argument to `-exec'   and:  find dir1 -name '*.png' -exec cp {} dir2 \+ find: missing argument to `-exec'   After looking at , I even tried:  find dir1 -name '*.png' -exec cp {} dir2 {} + find: Only one instance of {} is supported with -exec ... +    says that:     -exec {} + was added in [version] 4.2.12 in 2005   My version of find is 4.4.2.  What am I doing wrong? Default port for SQL Server I need to know the default  port settings for the following services   SQL Server SQL Browser SQL Reporting services SQL Analysis services   I neeed to know the port settings for these services for different versions of SQL Server (2000,2005,2008)  Also let me know whether the default port setting will change based on sql server versions  Thanks in advance. If I multiclass into 2 or more spellcasting classes, how do I determine my known/prepared spells?