What's the word when you learn something good unintentionally accidentally along the way stuff?
Word or phrase for mere coincidence that brings happiness
My friend installed Windows 7 on his computer (upgrading from XP).  He wasn't sure if his processor supported x64 or not, so he installed the x86 version, just to be on the safe side.  After I looked it up, it turns out his processor does support x64, and now we want to upgrade.  The question is: is it possible to upgrade without completely re-installing the OS?  Can I? Or do I have to format the C: partition again? plist or sqlite making low-quality PDF files clearer Some research papers available on the web have low-quality outputs. For example, see . It seems that it was converted from a PS file and typeset in (La)TeX. However, it has all the font information. Is there any way to make the PDF file clearer? Is it possible to create matter? Is it possible to create matter? In a recent discussion I had, it was suggested that with enough energy in the future, "particles" could be created.   It seems like this shouldn't be possible due to conservation but perhaps I could be wrong. Would any of you Physics masters care to elaborate?  (Note... I will understand the basics, I am by no means an expert.) I was trying to give the wording "isomorphic groups have the same structure" a precise interpretation, at least for the finite order case. A positive answer to my final question would reach the goal, as it would translate a "$\cong$" between groups into a "$=$" between sets.    For $n$ positive integer, let be:   $I_n:=\{1,\dots,n\}$; $G$, $\overline G$ groups of order $n$; $\psi\colon G \rightarrow \overline G$ isomorphism; $f$, $\bar f$ bijections; $\theta$, $\bar \theta$ embeddings; in general, $\varphi^{(\alpha)}$ the isomorphism between symmetric groups on sets of the same cardinality, defined by $\sigma \mapsto (g \mapsto (\alpha\sigma\alpha^{-1})(g))$, where $\alpha$ is a bijection between the sets; $S_n$ the symmetric group of degree $n$.   Visually:  $$ \newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c}    I_n     &amp;   &amp;    I_n      \\     \da{f}         &amp;           &amp;     \da{\bar f}          \\    G &amp; \ras{\psi} &amp;    \overline G  \\     \da{\theta}         &amp;           &amp;     \da{\bar \theta}          \\    \operatorname{Sym}(G) &amp; \ras{\varphi^{(\psi)}} &amp;    \operatorname{Sym}(\overline G)  \\     \da{\varphi^{(f)}}         &amp;           &amp;     \da{\varphi^{(\bar f)}}          \\    S_n &amp;  &amp;    S_n  \\ \end{array} $$     Is it $(\varphi^{(f)}\circ\theta)(G)=(\varphi^{(\bar f)}\circ\bar \theta)(\overline G)$? I recently tried to change a bulb using an adaptor (converting from a E27 to B22 fitting). After switching on the light fitting the fuse had blown. After inspection it turns out the adapter is faulty and must have caused a short-circuit.  After removing the bulb and resetting the switch in the main fuse box, I am now unable to operate the switch on the wall to turn the fitting off (the switch physically won't click).   Is there something I can do to fix the problem?  Thanks, Adverb position in "Listen carefully to what I say" In Iron Man 3,      When the real villain kidnapped the President of the United States,   it is never explained why none of the other Avengers came to help, or why Tony didn't call them. The only explanation so far is: "it was an Iron Man movie...", but an in universe answer would be good. Why so many pins for the MOSFET's drain? Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties. Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties:  (1) $f$ has derivative of all orders, and  (2) for all $x,y \in \mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.  Which of the following sentences is true?  (a) Any $f\in A$ is a polynomial of degree less than or equal to 1  (b) Any $f \in A$ is a polynomial of degree less than or equal to 2  (c) $\exists f \in A$ which is not polynomial  (d) $\exists f \in A$ which is a polynomial of degree 4  It is a problem from TIFR GS-2018. I have found polynomials upto degree 2 hold this properties but have to exclude the possibility of (c) and (d). I also found out if $f'(x)$ has at most one real root and clearly graph of $f$ is symetric about that root. How can I proceed further? Proving the geometric sum formula by induction $$\sum_{k=0}^nq^k = \frac{1-q^{n+1}}{1-q}$$  I want to prove this by induction. Here's what I have.  $$\frac{1-q^{n+1}}{1-q} + q^{n+1} = \frac{1-q^{n+1}+q^{n+1}(1-q)}{1-q}$$  I wanted to factor a $q^{n+1}$ out of the second expression but that 1- is screwing it up... English proverb for when a solution comes too late In Flemish we have a saying "Vijgen na pasen". Translated: "figs after Easter". It means a solution comes too late to be of any use.  What is the English equivalent for this?  Some googling gives me "Closing the barn door when the cow has bolted", but the explanation seems to point more towards the wrong solution for a problem. I have to give low and high bounds for the following:  $$ \sum_{n=1}^\infty \frac{1}{2^n - 3^n } $$  How do I determine an upper bound? How can I show this sum exists?  edit: removed erroneous conclusion. Partitions for Ubuntu and Windows 7 dual boot on SSD + HDD setup I have purchased a new SSD (120GB) and I had a HDD (1TB) already installed. I will install both Ubuntu and Windows on these drives. As the file space on the SSD is limited and I've read that writing to an SSD drive is discouraged, I would like to store most of the data on the HDD drive. I am aiming at faster system but also keeping the valuable data safe on the more reliable HDD and expanding the life span of the SSD.  I have already set up Windows on these drives (system files and applications on SSD, Users folder and ProgramData on HDD) and now it's time to set up Ubuntu.  So my question is which partitions I should create, on which drive should they be placed and how large they should be.  Based on my experience and articles I've read, I will probably create /swap, /home and / partitions.   Does it seem reasonable to place /swap and /home partitions on the HDD and / on the SSD?  And how large should / partition be?  Also, will the applications be installed in / partition? If so, is it rational to store the applications on separate partitions on the HDD instead of / on the SSD, and what should the size of the / partition be in this case? We are planning a trip with a bike rack attached to our trailer hitch. What is the best way to lock the bikes while the vehicle is unattended? I see some racks have locks on the knobs but they don't seem very strong.   What do you suggest? Please help me to draw the following diagram:  \begin{displaymath} \xymatrix{ H\ar[r]&amp;H .1={}_HI_Q\supset 1.Q&amp;Q\ar[l]_\epsilon^{\cong}  \\  &amp;R\ar[u]\ar[r]&amp;R\ar[u] } \end{displaymath} Beamer: Change only several bullets in the list I would like to change the bullets in the second itemize to arrows.  \begin{alertblock}{The Aim of this Research}      \begin{itemize}         \item Item 1         \item Item 2             \begin{itemize}                 \item Item 2-1                 \item Item 2-2             \end{itemize}     \end{itemize}  \end{alertblock}   Form above, I would like to make:  ・Item 1  ・Item 2    =&gt; Item 2-1  (change bullet to arrow)    =&gt; Item 2-2   I've tried inserting \renewcommand{\labelitemii}{$\rightarrow$} before and after the second \begin{itemize} and , but they don't work. OS X computer name not matching what shows on terminal My computer name in System Preferences &gt; Sharing is set to &quot;archos&quot;, but is showing as &quot;iphone&quot; on Terminal. It just started doing this after I loaded Xcode for doing iPhone development: Last login: Mon Nov  7 14:46:55 on ttys001 iphone:~ travis$   Any ideas what could be causing this? Detect an internet connection activation with Delphi I've been using a 3G wireless card for a while and every time I connect, my anti-virus fires up the updates.  I'm wondering what is the Win32 API set of functions that I can use to, either, get notified or query about the event of an Internet Connection coming up?  And is there already a set of ported headers for Delphi? Sum of two squares modulo p