How to reject a PhD Scholarship offer after accepting it?
Rejecting a PhD offer after having accepted it. Advice?
Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? increasing impedance of headphones I have headphones with differential input and impedance of 8 ohms. I need to drive this headphone with some DAC with built in headphone driver (Currently I have chosen ADAU1772). DAC's datasheet mentions that the minimum impedance of headphones it supports is 16 ohms. Can I increase impedance of my headphones so that I can drive them through this chip? What will happen if I drive this 8 ohm headphone through a chip supporting minimum impedance of 16 Ohms? Example code required for how to access embedded .Net image resources in C# It's very easy to mark an image file to become an embedded resource however how does one access the image thereafter. Please can I have some example code? Is Using Digital Signatures to prove identity a zero knowledge proof? Suppose Alice publishes a book with a public key in it, and later wants to prove that she wrote the book. She could sign challenge messages with her private key, and others could verify those signed messages. Alice is proving she knows a private key without revealing anything about it.  Is this an example of an interactive zero knowledge proof?   Or does this example somehow not apply to zero knowledge proofs? Can one preserve food by periodically heating it? I get that keeping food temperature low will slow down chemical reactions and preserve food, but sometimes my sister preserves food by heating it every 6 hours or so (up to 1-2 days). Does this help preserving the food? If yes, how does it work?  If it matters, the food currently in question is , but I would appreciate a general explanation. In case anyone is wondering, she doesn't store it in the fridge because it is full. Under what assumptions can a semigroup $(S,*)$ be embedded into a group? I'm Australian and have obtained my Canadian working visa. However rather than flying direct to Canada I am meeting an American friend in LA and driving across the border. I have a return flight booked back to Australia from LA in March of next year too. Will the esta visa will be enough to enter the USA because I have my Canadian working visa and plan to travel there? Or will there be a problem because my return flight is booked later then the 90 days allowed on the visa. I have a Google Apps email (me@business.com) set to auto-forward all mail to my personal Gmail (me@gmail.com).  I have added the me@business.com address to my personal Gmail so I can send mail as me@business.com from within my personal email account.  It's a fairly common way to consolidate email accounts.  Here's my problem.  I want me@gmail.com to filter all emails being forwarded from me@business.com and apply the Business label.  So I tried it two different ways:  A) Use an Alias:   me@business.com forward to me+business@gmail.com Create filter "To:me+business@gmail.com" in me@gmail account Have filter apply Business label   B) Use Googlemail.com address:   me@business.com forward to me@googlemail.com Create filter "To:me@googlemail.com" in me@gmail account Have filter apply Business label   However, neither of these filters work.  I've tested it using a third email address, me@yahoo.com.  Whenever I send an email from me@yahoo.com to me@business.com, the email is properly forwarded to me@gmail.com, and I can reply to me@yahoo.com using the me@business.com address.  But Gmail does NOT apply the Business label.  This is case using either filtering method A or B.  But if I send an email from me@yahoo.com DIRECTLY to me+business@gmail.com or me@googlemail.com, the filter works properly and applies the label.  So there must be something going on with the forwarding of me@business.com to me@gmail.com.  Does anyone know how to get this working? I've been going through  and have a question on the inductive proof for the . Recurrent relation:  $J(1) = 1$ $J(2n) = 2J(n) - 1$ $J(2n+1) = 2J(n) + 1$  Closed form hypothesis:  $J(2^m + l) = 2l + 1$, for $m \ge 0$ and $0\le l&lt;2^m$  Inductive Proof (for the even case):  $J(1)=1$ $J(2^m+l)= 2J(2^{m-1}+l/2)-1=2(2l/2+1)-1=2l+1$  How do the authors get from $2J(2^{m-1}+l/2)-1$ to $2(2l/2+1)-1$? If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix?  NB: I am making this question to provide a unified answer to questions about eigenvalues of small matrices so that all of the specific examples that come up can be marked as duplicates of this post. See . Find a formula for a sequence $\{\sqrt{3},\sqrt{3\sqrt{3}},\sqrt{3\sqrt{3\sqrt{3}}},...\}$ How to resolve ambiguous column names when retrieving results? Someone mentioned it in the IRC as the slicing problem. Circular determinant problem I'm stuck in this question: How calculate this determinant ? $$\Delta=\left|\begin{array}{cccccc} 1&amp;2&amp;3&amp;\cdots&amp;\cdots&amp;n\\ n&amp;1&amp;2&amp;\cdots&amp;\cdots&amp; n-1\\ n-1&amp;n&amp;1&amp;\cdots&amp;\cdots&amp;n-2\\ \vdots &amp;\ddots &amp; \ddots&amp;\ddots&amp;&amp;\vdots\\ \vdots &amp;\ddots &amp; \ddots&amp;\ddots&amp;\ddots&amp;\vdots\\ 2&amp;3&amp;4&amp;\cdots&amp;\cdots&amp;1 \end{array}\right|$$ Thanks a lot. How can I draw simple trees in LaTeX? I need to draw a tree, and constantly using dotty and including the PNGs is starting to be a pain. Is there a way to get LaTeX to draw it's own trees that doesn't involve learning an entirely new language like TikZ? Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian I've been working on this problem listed in Herstein's Topics in Algebra (Chapter 2.3, problem 4):     If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in G$, show that $G$ is abelian.   I managed to prove it, but I'm not very happy with my result (I think there's a neater way to prove this). Anyway, I'm just looking to see if there's a different approach to this.   My approach:  Let $j=i+1, k=i+2$ for some $i\in \mathbb{Z}$.  Then we have that $(ab)^i = a^ib^i$, $(ab)^j = a^jb^j$ and $(ab)^k = a^kb^k$.  If $(ab)^k = a^kb^k$, then $a^jb^jab =a^jab^jb$.   We cancel on the left and right and we have $b^ja = ab^j$, that is $b^iba = ab^j$.  Multiply both sides by $a^i$ on the left and we get $a^ib^iba = a^jb^j$, so $(ab)^iba = (ab)^j$. But that is $(ab)^iba = (ab)^iab$.   Cancelling on the left yields $ab=ba$, which holds for all $a,b \in G$, and therefore, $G$ is abelian.  Thanks! Sources for common laboratory Saccharomyces strains? I'm used to working with bacteria - some of the more common laboratory strains sometimes come with an order from some vendors.  What's the most convenient source to order  for lab work? Give moderators the ability to mark reviews as "bad" From what I've seen, many SO moderators trawl completed reviews and watch out for robo reviewers; and manual banning them when they see enough (>= 3?) bad reviews. This is inefficient because:   It doesn't integrate into the audit system. There's no "memory" to it. If mod A finds that user X has made two (any number less than the threshhold) bad reviews, he may make a mental note but there probably won't be any permanent/shared memory of this. If moderator B finds this user at a later date, he will have to look through the entire history, including stuff that mod A has already checked.    What I propose is that moderators get the ability to mark individual reviews as "invalid" from the completed review page. This is only to be used for reviews where any reviewer who is paying attention (and knows the rules) would choose the opposite reason (spam and other low quality stuff).   These bad review flags are treated like audits for a user -- if the number of failed audits and bad reviews is enough, the user gets autobanned. The nice thing is that this system should (probably) easily be able to , fixing the "Why was I manual banned?" problem.  This could also be integrated well with . How to validate an email address in JavaScript What part of speech is handsome in the sentence:     The bridegroom looked so handsome.