Can there exist a holomorphic function defined on the unit disk $D = \{ z \mid |z| &lt;1\}$ such that $f(\frac1n) = \frac{1}{2^n}$ where $n \in \Bbb Z$?  I thought of using CR equations to find a contradiction but unable to do.
There does not exist an entire function which satisfies $f({1\over n})={1\over 2^n}$, what I tried is if possible then define $g(z)=f(z)-{1\over 2^{1\over z}}$ Then $g({1\over n})=0$ and so $g(z)$ is entire and its $0$ set has limit point in it and so $f(z)={1\over 2^{1\over z}}$ which is not analytic at $0$? Please help!  Edit: OOps! the way I defined $g(z)$ that is not entire! could any one give me hint?
Say we have a rod with a fixed axis of rotation. If we give it a push at some point other than the axis, it will start to rotate. Say our force is $F$, then we might write: $\Delta P=F\Delta t$. I’m wondering how to incorporate angular momentum here. We know that for a single particle it holds that $L=r\times p$. And for an extended object with a fixed axis of rotation, we generally might write $L=I\omega$. However, can I write $L=r\times p$ too for the extended object? I would think that I would need to integrate. So we have $p_\alpha=m_\alpha v_\alpha=m_\alpha r_\alpha\omega$, where we consider an infinitesimal mass $\alpha$. So should I write $L=\int r^2\omega\,dm$? I’m confused whether to work with this integral, or just use $L=r\Delta P$ at once.  The reason I'm asking, is because I'm trying to calculate the sweet spot of a baseball bat. So this is my attempt at solving the exercise. Is Dragonstone unoccupied after Stannis Baratheon's defeat? Is anything known about Dragonstone after:      Stannis Baratheon is killed in his bid for the Iron Throne?   As the seat of the naval defense of King's Landing and the birthplace of Daenerys Targaryen, she has a vested interest in taking it.   At the conclusion of Season 6 Episode 10, is Dragonstone unoccupied? So far I've been using public void run() {} methods to execute my code in Java. When/why might one want to use main() or init() instead of run()? Detect when browser receives file download Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$ Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$.   This is my proof to show that a ring is commutative if it has the property that ab = ca implies b = c.  We need to show that if x, y ∈ R then xy = yx. Let a = x, b = yx and c = xy. Then ab = x(yx) = (xy)x = ca. Thus, by the hypothesis, b = c, or xy = yx. Thus, for x, y ∈ R we have xy = yx.   How would I show when $a\neq 0$? Here are some options I thought of, not sure which is the right one.   There was an I/O error reading from the pipe. The process writing to the other end of the pipe died with a failure. All processes who could write to the pipe have closed it. The write buffer of the pipe is full. The peer has closed the other direction of the duplex pipe. Writing failed because there are no processes which could read from the pipe. A system call returned the EPIPE error, and there was no error handler installed. counting triangles in a graph or its complement Given a simple graph $G$ [no loops or parallel edges] on six vertices,  let $G^c$ denote its complement. It is known that either $G$ or $G^c$ must contain a triangle  $T$ in it. [An example of a Ramsey number I think.]   My question came up when I tried to find such a graph on six vertices  which had only one $T$ in it or its complement.  I believe if I did things right there are none of these. This I tried to check by drawing a copy of each graph $G$  on six vertices which has a $T$ in it,  and then looking at the complements to see if any had  no $T$'s in them. I found none.  One way to pose my question for a general number $n$ of vertices  is then as follows: Consider the set of all graphs $G$ on $n$ vertices,  and for each one construct $G^c.$  Count the number of $T$ in $G$ and add that to the number of $T$ in $G^c.$  Finally find the minimal such sum for the given $n$  and call that say $M(n).$ So my above six vertex conjecture  is that $M(6)=2.$   By considering a cycle graph on five vertices we find $M(5)=0.$   So going further one could ask if anything is known  about this minimum function $M(n)$ for larger $n$ [such as bounds, etc.] And does it have a name?  I would also like to see a simpler way to handle the six vertex case, one not involving drawing all the graphs. Community Wiki posts don't change usernames if the user does Reading Excel files from C# Provide a means to reward users and their answers (decoupled from bounties) How do I update Android on my device? Why XML-Serializable class need a parameterless constructor Smaller parskip than half for KOMA-Script? I am using the documentclass scrbook. To mark paragraphs I need both vertical spacing and indentation.  I know that KOMA-Script supports the option parskip or halfparskip, however, this disables the indentation. In order to get it back I declared \setlength{\parindent}{1.5em} in the preamble.  Since I am using both vertical spacing and indentation, I would like to have even smaller spacing than halfparskip. Is it possible to get an even smaller parskip than half in KOMA-Script? Question:   Prove that a norm is always non-negative.   Assume that the norm is negative.   Define: $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{-}_{0}$  $v \mapsto \left ( \vec{v} \right )f=\left \| \vec{v} \right \|$  $\left ( \vec{u}+\vec{v} \right )f=\left \| \vec{u}+\vec{v} \right \| \in \mathbb{R}^{-}_{0}$ and  $\left ( \vec{u} \right )f=\left \| \vec{u} \right \|,\left ( \vec{v} \right )f=\left \| \vec{v} \right \|=\left \| \vec{v} \right \| \in \mathbb{R}^{-}_{0}$  At this point I am unable to progress further.  Any help is appreciated. NULL values inside NOT IN clause This issue came up when I got different records counts for what I thought were identical queries one using a not in where constraint and the other a left join. The table in the not in constraint had one null value (bad data) which caused that query to return a count of 0 records. I sort of understand why but I could use some help fully grasping the concept.  To state it simply, why does query A return a result but B doesn't?  A: select 'true' where 3 in (1, 2, 3, null) B: select 'true' where 3 not in (1, 2, null)   This was on SQL Server 2005. I also found that calling set ansi_nulls off causes B to return a result. Why is the time limit for editing comments 5 minutes? What is the reasoning behind this? What does "at 19.45 hours" mean? I was reading a book and this sentence seemed a little odd to me:     At approximately 19.45 hours the two men rose, stretched and yawned. They picked up their gear and stood at the door, ...   What does the first part mean? If it's the exact time, shouldn't it be 19:45? And if it's duration, shouldn't it be "After 19.45 hours"?  (The story here happens at an airbase, if it's of any relevance)    Edit: The book I've quoted is this:   by John Creed.   The author is Irish and the text is set in northern England/Ireland. Proof of triangle inequality I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :) What's the word for the when you suck snot back up your nose? My mom and I say 'soup' like: "Why are you souping the snot back up your nose?" But I realise that this is not accurate. So what's the word? I have a batch file which is in a directory and must be run from there as well because it updates files within this directory. This works perfectly fine, except when the user runs the batch file as administrator (required on Vista). Then the starting directory is C:\Windows\System32. Is there any way to still be able to know from which directory the batch file was run? I dont want the user to enter the directory manually.