Clearly $f$ has zeroes everywhere $g$ does, and since $|f/g| &lt; 1$ in a deleted neighborhood of each of $g$s zeroes any singularities are at worst removable, but I haven't got a clue where to go from there.
Suppose $f$ and $g$ are entire functions, and $|f(z)| \leq |g(z)|$ for all $z \in \mathbb{C}$, Prove that $f(z)=cg(z)$.   My try :  I consider $h(z)=\frac{f(z)}{g(z)}$. If I prove that $h(z)$ is entire, using the fact that $|h(z)|\leq 1$, the result follows immediately from Liouville's theorem.  To prove that $h(z)$ is entire, I have to prove that $h(z)$ has removable singularities at possible zeros of $g(z)$.  Suppose $g(z_0)=0 \Rightarrow |g(z_0)|=0 \Rightarrow |f(z_0)|=0$, BUT what if $z_0$ be a root of of order $k$ of $g(z)$ (i.e. $f^{(n)}(z_0) = 0 $ for all $ 0 \leq n \leq k-1$) and be a zero of order $m$ of $f(z)$ and $k&gt;m$ ?  And if that's not the case, how to rigorously show that $z_0$ is a removable singularity ?  Thank you all in advance.
I created a simple service layer polygon in network analyst (ArcGIS Desktop 10.2) with no hierarchies set in the street network. Now I don't understand why is this polygon created along some particular streets?   Why isn't is properly spread out? For example, why doesn't the polygon cover the areas/streets I have marked in red? Obviously those marked areas are also within 1 km.  The impedance is 1 km and 2 km respectively.   The streets are also well connected individual line elements as you can see the junctions. There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (given in the problem). There are 'n' candies and 't' boxes. Find the number of ways to place the candies in the boxes for each of the conditions (all the candies must be spread out) : (a) candies and boxes are different; (b) candies of the same diiferent boxes should not be empty cartons: (c) candies equally new, boxes are different; Edit :(d) candies are different, the boxes are the same, there should be no empty boxes; (e) candies of different boxes alike.(Edit : candies are different, boxes are equal) Specify the display type that matches the placement, if possible. My Answers : (a) Each layout is encoded with a word of $n$ letters from the alphabet of 't' letters $\implies$ possible $n^t$ variants. (b) Write, $n$ candies in the form of balls ina line, we need to put $(t-1)$ partition in $(n-1)$ place, but we can't put two partitions on one place, so we get : $^{t-1}C_{n-1}$. (c) First we need to choose a candy in a box (in the first box $n$ ways, in the second box $(n-1)$, $\cdots$ in the $t-th$ : $(n-t+1)$ ways $\implies$ total $n!/(n-t)!$ methods, and then we distribute the remaining $(n-t)$ candies into $t$ boxes, this is encoded with owrds from the alphabets (for each candy $t$ variants) $\implies$ $t^{n-t}$. The Answer is : $\frac {n!}{(n-t)!}*t^(n-t)$ Are my answers (a), (b) &amp; (c) are correct? For (d) &amp; (e) I don't know how to proceed? Please help me. I was reading the Math.random() javadoc and saw that random is only psuedorandom.    Is there a library (specifically java) that generates random numbers according to random variables like environmental temperature, CPU temperature/voltage, or anything like that? When NASA sends astronauts to space, they have to wear uncomfortable suits. Why aren't the humans in Star Wars spacecraft doing the same thing? What technology do they have on board that remove this need? Show that $\textbf{ZF}$ shows $\phi$ is $\Delta_1$ Is there a word for someone who somewhat delights in their illnesses/medications? Family of irreducible polynomials Where should I copy-paste-install software to? I'm installing IntelliJ, and the readme tells me to choose an installation folder. Where is a good, standard place to put installed programs? The search parameter isanswered ignores acceptances The SE definition of "answered" question is: has an answer that is either accepted or has positive score. But the search parameter isanswered appears to ignore the "accepted" clause: when set to "no" it still returns questions with an accepted answer, as long as it has 0 score.    to reproduce the bug (unless meta effect kicks in and the answers get upvotes).     By the way, it seems that isanswered works only when is:question is present, but I'm not complaining about that... yet.   Also, this parameter is not listed on the , but I'm not complaining about that either... yet. How can I clone an ArrayList and also clone its items in Java?  For example I have:  ArrayList&lt;Dog&gt; dogs = getDogs(); ArrayList&lt;Dog&gt; clonedList = ....something to do with dogs....   And I would expect that objects in clonedList are not the same as in dogs list. I'm new to regular expressions and would appreciate your help.  I'm trying to put together an expression that will split the example string using all spaces that are not surrounded by single or double quotes. My last attempt looks like this: (?!") and isn't quite working. It's splitting on the space before the quote.  Example input:   This is a string that "will be" highlighted when your 'regular expression' matches something.   Desired output:  This is a string that will be highlighted when your regular expression matches something.   Note that "will be" and 'regular expression' retain the space between the words. Is clockwise or counter-clockwise rotated text easier to read? When rotating text with css, should it be rotated clockwise or counter-clockwise?  Is one more readable than the other?  Counter Clockwise:   &nbsp;&nbsp;&nbsp;&nbsp;&mdash;&nbsp;&nbsp;&nbsp; Clockwise: Kaplansky's theorem of infinitely many right inverses in monoids? There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, and then showing that the elements $v+(1-vu)u^n$ are right inverses for all $n$ and distinct.  To see they're distinct, I suppose $v+(1-vu)u^n=v+(1-vu)u^m$ for distinct $n$ and $m$. I suppose $n&gt;m$. Since $u$ is cancellable on the right, this implies $(1-vu)u^{n-m}=1-vu$. Then $(1-vu)u^{n-m-1}u+vu=((1-vu)u^{n-m-1}+v)u=1$, so $u$ has a left inverse, but then $u$ would be a unit, and hence have only one right inverse.  Does the same theorem hold in monoids, or is there some counterexample? Backup GNOME-Terminal Why is it “a defeated Napoleon, not “the defeated Napoleon” who rode off the battlefield and into exile? I noticed an infinite article; ‘a’ was used before ‘defeated Napoleon’ in the following sentence of Jeffery Archer’s novel, "Fales Impression" :  “General Sir Harry Wentworth sat at the right hand of the Iron Duke that night, and was commanding his left flank when a defeated Napoleon rode off the battlefield and into exile."  I’m curious to know why it should be “a defeated Napoleon,” not “the defeated Napoleon” or simply “defeated Napoleon.” Isn’t it only one Napoleon who was defeated the battle and exiled (to St. Helena). It looks like as though there were many (defeated) Napoleons who were exiled.  It’s always headache for me to handle articles as we Japanese (I think Chinese too) don’t have article as the basic part of speeches in our language system. I’ve studied it on English grammar text books at school, but have never reached full understanding of the usage of articles, because we don’t have that practice. Is there any special knack to master how to use articles? Catch Regex too complicated when splitting a string in salesforce I have created a function wherein I split a string from the csv filecsvAsString.split('\n') . It is working when the string is not so big. But when I try to split a very large string, an error will occur      regex too complicated   what I want to do is to catch if an error occur during the split process. How will I do it ? I've tried to use try and catch but the problem still occur. Is there any other possible solution? Are there any versions of t-SNE for streaming data? My understanding of  and the Barnes-Hut approximation is that all data points are required so that all force interactions can be calculated at the same time and each point can be adjusted in the 2d (or lower dimensional) map.  Are there any versions of t-sne that can efficiently deal with streaming data? So if my observations are arriving one at a time, it will find the best location on the 2d map to place the new observation, or continuously update all points on the 2d map to account for ht new observation.  Would this even make sense or does it go against the setup of t-sne. If every compact set is closed, then is the space Hausdorff? PHP Echo Line Breaks Is there a way to refresh Notepad? Say, for example, that there's one process that's writing to a file. While the process is running, I opened the file in notepad. The process keeps  writing to the file.   Other than closing and reopening the file, is there any way for me to "refresh" the data that notepad is showing?