$\mathcal{l}^\infty$ contains all separable normed spaces
Isometric Embedding of a separable Banach Space into $\ell^{\infty}$
What are PATH and other environment variables, and how can I set or use them? Questions about    setting environment variables the PATH   are very common here, and in most cases the answers are very similar to each other. In the future it would be nice to have a good Q/A for this.  So the question is: What are environment variables, like the executable PATH, and how can I change and use them on major operating systems?  A good answer would include a simple explanation of what environment variables and especially PATH mean to the OS, as well as simple guidelines on how to set and read them accordingly. Bernoulli maximum likelihood Suppose $X_1, X_2, \dots, X_n$ are iid Bernoulli(p) random variables. How do you find the restricted maximum likelihood for p where $0&lt;p&lt;0.5$?  My work so far:  Write out the likelihood: $L=p^{\sum x} (1-p)^{n-\sum x}$  Take the log: $\ln(L) = \sum x\ln(p)+(n-\sum x)\ln(1-p)$  Take the derivative and set equal to zero: $\frac{\sum x-np}{p(1-p)}=0$.  Now I'm stuck. I don't know how to incorporate the restrictions for $p$. Could the Olympians use modern celestial weapons? Instead of using short range celestial bronze (CB) weapons like swords and arrows why not make advanced weapons like machine guns or grenades with CB shrapnel in it? pdfTeX error (font expansion): auto expansion is only possible with scalable I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. I have a professor at my university who uses content from an un-cited textbook that he has never referenced anywhere, including his green sheet (syllabus), as an additional resource. At least some slides he has used include parts of content that have been re-worded and others that are 100% the same. I noticed this by Googling for textbooks on the material we're studying in class and found that examples given in one that I found were exactly the same as what he has provided.  Would this technically be plagiarism? Should he be given the benefit of the doubt in this situation? I need to calculate the Cramer Rao lower bound of variance for the parameter $\theta$ of the distribution $$f(x)=\frac{1}{\pi(1+(x-\theta)^2)}$$   How do I proceed I have calculated $$4 E\frac{(X-\theta)^2}{1+X^2+\theta^2-2X\theta}$$  Can somebody help Is a NVIDIA GeForce with Optimus Technology supported by Ubuntu? Why is the gravitational force always attractive? Is there another way to explain this without the curvature of space time?   PS: If the simple answer to this question is that mass makes space-time curve in a concave fashion, I can rephrase the question as why does mass make space-time always curve concavely? Calculating $\lim_{x\to0} \left\lfloor\frac{x^2}{\sin x \tan x}\right\rfloor$ Arithmetic progression topology How can a "proper" function have a vertical slope? Plotting the function $f(x)=x^{1/3}$ defined for any real number $x$ gives us:   Since $f$ is a function, for any given $x$ value it maps to a single y value (and not more than one $y$ value, because that would mean it's not a function as it fails the vertical line test). This function also has a vertical tangent at $x=0$.   My question is: how can we have a function that also has a vertical tangent? To get a vertical tangent we need 2 vertical points, which means that we are not working with a "proper" function as it has multiple y values mapping to a single $x$. How is it possible for a "proper" function to have a vertical tangent?  As I understand, in the graph I pasted we cannot take the derivative of x=0 because the slope is vertical, hence we cannot see the instantaneous rate of change of x to y as the y value is not a value (or many values, which ever way you want to look at it). How is it possible to have a perfectly vertical slope on a function? In this case I can imagine a very steep curve at 0.... but vertical?!? I can't wrap my mind around it. How can we get a vertical slope on a non vertical function? A command-line clipboard copy and paste utility? In Windows I used command-line clipboard copy-and-paste utilities...      pclip.exe and gclip.exe     These were UnixUtils ports for Windows (but they only handled plain text). There were a couple of other native Windows utilities which could write/extract any format.    I've looked for something similar in Synaptic Package Manager, but I can't find anything.    Is there something there, that I've missed? ... or maybe this is available in Bash scripting?    The type of utility I'd like will be able to read/write via std-in/std-out or file-in/file-out, and handle Unicode, Rich Text Format, picture, etc. clipboard formats...    NB: I'm not after a clipboard manager. Can anyone explain IEnumerable and IEnumerator to me? Can anyone explain IEnumerable and IEnumerator to me? For example, when to use it over foreach? what's the difference between IEnumerable and IEnumerator? Why do we need to use it? XPath select node with namespace What does "apostrophe s" stand for in "What's it all mean"? I heard someone on a TV show say:     What's it all mean?   As far as I know, 's could be the short form of has and is. But in this case, it seems more like to be the short form of does. Am I correct? I assume it doesn't connect to anything (other than the satelite I guess), is this right? Or it does and has some kind of charge? Highlight a word with jQuery Is it possible to raise stats? I found a lot of dialogue choices requiring high Perception and I started the game with just 10 plus 1 for being an elf. With a magical helm and tower rest bonus I can temporarily boost it to 13, but is it possible to raise it permanently? Expected Value Of Number of Removals From Urn Here's the question: "An urn contains b blue balls and r red balls. You repeatedly and independently remove balls from urn (without returning them) until the first blue ball is drawn. All balls currently in the urn have an equal probability of being selected each time a ball is removed. Define the random variable X as the number of balls that are drawn (number of red balls that are removed plus the first blue ball that is removed). Find E[X], the expected value of random variable X."  I'm having trouble finding a closed form expression for the expected value. Here's the work I've done: