Let $M(m,n,r)$ be the matrix space of real matrices $m\times n$ with $rank \leq r$.  Is $M(m,n,r)$ an open set? or closed set? or Does it have some property?  Regards
Let $M$ be the space of all $m\times n$ matrices. And $C=\{X\in M|\operatorname{rank}(X)\leq k\}$ where  $k\leq \min\{m,n\}$. Check whether the set $C$  is:   Closed Connected Compact Open   What are some other good properties of the set $C$,for example is it a manifold?  Clearly the set $C$ is closed  if someone is interested  a good proof can be found , hence $C$ is not open. Also as $C$ is unbounded therefore not compact. How to check whether the set $C$ is connected or not?
Words that mean ''to say something unintentional'' How many digits does $2^{1000}$ contain? I tried this way, I only need to know if this is correct or if there are better ways to solve this:  $2^{1000}$ does not have a factor of $5$ obviously therefore we can assume  $$ 10^{m} &lt; 2^{1000} &lt; 10^{m+1}$$ for some  $m$   Assume $ k = 2^{1000}$, then take log on both sides $\log k = 1000  \log 2 \approx 301.02999 &gt; 301$  Therefore $2^{1000}$ has $302$ digits. (This question is meant to be a resource to direct others to given that this is a common question.)  Given the task of creating an image that will be printed at a certain size, what pixel dimensions and resolution should I be setting my file at? For one of the clients, I have spin up a modern communication site collection and on the landing page of the same, I am seeing 'Get the mobile app' showing up, I was able to hide the 'feedback' button using css but no luck with Get the mobile app button.  Can someone please share css to hide 'get mobile app' button, appreciate the help. Firstly, I apologise if my thinking is completely incorrect - I am by no means a physicist (yet!), I have included layman illustrations to hopefully assist in explaining my thinking.  I am imagining the big bang to be, for lack of a better word, an explosion, thus the universe to be progressively expanding:    Therefore, to me it seems logical to suggest that there is an edge.  If this is the case, then is it reasonable to suggest that explosion + edge = 'bright edge'? Where light projects in all directions at the edge, not just outwards into the 'nothingness'.     So, this brings me to my question: why, when someone looks deeply enough 'at the edge' do they see an absence of light?   Again, apologies if I'm jumping to irrational conclusions &amp; am way off. Differences among Doctoral Degrees/titles Confirm and agree to negative questions Why with two classes, LDA gives only one dimension? I am working with dimensionality reduction algorithms. Linear Discriminant Analysis (LDA) is a supervised algorithm that takes into account the class label (which is not the case of PCA for example). I am using Python to do a comparative study between some algorithms.  Why with two classes (k = 2), regardless of the data dimensionality, LDA gives one dimension, i.e. the new subspace is composed of only one dimension? For example if I try to project my dataset (of 100 attributes and 2 classes) to 10-dimensional space by using LDA(n_components=10) I only obtain 1 dimension?  For k&gt;2 I can obtain the number of discriminant vectors that I specified as n_components. How can I get n_components dimensions for k=2? Functions satisfying $f\left( f(x)^2+f(y) \right)=xf(x)+y$ Is the sentence "There is a large number of labourers who want to migrate to Japan for work." correct? Possible Duplicate:             There is a large number of labourers who want to migrate to Japan for work.   I type this sentence in a word processor, and it reports an error that "is" should be "are" (subject-verb agreement).  I doubt this is a valid error. Could anyone confirm this? Why is this true: The only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the identity Can anyone explain me please how to see this statement: the only orthogonal projection that is also unitary from $\Bbb C^n$ to $\Bbb C^n$ is the Identity.  how can I prove formally that? or how can I see that only the Identity satisfy it? a / an - adjective - noun The article a becomes to an when the following noun starts with a vowel.     Not a adventure but an adventure.   But how is it, when there's an adjective before the noun?     Not a thrilling adventure or not an thrilling Adventure?   And how is it, when the adjective starts with a vowel but the noun does not?     Not a exciting tournament or not an exciting tournament? In some cases, questions and answers that have been deleted are still showing up on /review pages. They do not have "Improve" buttons, but they can still be approved or rejected. I'm not sure what causes a post to end up in this limbo state instead of being removed from the queue entirely, which I think is what's supposed to happen.  The suggested edit I noticed this on was , but it just leads to a "Page Not Found" message now.  Tim Stone provided the following screenshot of another instance : My Prof wrote that   $\sum_{n=2}^{\infty} n(n-1)x^{(n-2)} = \frac{2}{(1-x)^3}$  when $-1&lt; x&lt;1$  I do not understand where the above formula comes from. Can someone please explain the steps or provide a hint?   Thanks! Single word for people who are like "a frog in the well" Is there a single word describing someone who is like a frog in a well? The frog believes the well is the entire world. How can I describe people who think that their own small environment is the entire world? I had recently taken the hard drive out of my computer so that I could install a bigger one. But when I reconnected it to the computer, all the files on the E:\ partition and some of the files on the C:\ partition had strange names like TTv5yV5354$YVC5Wy on them. I did on one occasion move an electromagnet near the drive. Could that have caused this? Please help me, some of those files were valuable. How to reduce memory usage on a Unix webserver I'm currently using a Joyent Accelerator to host my webapps, and it's working fine, however I need to reduce costs so I'm downgrading my current plan and that imposes some new memory limits (256M rss, 512M swap). I wasn't too far over them yesterday, but after restarting Apache several times today, I'm now 411M rss, 721M swap (prstat -Z -s cpu).  Searching in Server Fault only gives me lots of ways and specific tools to monitor the server, but no advice on how to reduce/optimize it's memory usage. I've also seen , but I don't think it's good for this particular (or may I say generic?) situation.  The server is running Solaris on a shared CPU, and I'm using a Apache + MySQL + PHP stack.  I'm interested in knowing the steps one can take to troubleshot this and solve the issues. However, I'm also running out of time to lower my memory foot print and downgrade the plan before the current ends, so anything that can make magic and save the day is welcome as well :) 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. What is the significance of the name Bleach? The name "Bleach" seems to be having no relevance to the plot unlike most other series. Was it just chosen at Kubo-sensei's whim or does it have some significance? Maybe some cultural significance associated with shinigami, etc. that I am now aware of? Neutrino Oscillations and Conservation of Momentum I would like to better understand how neutrino oscillations are consistent with conservation of momentum because I'm encountering some conceptual difficulties when thinking about it. I do have a background in standard QM but only rudimentary knowledge of particle physics.   If the velocity expectation value of a neutrino in transit is constant, then it would appear to me that conservation of momentum could be violated when the flavor eigenstate at the location of the neutrino source is different from that at the location of the interaction, since they are associated with different masses.  For this reason I would think that the velocity expectation value changes in transit (for instance, in such a way to keep the momentum expectation value constant as the neutrino oscillates), but then it seems to me that the neutrino is in effect "accelerating" without a "force" acting on it (of course, since the momentum expectation value is presumed constant, there may not be a real problem here, but it still seems strange).   Any comments?