How to show that $3^x+4^x=5^x$ has only one solution?
Proving that $ 2 $ is the only real solution of $ 3^x+4^x=5^x $
Why is the sum of two random variables a convolution? For long time I did not understand why the "sum" of two random variables is , whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum and not their convolution. The exact phrase "the sum of two random variables" appears in google 146,000 times, and is elliptical as follows. If one considers an RV to yield a single value, then that single value can be added to another RV single value, which has nothing to do with convolution, at least not directly, all that is is a sum of two numbers. An RV outcome in statistics is however a collection of values and thus a more exact phrase would be something like "the set of coordinated sums of pairs of associated individual values from two RV's is their discrete convolution"...and can be approximated by the convolution of the density functions corresponding to those RV's. Even simpler language: 2 RV's of $n$-samples are in effect two n-dimensional vectors that add as their vector sum.           Please show the details of how the sum of two random variables are a convolution and a sum. I remember reading a while back about arithmetics with only a single operator that did all the work of the familiar $+ - \times /$ operators. If there is such a thing, could someone point me to it?  As background: I have a side project where I'm playing with neural nets and genetic search algorithms. Right now it's the good old kind, with matrix multiplication and addition. I thought it would be interesting to implement a sort of neural network that used a combined or "all in one" operator.  I've googled like heck and can't pick up the trail of where I thought I saw that before.  Is there study in that area, or am I mis-remembering, or worse, just making it up? I'd appreciate any pointers. Find the area of the shaded region in the figure below: Find the area of the shaded region in the figure below:    I am completely stuck on how to start off this question. Please help on some guidance on how to start it off. Does the rank-nullity theorem hold for infinite dimensional $V$? Prove every odd integer is the difference of two squares I know that I should use the definition of an odd integer ($2k+1$), but that's about it.  Thanks in advance! How to make a half cylinder? I am having a tough time building a half cylinder and have not found any tutorial showing me how to do that. I've been working with the Fermat numbers recently, but this problem has really tripped me up. If the Fermat theorem is set as $f_a=2^{2^a}+1$, then how can we say that for an integer $b&lt;a$, the $\gcd(f_b,f_a)=1$? Prove that $\phi(n) \geq \sqrt{n}/2$ Zoom lenses and cameras with superzooms are often sold with a times ("×") zoom number — like a 12x zoom or 30x zoom. The question  explains how this relates to focal length numbers, like 70-200mm or 18-55mm. But how does ×-zoom relate to magnification, when used for macro or just to make something far away look bigger?  How can two 70-200mm lenses have different magnification? How to remove blank lines from a file (including tab and spaces)? I want to remove all empty lines from a file. Even if the line contains spaces or tabs it should also be removed. sql server linked server to oracle returns no data found when data exists Do I need to change my ESTA address when visiting the US? I have an ESTA for my visit last Sept to NY and I know it is valid for 2years.  I am going to Las Vegas in May, do I have to update the address at which I will be staying? Why is my variable unaltered after I modify it inside of a function? - Asynchronous code reference How can I check if share certificates I found have value? I have been clearing my parents flat and found share certificates from    P and O  European Ferries   Can I find out if they are still valid? If not, is there a market for them as collectibles or the like? I've heard the number 7% thrown around a bunch of times, but I am struggling to find reputable data to back it up.   All I found was a quote from Warren Buffet that I couldn't find an original source for and data assembled by individuals on their personal websites. Is there something from an organization such as Market Watch making a statement about this?  I'm hoping to specifically find what the 10 or 20 year return is for investing in the S&amp;P 500 index fund or another diversified portfolio option. Electric fields and potential E = electric field V = electrostatic potential r = distance ' = derivative   E = - (V)'   therefore if E increases V decreases Also if r increases E decreases                                                      because  E=kq/r^2 From 1 &amp; 2, if r increases then V increases But V=kq/r, so if r increases V decreases   Why does the 3rd point contradict with the 4th ? Possible Duplicate:          I live in a small apartment building and we can catch each-others wifi networks. We have different service provides and were wondering if we could amalgamate our networks so that we could use each others bandwidth transparently.   I would assume this would reduce latency.  Does anybody have a tutorial on how to fuse together wifi networks for a throughput increase? The target machine is a Linux computer.   EDIT  My work makes me do lots of 1TB flat file-transfers, so I believe my motivation is sound. Splitting all polygon features in a shapefile into smaller equal width polygons I have a shapefile of thousands of polygon features, that i would like to break up into smaller pieces (i.e. break a square into 8 equal pieces). I have read all over these boards for the right solution and haven't found one yet unfortunately. I've tried parcel editor but that only allows you to manually edit one feature at a time, and I've tried fishnet but that doesn't run through each individual feature it just processes across the entire extent of the shapefile. I have the full license for ArcGIS so all options are game. Ultimately I would like to be able to split this initial shapefile of features into a specified amount of equal width slots that each would be their own feature in the resulting shapefile.  Any thoughts? The Drupal 8 REST UI configuration page asks me to select Authentication providers for GET method; otherwise, it doesn't allow me to export the REST request.  How can I provide anonymous user to access my REST GET API without providing any auth information? bash if -a vs -e option Both about -a and -e options in  is said:  -a file     True if file exists. -e file     True if file exists.    Trying to get what the difference is I ran the following script:  resin_dir=/Test/Resin_wheleph/Results  if [ -e ${resin_dir} ] ; then     echo "-e "; fi  if [ ! -e ${resin_dir} ] ; then     echo "! -e"; fi  if [ -a ${resin_dir} ] ; then     echo "-a"; fi  if [ ! -a ${resin_dir} ] ; then     echo "! -a"; fi   /Test/Resin_wheleph/Results exists and is a directory. And this is what I get:  -e -a ! -a   which seems to be a little strange (notice -a and  ! -a). But when I use double brackets (e. g. if [[ -e ${resin_dir} ]]) in the similar script it gives reasonable output:  -e -a   So:    What is a difference between -a and -e options? Why -a produces a strange result when used inside single brackets?