Let f be an entire function. Suppose $\exists$ $M&gt;0$ such that   $\mid$$f(z)$$\mid$ $\leq$ $M$$\mid$$z$$\mid$ $\forall$   $z$$\in$$\Bbb{C}$. Show that $\exists$$\lambda$$\in$$\Bbb{C}$ such   that f(z)=$\lambda$$z$ $\forall$ $z$$\in$$\Bbb{C}$.   This problem is related to Liouville's theorem (which is a corollary of the cauchy inequalities) and someone told me use function g(z)=f(z)/z. But since g(z) is not 'entire' function, I don't really know how to use this function to solve this problem. How can I solve this matter?
Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r&gt;0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|&lt;M\left|z\right|^n$ for all $z \in \mathbb{C}$ with $\left|z\right|\geq r$.  Show that $f$  polynomial of degree at most $n$.  Remark: I tried to follow the proof of Liuville's Theorem but I have complications with the condition  $\left|z\right|\geq r$..
Java - Method name collision in interface implementation How do I prove that a graph is a topological manifold Geometry with Trigonometry I would like to take control of my app on an iOS device (mostly an iPhone) from my MacBook Pro or Windows, doesn't matter which one.  I know that it can be accomplished via Veency, and that's exactly what I need - but I do NOT want to jailbreak my iPhone.  Is there a similar legal alternative?  If not - is it something I could develop perhaps, or will I run into some SDK limitations for such a thing? How many fixed points in a permutation I am taking a probability course and we have been of course learning the classic problems of men with hats or variations of that in some other form. The problems being of the type if $n$ men have $n$ hats and mix them all together, whats the probability $0, 1, 2,$ or $k$ men choose their hat.   But from what I can gather is that they are all problems of the same type, for any given random permutation of $k$ objects how many fixed points are there. My professor taught us to solve this using conditional probabilities (conditioning on whether the first person matched and then generating a system of equations) and using the inclusion-exclusion principle. But is there any other way to reason this? Theorem. Let $X$ and $Y$ be sets with $X$ nonempty. Then (P) there exists an injection $f:X\rightarrow Y$ if and only if (Q) there exists a surjection $g:Y\rightarrow X$.  For the P $\implies$ Q part, I know you can get a surjection $Y\to X$ by mapping $y$ to $x$ if $y=f(x)$ for some $x\in X$ and mapping $y$ to some arbitrary $\alpha\in X$ if $y\in Y\setminus f(X)$. But I don't know about the Q $\implies$ P part.  Could someone give an elementary proof of the theorem? Probability of Various Combinations of Weather There's a 10% chance it will rain today. There's a 15% chance it will snow today.   What percentage chance is it that it will:   Both rain and snow (it can both rain and snow at the same time)? Neither rain nor snow? -   Raining and snowing are independent of each other.  EDIT: (ADDED)   Rain or snow but not both? Best way to randomize an array with .NET What is the best way to randomize an array of strings with .NET? My array contains about 500 strings and I'd like to create a new Array with the same strings but in a random order.  Please include a C# example in your answer. Local file access with JavaScript Is there local file manipulation that's been done with JavaScript? I'm looking for a solution that can be accomplished with no install footprint like requiring .  Specifically, I'd like to read the contents from a file and write those contents to another file. At this point I'm not worried about gaining permissions and am just assuming I already have full permissions to these files. How to use "text" as a verb How to solve "sudo: /etc/sudoers.d is world writable" Is stats maths or not?   Given that it's all numbers, mostly taught by maths departments and you get maths credits for it, I wonder whether people just mean it half-jokingly when they say it, like saying it's a minor part of maths, or just applied maths.   I wonder if something like statistics, where you can't build everything on basic axioms can be considered maths. For example, the $p$-value, which is a concept that arose to make sense of data, but it's not a logical consequence of more basic principles. Below is experimental data for the ratio $$R=\frac{\sigma(e^+e^-\rightarrow hadrons)}{\sigma(e^+e^-\rightarrow\mu^+\mu^-)}$$ as a function of the centre of mass energy $\sqrt s$. I am interested in the peak at around $100GeV$ which corresponds to the resonance of the $Z^0$ boson. There are two ways of looking at this that I have in mind:  We can effectively ignore the electromagnetic process around this energy, and so each process should have it's own Breit-Wigner peak centred on the mass energy of the $Z^0$. The ratio of these is just a constant. Thinking in terms of Feynman rules and again ignoring the electromagnetic process around this region, each process has the same propagator and vertex factors (roughly - ignore quark mixing) and there are some extra factors due to different quarks being possible and colour degeneracy, but still the ratio should be a constant (one possible issue here is interference between Feynman diagrams which I have neglected?).  So my question is why does the peak exist in the data? Selectively disable iCloud keychain syncing for WiFi passwords I have iCloud Keychain sync enabled, but I want to stop it syncing WiFi passwords.  I have a portable WiFi hotspot which I use for my laptop and iPad, but I don't want my iPhone to connect to it since it has it's own 3G  Because iCloud is syncing the settings, within about 30 seconds of connecting my laptop to the 3G hotspot, my iPhone will connect itself. If I go into wifi settings on the iPhone and manually disconnect, about 30 seconds later my laptop will disconnect too. It seems with iCloud wifi sync on, I can’t connect my laptop to WiFi without my phone also connecting.  The only way to get around it seems to be disabling Keychain sync in iCloud. I’ve tried Googling, but got no real help on how to disable it without turning off Keychain, which is what I’ve done at the moment.  I also don't want to enable 'Ask before joining networks' on my iPhone, since I'll be manually re-connecting it constantly. Do I need a Kuwait transit visa as an Indian national? I am an Indian national and passport holder, traveling from New Delhi to Frankfurt (via Kuwait) and back with Kuwait Airways. I see no Terminal change for the onward and return journeys. I have no plans to leave the airport due to short layovers.  Do I need a Kuwait transit visa? I need to extrude a circular shape from a surface that's basically just a grid of quads. What's the best practice to do this without screwing up the topology too much?  I basically want to have a circular face I can extrude, in order to have a cylinder like protuberance on the object, something like this: What are the differences between Holmes, Moldvay, and Mentzer D&D? Can you teleport farther away with misty step if you have a double from Invoke Duplicity? The Channel Divinity feature says:     you can cast spells as though you were in the illusion's space, but   you must use your own senses. (PHB 63)   If you were where your duplicate is, you could step to a space 30ft from its position. So if it is 60ft from you, you could teleport to a space 90ft from your actual position. But the spell says that:     you teleport up to 30 feet (PHB 260)   This would put a hard limit on the distance you can travel.  Which rule takes precedence here? Can I attack a specific player? I was in a clan and someone kicked me from it.   I think I'm stronger than him. Is there any way I can attack him? The only method I see to attack is random players. Can I attack a specific player? How to Implement Page Object and Page Factory Pattern in Selenium Webdriver? I am a beginner in learning selenium Webdriver. I am not able to understand how page object and factory models works and how it can be beneficial?    Kindly help me explain it with detailed explanation.