I'm Trying to find an example for two non isomorphic,  finitely generated,    residually finite groups $G_1$ and $G_2$ such that there exists $N_1\unlhd G_1$,$N_2\unlhd G_2$ while $G_1/N_2 \cong G_2/N_2$ and $|[G_1:N_1]|=|[G_2:N_2]|&lt;\infty$.  It's my understanding that residually finite is not the same as finitely presented. Thank's in advance to all responders.
I'm looking for an example of $2$ non isomorphic groups $G_1,G_2$  that are finitely generated and presented that have the same finite quotients (up to isomorphism) .  Thanks
Combining two covariance matrices Undergraduate research field different from PhD field I am a Physics undergrad who is interested in pursuing a PhD in pure maths in the future (algebraic geometry/topology) but I am a bit unsure. My question is quite general, and I don't wish to provide more background for fear of bias in the answers.  My question: Is it is advisable/possible/unfavorable/favorable to apply for a PhD in a field, different from that in which you have done most of your undergraduate research?  My research (includes just reading and understanding papers, writing summaries until now, I havent published anything) mostly includes Quantum field theory, and gauge theory. Would the selection committee turn down an application to a pure maths field, if I have no research experience whatsoever?  I would also like to ask the question other way round. What if I concentrate my undergraduate research SOLELY on topics in pure maths such as algebraic geometry/topology and take other physics courses, would I be able to apply to a string theory PhD with a high chance of success?  Should I consider spending time on both of these (which is almost an impossible task), to improve my chances in both the areas or would research in one area, and grad level courses in the other suffice?  In brief: Should the PhD field you are applying to be the same as your undergraduate research area? Is it possible to simulate key press events programmatically? Is it possible to simulate key press events programmatically in JavaScript? On Mac, are there keyboard shortcuts to create a new Space? Or at least any way to automate this process with keyboard shortcuts? I want to not have to use the mouse to create Spaces. I want to use the keyboard for that. It is stated as a problem in Spivak's Calculus and I can't wrap my head around it. Do harmonic numbers have a “closed-form” expression? How to retrieve the content/file in JSP? What does the leading semicolon in JavaScript libraries do? Let's say I am building a logistic regression model where the dependent variable is binary and can take the values $0$ or $1$. Let the independent variables be $x_1, x_2, ..., x_m$ - there are $m$ independent variables. Let's say for the $k$th independent variable, the bivariate analysis shows a U-shaped trend - i.e., if I group $x_k$ into $20$ bins each containing roughly equal number of observations and calculate the 'bad rate' for each bin - # observations where y = 0 / total observations in each bin - then I get a U shaped curve.  My questions are:   Can I directly use $x_k$ as input while estimating the beta parameters? Are any statistical assumptions violated which might cause significant error in estimating the parameters? Is it necessary to 'linearize' this variable through a transformation (log, square, product with itself, etc.)? My wife has a Nikon D3100 that I've been using to film, but the battery typically won't last through long shoots.  I've gathered from searching around that I can't charge the camera with USB, but I wondered if there's ANY way to do that. It seems at least possible that an adapter could exist that would fill in for the battery and plug into the wall, but I haven't been able to find such a thing.  Is there such a product or technique? It would be nice to not buy a second battery, I'm really hoping for a "set it and forget it" solution. The meaning of imaginary time What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in physics, why should imaginary (or complex) time make sense?    By imaginary I mean a multiple of $i$, and by complex I mean having a real and an imaginary part, i.e., $\alpha + i\beta$, where $\alpha, \beta \in {\mathbb R}$. Run VirtualBox in background, without a window? I was wondering if it's possible to run an Ubuntu virtual machine using VirtualBox in background, I mean, without any window open.  The idea is to connect via SSH to the Linux host, that would be running in background.  Has anyone ever done something like this? Is it possible? How can I identify hardware components in a laptop? I just reinstalled my Windows and now I don't have all the hardware components recognized. I have an  laptop and  website seems to have all the necessary drivers for my model. The problem is that I don't know exactly which components I have. For example, in my laptop video card case the site has available drivers for ATI and Intel chips.   How do I know what are the right drivers for my machine? real memory usage Problems after upgrading to 14.04 (only background and pointer after login) Using iOS 5 rich text editor "If" and "Whether" - Is interrogative IF always replacable by WHETHER? What dried food can I take across the American border from Canada? General term of Taylor Series of $\sin x$ centered at $\pi/4$ I thought people would be working on little code projects together, but I don't see them, so here's an easy one:  Code that validates a valid US Zip Code. I know there are ZIP code databases out there, but there are still uses, like web pages, quick validation, and also the fact that zip codes keep getting issued, so you might want to use weak validation.  I wrote a little bit about zip codes in a side project on my wiki/blog:    There is also a new, weird type of zip code.     I can do the javascript code, but it would be interesting to see how many languages we can get here.