prove that for any integrable function $f$ defined on $(-\infty,\infty)$ (or maybe square integrable function.. my professor didn't gave exact imformation about $f$.) $$\lim_{n\to\infty}\int_{-\infty}^\infty \frac n {\sqrt {\pi}}e^{-(nx)^2} f(x) dx = f(0)$$  I think I have to get some $\delta&gt;0$ and divide integral into three terms.    $$\int_{-\infty}^{-\delta} f*g_n +\int_{-\delta}^{\delta}f*g_n + \int_\delta ^\infty f*g_n$$ ($g_n(x)=\frac n {\sqrt {\pi}}e^{-(nx)^2}$)  since $f*g_n$ converges uniformly to $0$ if $x\neq0$, first and third term vanish. so what I have to do is to take $N$ and $\delta$ (from the given value $\epsilon&gt;0$) so that if $n&gt;N$  $$\left|\int_{-\delta}^{\delta}\frac n {\sqrt {\pi}}e^{-(nx)^2} f(x)dx - f(0) \right|&lt;\epsilon$$  I can't go any further.
Consider the sequence $(\phi_n)_{n\in \mathbb{N}}$ of test-functions $\phi_n\in \mathcal{D}(\mathbb{R})$ defined by  $$\phi_n(x) = \dfrac{n}{\sqrt{\pi}}e^{-n^2x^2}.$$  I want to show that this is a delta sequence, in the sense that:  $$\lim_{n\to\infty}(\phi_n,\psi)=(\delta,\psi)=\psi(0), \quad \forall \psi\in \mathcal{D}(\mathbb{R}).$$  In that case we know that we have  $$(\phi_n,\psi)=\dfrac{n}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-n^2x^2}\psi(x)dx.$$  The problem here is to compute this integral. The function $x\mapsto e^{-n^2x^2}\psi(x)$ has no singularities, hence contour integration wouldn't help. Integration by parts or substitution also doesn't seem to be of great help.  The major problem is that there is a function $\psi$ inside the integral about which we don't know really anything apart from the fact that it is $C^\infty$ and with compact support.  How can we compute this integral and in the end show that the sequence $(\phi_n)$ is a delta sequence?  Just to make clear, I'm here doing all of this in the context of the Riemann integral.
shorter proof of generalized mediant inequality? Is there a Java equivalent or methodology for the typedef keyword in C++? If $2^n - 1$ is prime from some integer $n$, prove that n must also be prime. I found native people prefer to use passive voice in a sentence. For example:  Tonight's moon can be seen from anywhere worldwide. We have reason to believe that progress can be made.  Be honest, it's difficult for an English learner to understand. Why not phrase it this way:  We can see the moon from anywhere worldwide.  In which cases do people prefer to use passive voice rather than active voice? I need help to advance in the resolution of that limit: $ \lim_{n \to \infty}{\sqrt[n]{\frac{n!}{n^n}}} $ When using the \cite command in figure captions and the ieeetr bibliography style, then  BibTeX numbers the citations as appearing earlier than they do in reality (i.e. the first citation in the document ends up being [42]).  Is there any way to get BibTeX to ignore the \listoffigures section when numbering? When you attempt to modify a file without having write permissions on it, you get an error:  &gt; touch /tmp/foo &amp;&amp; sudo chown root /tmp/foo &gt; echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Sudoing doesn't help, because it runs the command as root, but the shell handles redirecting stdout and opens the file as you anyway:  &gt; sudo echo test &gt; /tmp/foo zsh: permission denied: /tmp/foo   Is there an easy way to redirect stdout to a file you don't have permission to write to, besides opening a shell as root and manipulating the file that way?  &gt; sudo su # echo test &gt; /tmp/foo Title tag different from title appearing in Google? Italics markup not working in comments If I enter the following mark-down source:  “*blah*”   into a question or answer, then it works, but in a comment it doesn’t. The asterisks remain literal and don’t turn into italics. How to disable sticky corners in Windows 10 It seems Microsoft has tried to solve this problem .  In Windows 7, there is no boundary between monitors and the mouse can move freely across the top of the screen between my 3 monitors, provided I had set their heights equal in display settings. Windows 8 introduced sticky corners. See .  Windows 10 got sticky corners as well. There are a few pixels at the top corners of each display where the mouse cannot cross over onto the other display. One must move the cursor down to avoid this region in order to get to the next display.  The image shows roughly the region where mouse movement is not permitted in Windows 10, but is allowed in Windows 7.    Personally, I had no problem with unrestricted mouse movement across the top of my screens - I got used to "aiming" for the x, and the convenience of unrestricted cursor movement. Like all the people who wanted to disable it in W8, I'm wondering if there is a way to disable it in W10.  Edit to address possible duplicate:  Although the problem is identical to the one in , solutions to solve the problem in W8 involving editing the registry key MouseCornerClipLength do not work in W10, since that registry key is not present in W10. Also adding that key and setting the value doesn't work. I searched the entire registry and couldn't find it in another location. No other W10 keys in the node referenced in the W8 solution are obvious replacements.  Edit to address possible solutions in comments  harrymc's  and  that worked for for Windows 8.1 do not work in Windows 10. Class method differences in Python: bound, unbound and static What is the difference between the following class methods?  Is it that one is static and the other is not?  class Test(object):   def method_one(self):     print "Called method_one"    def method_two():     print "Called method_two"  a_test = Test() a_test.method_one() a_test.method_two() There is the next characterization for valuation domains:  Let $D$ be an integral domain and $K$ its field of fractions. The following are equivalent:   For every nonzero $x$ in $K$, either $x$ in $D$ or $x^{−1}$ in $D$. The ideals of $D$ are totally ordered by inclusion. The principal ideals of $D$ are totally ordered by inclusion (i.e., the elements in D are totally ordered by divisibility.) There is a totally ordered abelian group $Γ$ (called the value group) and a surjective group homomorphism (called the valuation) $v :$ $K^×$ $→ Γ$ with $D = \{ x ∈$ $K^×$ | $v(x)$ ≥ $0 \}$ $∪$ {0}.   The equivalence between 1, 2 and 3 are clear, and I understand that the fourth can be found in Krull's article (1936), but I don't speak german and I can't find a proof anywhere. Any link or hint will be much appreciated. Proof on Catalan numbers Add layer to a QGIS group using Python I am trying to add new layers into a specific group which I can get to work if I know the index, but how can I get Python to find the index of a group so it can load the new layer into the group wherever it is in the TOC?  qgis.utils.iface.legendInterface().moveLayer( layer, index ) Proving that if $f: \mathbb{C} \to \mathbb{C} $ is a continuous function with $f^2, f^3$ analytic, then $f$ is also analytic Let $f: \mathbb{C} \to  \mathbb{C}$ be a continuous function such that $f^2$ and $f^3$ are both analytic. Prove that $f$ is also analytic.   Some ideas: At $z_0$ where $f^2$ is not $0$ , then $f^3$ and $f^2$ are analytic so $f = \frac{f^3}{f^2}$ is analytic at $z_0$ but at $z_0$ where $f^2$ is $0$, I'm not able to show that $f$ is analytic. Why does editing posts on SO require a minimum of 6 charaters? I have just recently started editing the post on Stack Overflow.  I found some posts which only required 2-3 character changes like they had correct grammar but some mistakes in spelling.  When I tried to correct the spelling mistakes, and clicked on "edit", it said:  Edits must be at least 6 characters   I don't know why Stack Overflow added this limit. How can I edit only 3-4 characters in a post? What would happen if I used regeneration on a creature that was the target of Annihilating Fire? For example, what would happen if a freshly cast  were to be regenerated in response to ?  My hypothesis is that it depends on whose turn it is. The regeneration being essentially a replacement effect, when two replacement effects were to be applied simultaneously, they are to be chosen in APNAP order. Is this correct?  I haven't thought this through to the end, though, so I'm not sure what would happen in the end. Exception thrown inside catch block - will it be caught again? This may seem like a programming 101 question and I had thought I knew the answer but now find myself needing to double check. In this piece of code below, will the exception thrown in the first catch block then be caught by the general Exception catch block below?  try {   // Do something } catch(IOException e) {   throw new ApplicationException("Problem connecting to server"); } catch(Exception e) {   // Will the ApplicationException be caught here? }   I always thought the answer would be no, but now I have some odd behaviour that could be caused by this. The answer is probably the same for most languages but I'm working in Java. "A flea hops randomly on the vertices of a triangle with vertices labeled 1,2 and 3, hopping to each of the other vertices with equal probability. If the flea starts at vertex 1, find the probability that after n hops the flea is back to vertex 1."  Could someone provide a hint to help me start this? Python idiom to return first item or None