Prove that $f$ is constant if it is entire on $\mathbb{C}$ and it does not receive any value from {$z \in \mathbb{C}: \text{Re } z>0$}
An entire function whose real part is bounded must be constant.
Is there an integer $N>0$ such that $\varphi(n) = N$ has infinitely many solutions? Let $\varphi: \mathbb{N} \to \mathbb{N}$ be the totient function.   Is there an integer $N &gt; 0$ such that there are infinitely many integers $n &gt; 0$ such that $$\varphi(n) = N?$$ How to enable sleep/standby on Windows Server 2012 Hyper V What makes you lose the Field Upgrade Score? When playing in a squad, you can get "Field Upgrades". However, sometimes when I die, it tells me that I have lost the "Field Upgrade Score".   Where can I see this score and what are the events that make me lose it? Is there a proof of existence of time? It seems to me that there is no such thing as time. There  is only movement in the universe and we compare our own movement to a different object to have a sense of time. It can be a clock or a atomic vibration.  Does this view of time work within the current framework of phsics?  Do physicists have an explanation/proof about time's existence? Show that the following series of function defines a continuous differentiable function function in $\mathbb R$. $$\sum_{n=1}^{\infty}\frac{\sin (nx^2)}{1+n^3}.$$  We have , $|f_n(x)|=\left|\frac{\sin (nx^2)}{1+n^3}\right|\le \frac{1}{1+n^3}\le \frac{1}{n^3}=M_n \text{ (say) }$  As, $\sum M_n$ is convergent so, the given series is uniformly convergent. Also , each $f_n(x)$ is a continuous function in $\mathbb R$. So, the given series converges to a continuous function , say $f(x)$.  But how we can show that $f(x)$ is differentiable function ?  Please help... Why is the Java plugin (JRE) disabled in Chrome? Why is the Java plugin (JRE) is disabled in Chrome? It is some security concern?  From official Java website:     Chrome no longer supports NPAPI (technology required for Java applets)   The Java plug-in for web browsers relies on the cross platform plugin architecture NPAPI,   which has been supported by all major web browsers for over a decade.   Google's Chrome version 45 (scheduled for release in September 2015) drops support   for NPAPI,   impacting plugins for Silverlight, Java, Facebook Video   and other similar NPAPI based plugins.   But anyone knows why? How it could be dangerous for Chrome user with latest version of Java JRE installed? what are all the open subgroups of $(\mathbb{R},+)$ How to derive $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$ indicates that the following relation is &quot;easily shown&quot;: $[x_i, F(\vec p)] = i \hbar \frac {\partial F(\vec p)}{\partial p_i}$, however I'm having some trouble showing it.  I think I'm just messing up the multivariable Taylor expansion (of $F(\vec p)$).  Can one of you walk me through it or link me to site that will?  Thanks.  Edit: Here's what I get (without using $x=i\hbar \frac \partial {\partial p}$ which I haven't proven yet): $$F(\vec p) = F(\vec 0) + \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}p_j + \frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} p_j p_k + \dots$$ so $$(x_iF(\vec p) - F(\vec p)x_i)\psi$$ $$= x_i[F(\vec 0)\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}(\nabla \psi)_j + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} (\nabla \psi)_j (\nabla \psi)_k + \dots]$$ $$- [F(\vec 0)x_i\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}(\nabla x_i\psi)_j + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} (\nabla x_i\psi)_j (\nabla x_i\psi)_k + \dots]$$ where $$(\nabla x_i\psi)_j=\frac {\partial x_i}{\partial x_j}\psi + x_i\frac {\partial \psi}{\partial x_j} = \delta_{ij}\psi + x_i\frac {\partial \psi}{\partial x_j}$$ and $$(\nabla x_i\psi)_j(\nabla x_i\psi)_k=(\delta_{ij}\psi + x_i\frac {\partial \psi}{\partial x_j})(\delta_{ik}\psi + x_i\frac {\partial \psi}{\partial x_k}) = \delta_{jk}\psi^2 + x_j \psi \frac {\partial \psi}{\partial x_k} + x_k \frac {\partial \psi}{\partial x_j} \psi + x_i^2 \frac {\partial^2 \psi}{\partial x_j \partial x_k}$$ Thus: $$(x_iF(\vec p) - F(\vec p)x_i)\psi$$ $$=  x_i[F(\vec 0)\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}\frac {\partial \psi}{\partial x_j} + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} \frac {\partial \psi}{\partial x_j} \frac {\partial \psi}{\partial x_k} + \dots]$$ $$- [F(\vec 0)x_i\psi -i\hbar \sum^3_{j=1} \frac {\partial F(\vec 0)}{\partial p_j}(\delta_{ij}\psi + x_i\frac {\partial \psi}{\partial x_j}) + \hbar^2\frac 12 \sum^3_{k=1} \sum^3_{j=1} \frac {\partial^2 F(\vec 0)}{\partial p_k \partial p_j} (\delta_{jk}\psi^2 + x_j \psi \frac {\partial \psi}{\partial x_k} + x_k \frac {\partial \psi}{\partial x_j} \psi + x_i^2 \frac {\partial^2 \psi}{\partial x_j \partial x_k}) + \dots]$$ From here it doesn't look like those higher order terms are all going to cancel out. How should I structure my URLs for both SEO and localization? When I set up a site in multiple languages, how should I set up my URLs for search engines and usability?  Let's say my site is www.example.com, and I'm translating into French and Spanish. What is best for usability and SEO?  Directory option:  http://www.example.com/sample.html http://www.example.com/fr/sample.html http://www.example.com/es/sample.html   Subdomain option:  http://www.example.com/sample.html http://fr.example.com/sample.html http://es.example.com/sample.html   Filename option:  http://www.example.com/sample.html http://www.example.com/sample.fr.html http://www.example.com/sample.es.html   Accept-Language header:  Or should I simply parse the  header and generate content server-side to suit that header?  Is there another way to do this? If the different language versions don't have different urls, what do I do about the search engines?    UPDATE 2011-12-06  Google has new recommendations for meta tags for explicitly pointing to other language content: .  UPDATE 2012-05-25  Related but not precisely:   UPDATE 2013-06-12  includes discussion of several URL schemes directly relevant to the question. What do 'real', 'user' and 'sys' mean in the output of time(1)? Is there any value in full 'beginner's kit bundles'? How can a network with only ReLU nodes output negative values? I'm trying to use an api with a feedforward neural network for time series forecasting.  For dense aggregate data it works fine, but for sparse data it sometimes forecasts negative values, even though my historical data has only positive values.   The source code is very dense, and I might be missing a line or two, but as far as I can tell, the input layers and the hidden layers are all ReLU nodes.   Assuming I am correct, how can a network with only ReLU layers lead to negative values, especially if none of the training data has negative values? What would happen if the earth would stop spinning? How much heavier would we be? I mean absolutely stop spinning. How much does the centrifugal force affect us?  If you give technical answers (please do), please explain them in laymen's terms, thanks.  Edit: I am asking what would be different if the earth were not spinning. Nevermind the part about it stopping. Frobenius Inequality Rank I was looking for an answer for this problem in terms of matrices, but I really don't know how to prove this result. The proposition says that:  Let $A\in M_{m\times k}(\mathbb{C})$, $B\in M_{k\times p}(\mathbb{C})$ and $C\in M_{p\times n}(\mathbb{C})$, then $\textrm{rank}(AB)+\textrm{rank}(BC)\leq \textrm{rank}(B)+\textrm{rank}(ABC)$ Why not group search results by thread? I like the new search, but the listing can become quite verbose. That is generally excellent because Stack Exchange is searching in more places. But now that you  to the original question, it becomes quite difficult to remember "that's not the question(s) I want &hellip; that's not the question(s) I want" when the questions' answers appear dozens of times scattered throughout the listings.  Why not group same-question entries together in the search results?  I realize that each individual post wont be sorted strictly "by relevance", but the most relevant question threads will be listed first&hellip; making it easier to skip over those entries once you decide that's not the question you're looking for.  Consider what we have now &mdash;  Apple SE > Search "Exchange contacts"   A: Exchange contacts do not appear in iCloud A: Exchange contacts do not appear in iCloud Q: iPhone Contact Sync A: How to use Gmail contacts on an iPod Touch A: Exchange contacts do not appear in iCloud A: Exchange contacts do not appear in iCloud A: How to use Gmail contacts on an iPod Touch A: Exchange contacts do not appear in iCloud A: Exchange contacts do not appear in iCloud A: iPhone Contacts Sync Q: Best way to transfer iphone contacts away from Exchange server Q: Can I use Google's new CardDAV-based contacts sync to sync between my Exchange contacts and … A: How to delete contacts from iPhone? A: Can I use Google's new CardDAV-based contacts sync to sync between my Exchange contacts and … : etc, etc. :   Once I realize I don't want that iCloud question&hellip; or that iPod Touch question&hellip; or that sync question &mdash; or a dozen others &mdash; it becomes extremely cumbersome to remember and search the listings for questions I didn't eliminate yet.  I'm not suggesting rolling up the results back to the original question&hellip; just grouping these posts together (I like seeing each answer-match with the text excerpt underneath). But once I eliminate the question entirely, it would be nice to scroll past those entries to continue my search&hellip; without all the noise of those entries reappearing over and over again in the search results. How do I give myself an item that can only be placed on a block with a certain damage value? I'm currently in the process of making a custom map and I want to add blocks that can only be placed on a certain colour clay.  This is the command I tried using.  /give @p cobblestone 1 0 {CanPlaceOn:["hardened_stained_clay"],Damage:3}   However, even with this command the cobblestone can be placed on any colour of stained clay. What command can I use for placing only on a certain block? Using CSS to affect div style inside iframe What will be the trajectory of the given motion If it is given that component of acceleration perpendicular to the velocity of a body has a constant, non-zero magnitude, how can we mathematically prove that the trajectory of the body will be circular?   I have basic knowledge of calculus, vectors and co-ordinate geometry. I can prove that the trajectory of a body in projectile motion is parabolic. I tried to follow a similar method to get an equation of circle but couldn't succeed. I tried forming parametric equations of x and y in terms of time t (the way we proceed when proving the parabolic path of a projectile) but could not reach the conclusion.  Edit: I have edited the question in order to improve it. However I still can not share my work on this question as I had asked the question almost 6 months ago and now I can not find where I tried solving it. Sincerely sorry for that. Order of the Rubik's cube group Associated to the Rubik's cube is a group as described in this : $G = \langle F, B, U, L, D, R\rangle$. For example, the element $F$ corresponds to rotating the front face clockwise by $90$ degrees.  According to the article the order of the group is: $2^{27} 3^{14} 5^3 7^2 11$.  I have come across several places (for example ) where this order has been found by just counting various ways to arrange edges and such. Is there a nice way to compute the order by appealing to more "group theoretic" tools? Why does boiling water in the microwave make a cup of tea go weird? When I boil water in the kettle, it makes a nice cup of tea.  Sometimes I need to use a microwave because a kettle isn't available.  I boil the water in the mug and it looks pretty normal, but when I drop in the teabag the water froths up and looks foamy.  I don't see what the chemical difference is here, so I assume it must be some physical difference.  I have noticed this with multiple types of tea and multiple microwaves, the results being consistent so it's not just a weird microwave or something like that.    What is the reaction here and how/why does it occur?  Here is a photo of the 'fizzy' looking tea just after dunking in the teabag.