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Is there any practical difference between saving at 300ppi or 72ppi? A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$.  Is there a characterization of the groups in which this property holds?  If this question is too broad, I might ask if such a characterization exists for $p$-groups.    History: I originally posed the opposite question, regarding groups for which $\exists N\unlhd G\,:\, \not\exists H \unlhd G\, \text{  s.t. } H \cong G/N$, and crossposted this to .  I received an answer there to the (now omitted) peripheral question about probability, which shows that most finite groups probably have this property.  After this, I changed the question to its current state, as this smaller collection of groups is more likely to be characterizable. I continue to find it amazing that something as “bulky” and macroscopic as a static magnetic or electric field is actually a manifestation of virtual photons.  So putting on your QFT spectacles, look closely at the space near the pole of a powerful magnet – virtual photons! Now look between the plates of a charged capacitor – virtual photons again!   But if it’s all virtual photons, how do we get the difference between a magnetic and electric field?  [This question needs the tag "magnetostatics" as well]. Using separation of variables to solve the wave equation I'm trying to teach myself separation of variables and have been following some notes for the wave equation, but there's one part which really confuses me and I'm not exactly sure how it makes the step.  For the wave equation   $$u_{tt} - c^2 u_{xx} = 0$$   with length $l$ and fixed ends, $u(0,t)=u(l,t)=0$ we seek a solution in the form $$u(x,t)=v(x)q(t)$$  and substituting this into the equation gives $$\frac{1}{q(t)}\frac{d^2 q}{dt^2}=c^2 \frac{1}{v(x)}\frac{d^2 v}{dx^2}=-\omega ^2$$   and I understand all of this so far. But when solving the equation  $$q''+\omega ^2 q = 0$$ the general solution is $$q(t)=A \cos(\omega t +\alpha)$$ where $A$ and $\alpha$ are constants - which is the part which I'm not exactly sure on how it gets to.   Could someone explain this step to me? Thanks! I've been told not to use for...in with arrays in JavaScript. Why not? apt-get stuck at 0 [Connecting to us.archive.ubuntu.com] How do "Dummy Variables" work? I do not understand how dummy variables work in math.   Suppose we have:  $$I_1 = \int_{0}^{\infty} e^{-x^2} dx$$  How is this equivalent to:  $$I_2 = \int_{0}^{\infty} e^{-y^2} dy$$  How does this dummy variable system work?   Since $y$ is the dependent variable for $I_1$ How can $y$ itself be and independent variable for $I_2$  ??  Thanks! Typical software & tools short-list for *manual* QA specialist (tester) working environment For QA workstations preparation (Windows-based PCs) our IT department uses typical batch deployment tools. I'm not at all interested in the details of such process, but IT guys ask me:  What typical set of software you need for everyday manual testing tasks? We could include these in our deployment so that you guys could save more time for testing, not for environment prep &amp; tuning.  Sure thing we do test various range of projects, from web applications to standalone solutions, and each application requires its own set of tools for manual testing, e.g.:  Browsers Virtualization utilities DB clients Screens capturing tools FTP clients XML editors etc.  However, and that's for sure - there's a list of tools every QA MUST have installed on its PC, regardless of project types under testing. My area of experience is mostly standalone / complex web apps, that's why my vision on the task is a bit subjective. So, your suggestions of what are tools / software an average QA specialist can't live without will be highly appreciated. 2 assumptions:  Manual testing is a point of interest. Let's limit the list to not more than 15 (better 10) items.  Thanks in advance! Intuitive interface for Composing Boolean Logic? I'm interested to know how people have, or would construct an interface which simplifies constructing logical boolean conditions? (Yes, it's a database query builder) In my mind a bad interface has them typing lots of AND, NOT IN, OR keywords etc. Or else has a million combo boxes.  I'm wondering if anyone has a good example I could glean ideas from? Or some suggestions?  At the moment I'm considering a split screen with a grid for composing the query and a canvas to display a visual representation of the sets of conditions and how they overlap etc. Like a Venn Diagram. This still gives me a problem with the usability/intuitiveness of the controls on the grid.  EDIT:  I'm very interested in ideas which simplify it for less technical users. However, ideas for a version of the UI for advanced users are very helpful as well. Application of Kolmogorov's Zero-One Law Most of the books I read, they only state some examples of tail events. One of them is  $[\sum_n X_n \ converges]$. My main problem is to show that this is indeed a tail event. How to arrive at $\int_{0}^{1} f^3(x) \, dx < (\int_{0}^{1} f(x) \, dx)^2$? Define $f : [0,1] \to \mathbb{R} $ differentiable satisfying $f(0)=0$ and $0&lt;f'(x)&lt;1$. Prove:  $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$ I found the following answer,  Define $F(x)=\displaystyle\int_{0}^{x} f^3(t) \, dt - \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)^2$ We have $$F'(x)=f^3(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)f(x)=f(x)\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]$$ and $$\left[f^2(x)-2 \left(\displaystyle\int_{0}^{x} f(t) \, dt\right)\right]'=2f(x)\left[f'(x)-1\right]$$ from which we can arrive at the conclusion.  But I felt a little confused. In the past I usually proved the inequality by classical inequalities and something like Taylor expansion to use the condition about derivative. But this answer is little out of my expectation. Do we have other proofs or ideas to arrive at $\displaystyle\int_{0}^{1} f^3(x) \, dx &lt; \left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2$? And other thoughts about solving these types of issues? I think when we talk about $\left(\displaystyle\int_{0}^{1} f(x) \, dx\right)^2 &gt;\ldots \,$ it's a little difficult to deal with. I don't quite know what tools to use. Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even How To Reduce Noise In Cycles Render Proof by induction of summation inequality: $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$ Prove by induction the summation of $\frac1{2^n}$ is greater than or equal to $1+\frac{n}2$.  We start with $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac1{2^n}\ge 1+\frac{n}2$$ for all positive integers.   I have resolved that the following attempt to prove this inequality is false, but I will leave it here to show you my progress. In my proof, I need to define P(n), work out the base case for n=1, and then follow through with the induction step. Strong mathematical induction may be used.  This is equivalent to $$\sum_{k=0}^n\frac1{2^k}\ge 1+\frac{n}2\;.$$  Let $P(n)$ be summation shown above.   Base case for $n=1$, the first positive integer,   $$\sum_{k=0}^1\frac1{2^k}=\frac1{2^0}+\frac1{2^1}=1+\frac12=\frac32\ge 1+\frac12=\frac32\;,$$  so base case is true.  Induction step: Assume $P(n)$ is true and implies $P(n+1)$. Thus   $$\sum_{k=0}^{n+1}\frac1{2^k}\ge\frac1{2^{n+1}}+\sum_{k=0}^n\frac1{2^k}\ge 1+\frac{n+1}2\;.$$  This can be written as   $$\sum_{k=0}^{n+1}\frac1{2^k}\ge \frac1{2^{n+1}}+1+\frac{n}2\ge 1+\frac{n+1}2\;.$$  I work the math out but I get stuck contradicting my statement. Please show your steps hereafter so I can correct my mistakes. How can I let a footnote continue on the next page? I have a rather long footnote which breaks a listing in two parts, unnecessarily.  Since the listing is more important then the footnote I'd rather lake to have the footnote split.   Can I customize (one or all) footnotes to have a maximal length before they are continued on the next page? Or can I manually split a footnote in two to get the effect? I just noticed that I lost a bunch of points from my reputation score, and I used the "reputation" tab on my user profile page to try and track down the cause.  During my investigation, I noticed there was an unusual event of type "reversal". In the normal place of a question title, it says "voting corrected".     What does this mean, and what caused it?  Did I do something wrong? Why did I lose all of that reputation? Is the system punishing me for leaving too many good answers?  Is there any way to earn it back? I need help!  I need to prove that for any $2 \le n,k$ positive integers $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$  Thanks! Match-3 algorithm to fill empty space left from removing matches I am currently creating a match 3 game.   I am at the point where I have objects dropping down in the column when only one item is removed. So if an item is removed, the above item moves into its old position etc.  The problem I am having now is that when more than one item is removed from a column, the items in that column only drop down to the first item that was removed. Instead of filling in both items that were removed it just drops down to fill in the first one.   So basically, if two(or more) items are removed from a column the second, third, fourth, etc positions are left blank.  The way I am designing my game is I am using units of 1. So each item takes up 1 unit by x and 1 unit by y. This means that I can position the items like in a grid, at 0x1, 0x2, etc.   What kind of algorithm could I use to move multiple items down to fill the empty space? Why Java needs Serializable interface? We work heavily with serialization and having to specify Serializable tag on every object we use is kind of a burden. Especially when it's a 3rd-party class that we can't really change.  The question is: since Serializable is an empty interface and Java provides robust serialization once you add  implements Serializable - why didn't they make everything serializable and that's it?    What am I missing? Is there a particular reason why vote times are omitted in the data dump?  IMO this could be a very useful data-set. Is this one of the steps that was done to anonymize the data, or does it simply not exist? This would be a very good addition to future dumps to help build time-based relationships, which are currently very rigid and full of assumptions.