Show that $\|.\|_{1/2}$ is not a norm. would anybody guide me that how can i prove or disprove?  thanks
please I tried to find counterexamples to see that $l^p$ is not norm with $p&lt;1$ in the triangle inequality but I have problems with convergence when I choose some successions. Thanks.
Apache doesn't log remoteIP when RemoteIPHeader X-Forwarded-For is present Arbitrary precision spline interpolation I tried to write out the aura if statement but I am getting stuck if someone can explain it to me in detail. thank you  &lt;aura:if isTrue=&quot;{!lessthan(v.Tl.time, v.Tl.End_Date_Time1__c)}&quot;&gt;                            &lt;/aura:if&gt; How fast does electricity flow? 50 question per month limit? I get a message:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Why is there a question limit? 50/month is just not enough :) How does Cycles work and how is it different from Blender Internal?  Which is better? Prove that any continuous map on $\mathbb{R}$ which is the identity on $\mathbb{Q}$ is the identity map, hence Aut$(\mathbb{R}/\mathbb{Q})= 1.$  proof: Suppose $\sigma$ is continuous on $\mathbb{R}$ which is an identity on $\mathbb{Q}$. let $a \in R $. Then since $\sigma$ is continuous, then by definition there is $\delta_1 &gt; 0$ such that $|a - b| &lt; \delta_1$.  And assume $|\sigma(a) -\sigma(b)| &lt; \epsilon/2$ .  Let $\delta $= min{$\epsilon/2, \delta_1$}.  And now let $b \in \mathbb{Q}$ such that $|a - b| &lt; \epsilon$. Then  $|\sigma(a) - a| = |\sigma(a)  - b + b - a| \leq |\sigma(a) - \sigma(b) | + |b - a| &lt; \epsilon/2 + |b-a|$.  Can someone please help me? I am not sure . Can I say $|b - a| &lt; \epsilon/2 $? I have a Fujitsu laptop and all was working fine until I ended teamviewer. On the keyboard it has a numberpad on some of the letters which you press the Fn button to use. The system is acting like I have the Fn key pressed and I get numbers shown when it should be letters. If I press the Fn key the numbers go to letters. The Fn key is not stuck down. Is there a way to reset keyboard settings or a known fix. The operating system is Ubuntu 11.04 Natty. Reflection to Identify Extension Methods How could I retrive full internal storage of an Android smartphone in order to restore data? Story about last 2 men on Earth. One very evil, and the other very ethical How much money do I need to show as proof of support when applying for a UK Standard Visitor visa? Displaying mysql data in html table PHP What is the integral of 0? Is there a (basic) way to understand why a limit cannot be brought inside of a sum, in certain situations? I have seen that the method of "dominated convergence" implies that sometimes a limit cannot be brought inside of a sum, but is there a more basic way to understand this, preferably without using integrals?   A conceptual or simple answer would be helpful. Thank you! Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$, with $f: (k_1,k_2)\to\Bbb R$ differentiable at $a\in (k_1,k_2)$, $\lim b_n=\lim a_n=a$, and $a_n&lt;a&lt;b_n$.   My attempt: Im trying to write the above equality as a particular case of this other equality that hold by any interior point $a$  $$f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a-h)}{2h}$$  Because $a_n&lt;a&lt;b_n$ then I can write  $$f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}=\lim_{n\to\infty}\frac{f(h_n+a)-f(a-j_n)}{h_n+j_n}$$  Where $h_n=b_n-a&gt;0$ and $j_n=a-a_n&gt;0$, where these two new sequences tends monotonically to zero.  Now my problem: my thought is that $h_n$ and $j_n$ are particular sequences then I can build a sequence $k_n$ such that $j_n$ and $h_n$ are subsequences so I can write:  $$\{h_n:n\in\Bbb N\}\subset\{k_n:n\in\Bbb N\}\text{ and }\{j_n:n\in\Bbb N\}\subset\{k_n:n\in\Bbb N\}$$  (I can build this $k_n$ by example doing $k_{2n}=h_n$ and $k_{2n-1}=j_n$) and then I have the desired expression  $$f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}=\lim_{n\to\infty}\frac{f(k_n+a)-f(a-k_n)}{2k_n}$$  My question:   Is this proof right? In particular Im unsure of the implicit statement "if the limit exist for $k_n$ then it must exist too for the particular case with $h_n$ and $j_n$". The use of two sequences, $h_n$ and $j_n$ in the limit make me unsure about the validity of this manipulation and assumption. There is a simpler/different way to prove the same?   Thank you in advance. Calculating optimal number of bins in a histogram I'm interested in finding as optimal of a method as I can for determining how many bins I should use in a histogram. My data should range from 30 to 350 objects at most, and in particular I'm trying to apply thresholding (like Otsu's method) where "good" objects, which I should have fewer of and should be more spread out, are separated from "bad" objects, which should be more dense in value. A concrete value would have a score of 1-10 for each object. I'd had 5-10 objects with scores 6-10, and 20-25 objects with scores 1-4. I'd like to find a histogram binning pattern that generally allows something like Otsu's method to threshold off the low scoring objects. However, in the implementation of Otsu's I've seen, the bin size was 256, and often I have many fewer data points that 256, which to me suggests that 256 is not a good bin number. With so few data, what approaches should I take to calculating the number of bins to use? I'm looking to view and archive all my old DMs in Twitter, since I would like to revive conversations that took place 1 year ago.  Acording to Twitter they don't delete old messages, they simply archive them in their database.  Is there a way to view and download all of these messages, just like you can view all your old public tweets with third party apps?  UPDATE: I found  but apparently it's not working for DMs now If $f(xy)=f(x)f(y)$ then show that $f(x) = x^t$ for some t Let  $f(xy) =f(x)f(y)$ for all $x,y\geq 0$. Show that $f(x) = x^p$ for some $p$.    I am not very experienced with proof. If we let $g(x)=\log (f(x))$ then this is the same as  $g(xy) = g(x) + g(y)$   I looked up the hint and it says let $g(x) = \log f(a^x) $   The wikipedia page for functional equations only states the form of the solutions without proof.  Attempt  Using the hint (which was like pulling a rabbit out of the hat)  Restricting the codomain $f:(0,+\infty)\rightarrow (0,+\infty)$ so that we can define the real function $g(x) =  \log f(a^x)$ and we have $$g(x+y) = g(x)+ g(y)$$   i.e $g(x) = xg(1)$ as $g(x)$ is continuous (assuming $f$ is).  Letting $\log_a f(a) = p$ we get $f(a^x) =a^p $. I do not have a rigorous argument but I think I can conclude that $f(x) = x^p$ (please fill any holes or unspecified assumptions) Different solutions are invited Named parameters in JDBC