Help Determinant Binary Matrix
Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
How to create or recover Windows Bootloader after deleting Ubuntu boot drive I have a computer that dual-boots (or tri-boots) Windows 8 Release Preview, Windows 7, and Ubuntu 12.04. Grub boots between Windows 8 and Ubuntu; for which I use primarily. Recently, I decided to remove Ubuntu, as I hardly used it. I deleted the Ubuntu partition accidentally before replacing the Grub bootloader. Now, whenever I want to boot the machine, it gives me the "grub-rescue" prompt -- I am unable to boot into either Windows (8 nor 7), nor Ubuntu (except via USB, of course). I do not have any Windows 7/8 recovery media, so that isn't an option. Please note that after I deleted the Ubuntu partition, I put the PC into hibernate, and then turned it on. This means the C:\ [Windows 8] drive cannot be mounted. I don't know if that is bad, but it definitely doesn't make things better.  I am currently booting Ubuntu via USB, in an effort to restore the Windows bootloader. I have looked into using boot-repair to solve the problem using the instructions , although after attempting to apply the changes, it gave the error: "Please install the [mbr] packages. Then try again." I don't know why I'm getting this error; is there a way to install the 'mbr packages?' I honestly don't know what exactly they are, nor how to install them. Are there any other options I have not yet exhausted to be able to boot back into Windows, in the case that there is a better way? I want to set the bootloader to boot into Windows 8, but booting into either Windows 7 or 8 is fine (I can use EasyBCD from there). Is there a simple solution to this? I've checked BIOS, and I haven't been able to find a way to boot into Windows. What do *args and **kwargs mean? What exactly do *args and **kwargs mean?  According to the Python documentation, from what it seems, it passes in a tuple of arguments.  def foo(hello, *args):     print hello      for each in args:         print each  if __name__ == '__main__':     foo("LOVE", ["lol", "lololol"])   This prints out:  LOVE ['lol', 'lololol']   How do you effectively use them? Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$ I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ }dx$, but I could not build a rigorous argument from that.  Any help greatly appreciated! We're given a sequence defined by the recursive relation: $$a_n = \sqrt{a_{n-1}a_{n-2}}$$  $a_1$ and $a_2$ are positive constants. We have to show the following:   The sequences $\{ b_n \} = \{ a_{2n-1} \}$ and $\{ c_n \} = \{ a_{2n} \} $ are monotonic, and if one is increasing, the other is decreasing. The limit of the sequence $\{ a_n \} $ is $ \left(a_1a_2^2\right)^{\frac13} $     Now, I have proved the first part. Besides that, I have also proved a few other things:  If $a_1 &gt; a_2$, then:  a. $ \{ b_n \} $ decreases, and $ \{ c_n \} $ increases.  b. $ c_n &lt; b_n $  If $a_1 &lt; a_2$, we just flip $\{ b_n \}$ and $\{c_n\}$  Besides, I have also shown that both the sequences : $\{b_n\}$ and $\{ c_n\}$ have the same limit. What I don't know, is how to evaluate the limit. I don't know if this is a famous problem but my professor assigned this for a research project and I am hitting a dead end( I don't know what to do next).   Note: In $\phi(n)$ ; $n=rows=columns$ and this function throws out total combination of matrices that satisfies the following rule:  Rule for this square binary matrix:  Every Row and Column has two 1's.  So,    $$\phi(2)=1$$ \begin{bmatrix}    1 1 \\    1 1 \\ \end{bmatrix} $$\phi(3)=6$$  I can't write out all the possible matrices for $n=3$ but this is one of them (btw i did this by hand): \begin{bmatrix}    0 1 1 \\    1 0 1 \\ 1 1 0 \\ \end{bmatrix}   According to a program made by my friend, he told me:   $$\phi(4)=90$$ $$\phi(5)=2040$$ Where can I download Windows 8 legally, from Microsoft? How to run/use popcorn-time? I got popcorn time installed using this link:    But I have no clue how to use or run Popcorn time.  Do I open my torrent client and download a movie and then it gets going? or do I have to open it? If I have to open it, how do I do that, I opened the dash and tried clicking on it but nothing happened. Please help me!! :D Does the Rule of Two Serve any Actual Purpose? I know Darth Bane came up with the Rule of Two to ensure the survival of the Sith, but, in reality (or in the Star Wars reality), does it (or any Sith rule) serve a true purpose?  And why does each successive Sith bother with it?  Due to the nature of the Sith, Sith Lords tend toward arrogance and self-importance.  Both the Sith master and Sith apprentice seem to regularly break the rule by training others.  Deceit and personal interest over loyalty is a way of life for the Sith.  What is to keep any Sith from following the Rule of Two (or any rule, for that matter) if it is against their own self interest?  If a Sith Master can stay in power longer by training a 2nd apprentice and pitting the two against each other, is he really going to worry about the Rule of Two more than his own plans and survival?  The Rule of Two also assumes that every Sith Master would place the survival of the Sith over his own survival, as opposed to attempting to ensure his own immortality in whatever way he could (like Plagueis).  When you have a group that, by default, is arrogant and self-serving, as well as deceptive, why would they want the Sith to survive themselves?  I can see how the Rule of Two, and any other Sith beliefs would be possible guidelines, but it terms of the nature of ambitious people (and the Sith are ambitious), it's hard to believe that each new Sith Master would follow this rule faithfully.  So does this rule, or any Sith rule, actually work?  Or is it more of a suggestion that is broken when convenient? How do I install the Tor Browser Bundle in Ubuntu? In Ubuntu 13.04 I used to use the ppa:upubuntu-com/tor64  PPA to install the 64-bit Tor Browser Bundle. However, in Ubuntu 13.10 it doesn't work anymore. How do I install the Tor Browser Bundle in Ubuntu? How to determine CPU and memory consumption from inside a process? Continuous function from $(0, 1]$ onto $(0, 1)$? How do I delete all my mail from my Gmail account? Reset Windows 10 from USB $\mathbb R = X^2$ as a Cartesian product I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism? Prove that $B[x] \cap B[x^{-1}]$ is integral over $B$ Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$ Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.   The ring that I am talking about is $\mathbb{Z}[x]$ so $\langle 2,x \rangle$ refers to $2g(x) + xf(x)$ where $g(x)$, $f(x)$ belongs to $\mathbb{Z}[x]$. Generating correlated binomial random variables I was wondering if it might be possible to generate correlated random binomial variables following a linear transformation approach?  Below, I tried something simple in R and it produces some correlation. But I was wondering if there is a principled way to do this?  X1 = rbinom(1e4, 6, .5) ; X2 = rbinom(1e4, 6, .5) ;  X3 = rbinom(1e4, 6, .5) ; a = .5  Y1 = X1 + (a*X2) ; Y2 = X2 + (a*X3) ## Y1 and Y2 are supposed to be correlated  cor(Y1, Y2) I accidentally opened too many menus (or windows) and I want to reset them to default.  It is really annoying, and effecting my animation.  Thanks When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$ When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$, where $f_{xy}$ and $f_{yx}$ denote the mixed (second) partial derivatives of a multivariable function $z=f(x,y)$. How to evaluate potential advisers on grounds other than their research/publications?