Prove that a function which is analytic in the whole plane and satisfies the inequality $|f(z)|&lt;|z|^n$ for some $n$ and all sufficiently large $|z|$ reduces to a polynomial.   The function is analytic, so $f^{n}(z)$ exists for all $n$, all $z$. We have the Cauchy's integral formula for higher derivatives  $$f^{(n)}(a)=\frac{n!}{2\pi i}\int_C\frac{f(z)}{(z-a)^{n+1}}dz.$$  So $$\int_C\frac{f(z)}{(z-a)^{n+1}}dz&lt;|z|^n$$ for all large $|z|$. How does that help?
Suppose that an entire function $f(z)$ satisfies $\left|f(z)\right|\leq k\left|z\right|^n$ for sufficiently large $\left|z\right|$, where $n\in\mathbb{Z^+}$ and $k&gt;0$ is constant. Show that $f$ is a polynomial of degree at most $n$.
I was recently reading some Lord of the Rings/Stargate SG-1 crossover fanfiction. It is a guilty pleasure, I admit :) In this story, it was suggested by Col. O'Neill to destroy the Ring using a nuclear device. Truth be told I stopped reading at this point, as it was getting to be a little much even for me.  Obviously, this is not canonical in either universe, but it made me wonder: When Elrond said that "only in the fires of Mt. Doom may it be unmade," was he speaking literally? Or was this merely the only known active volcano in Middle Earth? In other words, was whatever magic that protected the One Ring from damage location-based or did it just make it very resistant to destructive force, and Mt. Doom just happened to be hot enough to do this? I know there is (understandably) little information about such technicalities of how magic works in Middle Earth, but I'll accept any reasonable conjecture/consensus as an answer. Origin of the dot and cross product? Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product equation as a determinant of a carefully-constructed matrix,   but what I want to ask is... How did the dot and cross product come to be? When were they "invented"? Some detailed proofs? Did someone say: "Hey, wouldn't it be nice if we could construct a way to calculate a vector that is perpendicular to two given operands?"  Basically, how/why do they work?  I would appreciate explanations, links to other explanations, other web resources... I've been searching the Internet lately for explanations, but most of them are on how to use it and nothing that really gives substance to it. Old-style/Antique typesetting in LaTeX/TeX Adding a text node to tikz drawing How can I get a significant overall ANOVA but no significant pairwise differences with Tukey's procedure? I performed with R an ANOVA and I got significant differences. However when checking which pairs were significantly different using the Tukey's procedure I did not get any of them. How can this be possible?  Here is the code:  fit5_snow&lt;- lm(Response ~ Stimulus, data=audio_snow) anova(fit5_snow)  &gt; anova(fit5_snow) Analysis of Variance Table  Response: Response           Df Sum Sq Mean Sq F value  Pr(&gt;F)   Stimulus   5  73.79 14.7578  2.6308 0.02929 * Residuals 84 471.20  5.6095                   --- Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1   df&lt;-df.residual(fit5_snow) MSerror&lt;-deviance(fit5_snow)/df  comparison &lt;-  HSD.test(audio_snow$Response, audio_snow$Stimulus, df, MSerror, group=FALSE)  &gt; comparison &lt;-  HSD.test(audio_snow$Response, audio_snow$Stimulus, df, MSerror, group=FALSE)  Study:  HSD Test for audio_snow$Response   Mean Square Error:  5.609524   audio_snow$Stimulus,  means                  audio_snow.Response   std.err replication snow_dry_leaves            4.933333 0.6208034          15 snow_gravel                6.866667 0.5679258          15 snow_metal                 6.333333 0.5662463          15 snow_sand                  6.733333 0.5114561          15 snow_snow                  7.333333 0.5989409          15 snow_wood                  5.066667 0.7713110          15  alpha: 0.05 ; Df Error: 84  Critical Value of Studentized Range: 4.124617   Comparison between treatments means                                Difference   pvalue sig        LCL      UCL snow_gravel - snow_dry_leaves  1.9333333 0.232848     -0.5889913 4.455658 snow_metal - snow_dry_leaves   1.4000000 0.588616     -1.1223246 3.922325 snow_sand - snow_dry_leaves    1.8000000 0.307012     -0.7223246 4.322325 snow_snow - snow_dry_leaves    2.4000000 0.071587   . -0.1223246 4.922325 snow_wood - snow_dry_leaves    0.1333333 0.999987     -2.3889913 2.655658 snow_gravel - snow_metal       0.5333333 0.989528     -1.9889913 3.055658 snow_gravel - snow_sand        0.1333333 0.999987     -2.3889913 2.655658 snow_snow - snow_gravel        0.4666667 0.994348     -2.0556579 2.988991 snow_gravel - snow_wood        1.8000000 0.307012     -0.7223246 4.322325 snow_sand - snow_metal         0.4000000 0.997266     -2.1223246 2.922325 snow_snow - snow_metal         1.0000000 0.855987     -1.5223246 3.522325 snow_metal - snow_wood         1.2666667 0.687424     -1.2556579 3.788991 snow_snow - snow_sand          0.6000000 0.982179     -1.9223246 3.122325 snow_sand - snow_wood          1.6666667 0.393171     -0.8556579 4.188991 snow_snow - snow_wood          2.2666667 0.103505     -0.2556579 4.788991 What causes a black-body radiation curve to be continuous? The  (unlike the quantized emission seen from atomic spectra), is continuous over all frequencies.  Many objects approximate ideal blackbodies and have radiation curves very similar in shape and continuity to that of an ideal black-body (often minus some emission and absorption lines from the atoms in an object, such as radiation curves seen from stars).  I am wondering what exactly gives rise to a basically continuous black-body radiation curve in real objects? Since atomic energy states are quantized, it seems real life black-body curves would have some degree of measurable quantization to them (or perhaps the degree of quantization is so small the radiation curves look continuous). \overbrace and \underbrace with square bracket In $\mathbb{Z}\left [ x \right ]$, let $I = \left \{ f\left ( x \right ) \in \mathbb{Z}\left [ x \right ]: f\left ( 0 \right ) \text{ is an even integer} \right \}$. Is $I$ a Prime ideal of $\mathbb{Z}\left [ x \right ]$?  Let $f_{1}\left ( x \right ),f_{2}\left ( x \right ) \in \mathbb{Z}\left [ x \right ].$ Let $f_{1}\left ( x \right )f_{2}\left ( x \right ) \in I$. Clearly, I is a proper ideal of $\mathbb{Z}\left [ x \right ]$ since the elements in $\mathbb{Z}\left [ x \right ]$ evaluated at 0 can produce an odd integer but this cannot be true in I. Now, $f_{1}\left ( 0 \right )f_{2}\left ( 0 \right )$ is an even integer in I. Hence, either $f_{1}\left ( 0 \right )$ has an even constant integer or $f_{2}\left ( 0 \right )$ has an even constant integer. Thus, I is a prime ideal by definition. Is this correct? I would now like to show if I is a maximal ideal. Again, clearly, I is a proper ideal by the above argument. Let B be an ideal of $\mathbb{Z}\left [ x \right ]$ and I $\subseteq B\subseteq \mathbb{Z}\left [ x \right ]$. How should B be defined so that I would conclude either $B=I$ or $B=\mathbb{Z}[x]$? Centrifugal force acting on ring Take for example the case of a rod rotating about an axis passing through its centre of mass and perpendicular to it. It has a ring hung from one of its sides. The rotation of the rod causes the ring to move out wards and ultimately fall off the rod.   I asked my teacher what causes the ring to move out wards and he said the centrifugal force acting in the frame of reference of the ring causes motion. What exactly is the ring reference frame? Is it a rotating frame? How would u visualize motion in this frame of reference? Are confidence intervals open or closed intervals? How can I get around giving my mobile phone number to Google? Every time I open a folder from the desktop, a prompt is shown that the said folder is opened. How do I stop that? Ubuntu 18.10: visual artifacts (black bar) in dock if extend-height is disabled I am using the default Ubuntu dock with bottom placement and org.gnome.shell.extensions.dash-to-dock extend-height set to false. With Ubuntu 18.10 this produces a noticeable visual glitch, note the black empty spaces at the edges of the dock:    I am getting this both on an upgraded system and on a clean install inside a VM. This was working fine in Ubuntu 18.04. Also happens if the theme is set to something else than Yaru (the default theme in Ubuntu 18.10). Reboot does not help. Dock looks OK if extend-height is enabled.  Any ideas on how to fix this? Difference between "I have got" and "I have gotten" I see these two expressions are used almost identically in different contexts. Is there a difference between I have got and I have gotten? How do I update my graphics drivers? I am using a laptop and ubuntu 11.04. I would like the easiest way possible because I am new to linux.  here is my display information:  *-display              description: VGA compatible controller              product: Core Processor Integrated Graphics Controller              vendor: Intel Corporation              physical id: 2              bus info: pci@0000:00:02.0              version: 18              width: 64 bits              clock: 33MHz              capabilities: msi pm vga_controller bus_master cap_list rom              configuration: driver=i915 latency=0              resources: irq:41 memory:d0000000-d03fffff memory:c0000000-cfffffff ioport:e080(size=8)         *-communication UNCLAIMED              description: Communication controller Overall reputation scores are headed nowhere but up.  Pretty soon there will be thousands of mods.  What to do?  Thoughts on possibilities:   Thousands of mods is fine.  Carry on. Adjust the threshold values for high-level SO privileges. Do a straight-up inflation adjustment.  You had x rep, then at the turn of the year we applied the Consumer Rep Index to all scores, now you have (x / inflation) rep. Implement a rep decay mechanism, so that inactivity sheds some of the overall rep that's built up. Issue when promoting a second server as domain on R2 2008 I'm in an odd situation here.  I've upgraded my network to a 2008 domain from a 2003 with out any issue, this works fine and all the FSMO are in one DC,  I was about to install a second domain on a 2008R2 and this error is coming up again and again.  I The problem is I keep getting an area telling me that I need to run adprep /domainPrep /forestPrep, but when I run it on the first DC, it tells me this has already been run and updated and it is aborted. Does anyone have any ideas on why I can't add a Server 2008 machine as a domain controller? Were there any improvements to the second Death Star (other than the change in exhaust port) that we know of? Problem with babel and tikz using \draw I'm trying the next code:  \documentclass[pdftex,spanish]{article} \usepackage{babel} \usepackage{tikz}  \begin{document}  Simple arrow:   \begin{tikzpicture}     \draw[-&gt;] (-1.5,0) -- (1.5,0); \end{tikzpicture}  \end{document}   but it doesn't work. It throws the errors:    ! Argument of \language@active@arg&gt; has an extra }. ! Paragraph ended before \language@active@arg&gt; was complete.&lt;to be read again&gt;\par \draw[-&gt;]   What can I do if it works right without the babel package? About the anime version of Air:  Some sources on the net state that Kanna's curses were broken, and her next reincarnation would be a happy one.  Some say that the pair of children playing on the beach were the reincarnations of Misuzu-Kanna / Yukito-Ryūya.  What is the canonical explanation of this ending? Did the curses really get lifted? Please include sources in your answer.