Importing a large Excel file
Import Large Excel File
I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$.  I already know the logical Proof:  $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$  Hence summation can be expressed as:  $$\binom{n}{0}\binom{n}{n} + \binom{n}{1}\binom{n}{n-1} + \cdots + \binom{n}{n}\binom{n}{0}$$  One can think of it as choosing $n$ people from a group of $2n$  (imagine dividing a group of $2n$ into $2$ groups of $n$ people each. I can get $k$ people from group $1$ and another $n-k$ people from group $2$. We do this from $k = 0$ to $n$. Proof of for-all / there-exists "negation"? Show that every irrational number in $\mathbb{R}$ is the limit of a   sequence of rational numbers. Every rational number in $\mathbb{R}$ is   the limit of a sequence of irrational numbers.   How can I prove this? Academic papers using Stack Exchange data I want to run a script to simply change the current working directory:  #!/bin/bash cd web/www/project   But, after I run it, the current pwd remains unchanged! How can I do that? Sorting multiple keys with Unix sort Calculate first difference by group in R Java inner class and static nested class Is Bran able to have children after his fall? The thing that made me ask this question was that most of Stark family are dead and Bran is now the "heir" to Winterfell. So I guess he should have children to succeed him, otherwise the Stark line will end. (I'm assuming Arya will not come back to Westeros and Rickon's fate is sealed, though I could be wrong.) I don't remember reading anything about it. I am about to start a web project and have been working almost exclusively with a LAMP stack for the past 5 years.  I would like to build something larger and more scalable and hence have been looking into Java EE.  The division of logic between JSP and EJB seems ideal for what I want to do.  Can anyone recommend resources for getting familiar with Java EE?  And perhaps if anyone knows of technologies that are more scalable than PHP that I should consider as well, I'm all ears.  Thanks a lot! Is every complex (smooth) manifold a scheme? The question in the title doesn't quite make sense. I was always wondering if the scheme is the generalization of manifold. The precise statement should be like following:  If $X$ is a complex (smooth)manifold, is there a scheme $Y$ over $\mathbb{C}$ such that the corresponding analytic space $Y^{an}$ isomorphic to $X$?  I guess if $X$ is projective, then this is Chow's theorem. And I doubt the question has an affirmative answer if $X$ is non-compact (I guess that $X: y-e^x=0$ in $\mathbb{C}^2$ might be an example, but I do not know how to show this).  Edit 1:  There are also example (as @hunter pointed out) of a complex torus but not an abelian variety shows that a compact manifold might not be a variety, however, is it possible to be a scheme ?(according to Hartshorne, the category of varieties is a subcategory of quasi-projective, integral, separated scheme. But could there be some non-quasiprojective scheme($Y$ has to be integral, separated in our case according to GAGA, I guess) )?  Edit 2: Another question I think of is: is (Artin) Stack a generalization of manifold? I really want to know if manifold could be think of some subcategory of some category of with "algebraic objects". I have an HSA that allows me to invest the money in the account in the stock market. Any money not invested is available to be used for health expenses. I currently have ~50% of the total balance of the HSA invested, with the other ~50% on hand for expenses.  I've run into a situation where the ~50% on hand for expenses does not cover all of my current expenses. To cover the remaining costs, should I sell off my the HSA investments or should I leave those as is and use money from my savings account?  I'm trying to determine if the tax advantages of the HSA combined with the returns the invested money is seeing is worth me leaving that money alone - and instead, take the money from my savings account to cover the costs. Around 3 months ago, I submitted a paper to a Math journal for peer-review through an online submission system. The same day I received in my e-mail an acknowledgement saying that the paper had been received.  The problem is that the status of the paper on the online submission system is still "Manuscript submitted".  I think this means that the paper has not even been assigned to an editor yet (once this happens, the editor will probably start looking for reviewers), but as far as I know submissions are assigned to editors within a few weeks (I once submitted a paper to another journal and when an editor had been assigned, the status on the online system changed to "Editor handles" or something like that).  I know that publishing is a long and complicated process and I'm not trying to rush anybody, but sometimes manuscripts are lost in the system. In fact, this has once happened to me, so I just want to make sure that the paper was not lost in the system.  Should I send an e-mail asking for the status of the paper?  PS: I think that, at the time of submitting the paper, I didn't choose any specific editor to submit the paper to. What can cause "Android OS" process to use high percentage of battery? How to make a pattern follow a stroke, path or line? Here is my MS Paint job on a real bad attempt at what I'm trying to achieve:    I have access to Photoshop and Illustrator CS6, and I want to be able to draw a thick line (that is not straight) and have a pattern within the line show arrowheads within the line to indicate the direction of the line. Both sides differentiation expression with integral I have a problem with derivation of some function. This is related to computing the Poincare Map of Logistic Population model with Periodic harvesting. This function is:  $\phi(t,x_0) = x_0 + \int_{0}^{t} f(s,\phi(s,x_0)) ds $,  where $\phi(t,x_0)$ is a function: $\mathbb{R}  \times \mathbb{R} \rightarrow \mathbb{R}$.  I'm trying to differentiate  equation of this function both sides  with respect to $x_{0}$. In book from this equation comes from the result of this differentiation is:  $\frac{\partial \phi}{\partial x_0} (t,x_0) = 1 + \int_0^{t} \frac{\partial f}{\partial x_0} (s, \phi(s,x_0)) \cdot \frac{\partial \phi}{\partial x_0} (s, x_0)ds$.  I can't understand expression under the integral. In this book it is explained that it is clear from chain rule.Could someone explain to me this transformation in easy way? I will be grateful for your help Best regards How do I gunzip all files recursively in a target directory? I'd like to know what is the command to use to gunzip all files in a target directory recursively? I tried to use the unzip command but it didn't work.  I tried the command from How to type the Indian Rupee Symbol? I want to type the Indian Rupee Symbol. So I went to System Settings >> Keyboard Layout and clicked on the Options button. Under the Adding Currency signs to keys sections, I checked the Rupee on 4 option.    Later when I opened the Layout Chart of the Keyboard there was Rupee sign in the 4 key but I do not know how to type the symbol. What combination of the keys will type the Rupee symbol on the screen? amsalpha-like bibtex style with manual labels Are S.P.E.C.I.A.L. skills capped at 10? We know that the maximum level a dweller can train to on any SPECIAL skill is 10, however this does not consider bonuses given from their outfit.    Is there any additional benefit for going over 10? For example, a dweller with an Intelligence skill of 10 wearing a lab coat that boosts Intelligence by 3?   If it is a hard cap, would it be more useful to use outfits that raise a secondary characteristic instead?