So I can not figure out the combinatorial proof for Vandermonde's Identity for the example  $\sum_{i=0}^k \binom {k} {i}^2 = \binom {2k} {k}$ Any help would be appreciated.  Figured it out, thanks :)
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 $n$.
Batch normalize audio volume on .ogg-files Are there any simple tools to normalize an .ogg-library of 10'000+ songs so that the volume is the same throughout all songs?  Terminal or GUI doesn't matter, it only need to be simple.  One caveat though, I don't want soft interludes/intermissions and ballads blown out of proportion.  Preferably the process should find the overall gain of the album (I have all my CD's ripped into separate folders) and normalize the level thereafter. For a given ellipse, find the locus of all points P for which the two tangents are perpendicular.   I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof. Show that if $|f(z)| \leq M |z|^n$ then $f$ is a polynomial max degree n I can't prove this statement, can anybody show me how to prove it?    $$f:\mathbb{C}\rightarrow \mathbb{C} \in \mathcal{O}(\mathbb{C}), \exists n\in \mathbb{N}, R &gt;0 , M&gt;0 : |f(z)| \le M|z|^{n} \ \ \forall |z|&gt;R \Rightarrow \deg(f)\le n $$  To show is that if there exists such an $M$, that then $f$ is a polynomial of max degree $n$. I started like this:   $$f(z) = \sum_{n=0}^{\infty} a_n (z-z_0)^n$$ So if I put this into the inequality: $$|f(z)| = \left| \sum_{n=0}^{\infty} a_n (z-z_0)^n \right| \le M |z|^n .$$ How can I use QGIS to create a metro style map I'm pretty new to this and this is my first question here, any help would be greatly appreciated.  So basically what I want to achieve is a floor material as seen in this photo...    ...the reflections are very crisp but dim.  With a glossy shader I get the following results:    First picture: reflections are blurry  Second picture: roughness of the material looks right but reflection is bright red  Third picture: If tinted in any other color than white, it seems to produce the result I want. Reflection is sharp and gets tinted blue. What I need is the reflection to be tinted white.  Thanks for any help. How did the Swooping Evil Venom infused rain affect the magical community? I didn't see any wizards drenched in the rain at the end of Fantastic Beasts and Where to Find Them (don't recall, actually). But would the rain equally affect the wizards as it did no-maj's?   This event is something that doesn't necessarily need to be forgotten by the Wizarding community. How would they avoid being obliviated by the rain?  (They could avoid the rain by Apparating, but what about the showers and drinking water?)  This question is picking at my brain. Unicode identifiers in Python? Should "State" be capitalized on its own? Say we had the following:     Higher Education spending, clout, and influence in New York State is substantial. Within the State’s borders...   Should the latter instance of State be capitalized or not? Enable automatic deletion of old, unanswered, zero-score questions after a year? Related to Meta Super User efforts:  and Meta Server Fault efforts:  We already auto-remove negatively voted unanswered old questions automatically after 30 days, network wide, with no human intervention required. I was thinking we might extend that to remove old, unanswered zero score questions... based on the following criteria:  fewer than (question age in days * 1.5) views 0 score or lower, or 1 score or lower if the original author is deleted no answers 1 comment or fewer asked more than 1 year ago (thus based on creation date, not last activity date, so , or a user editing it, does not give a doomed question 365 more days of zombie &quot;life&quot;)  See the results of this query on:  (10509) |  (787) |  (650) This query has been refined based on the comments, and this is the final version that will roll out tonight. Every site in the network (except metas) will automatically delete old questions that meet this criteria from this point forward. Where can I find fantasy minis on the cheap? I'm looking to build a collection of fantasy miniatures, preferably pewter - but I don't want to buy them one at a time from my local hobby store... Much too expensive, and they don't often carry anything I'm interested in buying, which makes it a long term venture as well. So I'm looking for ideas on how to obtain many miniatures at once, preferably for cheap. D&amp;D Miniatures aren't a good source for me as they're collectibles, and the prices on those are also prohibitive.  By the way - they don't have to be pewter, that's just my preference. =) Display of content-type I've got custom content-type with several groups and fields.   I want to change default display of that content-type. In which file can I find appropriate code? states:  "In practice, explicit right outer joins are rarely used, since they can always be replaced with left outer joins and provide no additional functionality."  Can anyone provide a situation where they have preferred to use the RIGHT notation, and why? I can't think of a reason to ever use it. To me, it wouldn't ever make things more clear.  Edit: I'm an Oracle veteran making the New Year's Resolution to wean myself from the (+) syntax. I want to do it right Importing only certain rows between spreadsheet documents Document 1 has many rows of data.  How do I import into document 2 only those rows that have a certain value in column C? A natural number multiplied by some integer results in a number with only ones and zeros I recently solved a , which says that,     A positive integer can be multiplied with another integer resulting in   a positive integer that is composed only of one and zero as digits.   How can I prove that this is true(currently I assume that it is). Also, is it possible to establish an upper bound on the length(number of digits) of the number generated? How to backup an Android device? Will wolves attack Ocelots? ColA                  Col B              ColC Project A            A.pdf               A.pdf Project B                                No Link   I have a list of project records. One column (e.g., B) may have a reference to a pdf file. If I create a formula   =if(isblank(B2),"No Link",hyperlink(folderURL + B2),B2)   What I want is to place a hyperlink on only those rows where column B contains a pdf reference, and insert text "No Link". I get the proper text values, but in all rows have the hyperlnk default appearance and behavior -- Excel seems to be creating "phantom links" out of the "No Link" cells.     Clicking on the "No Link" value in Col C activates an attempt to open the file listed in Col B, which returns a No file found error. Basic circuit to keep LED either on or off depending on night/day How can I have a LED either on or off (not in between) depending on night or day with basic electronics? I have created the circuit below, but it only shines and in darkness it dims.    The photo-resistor is about 3kΩ when light shines and about 1MΩ when dark.  I want my LED to shine at night and be off at day.  What is the simplest way I can achieve this? How do I use large bitmaps in .NET? I'm trying to write a light-weight image viewing application. However, there are system memory limitations with .NET.  When trying to load large bitmaps (9000 x 9000 px or larger, 24-bit), I get a System.OutOfMemoryException. This is on a Windows 2000 PC with 2GB of RAM (of which 1.3GB is used up). It also takes a lot of time to attempt loading the files.  The following code generates this error:  Image image = new Bitmap(filename); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Point(0, 0)); }   As does this code:  Stream stream = (Stream)File.OpenRead(filename); Image image = Image.FromStream(stream, false, false); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Rectangle(0, 0, 100, 100), 4000, 4000, 100, 100, GraphicsUnit.Pixel); }   Also, it is enough to do just this:  Bitmap bitmap = new Bitmap(filename); IntPtr handle = bitmap.GetHbitmap();   The latter code was intended for use with GDI. While researching this, I found out that this is in fact a memory issue where .NET tries to allocate twice as much as is needed in a single contigous block of memory.    I know from other applications (Internet Explorer, MS Paint etc.) that it IS possible to open large images, and rather quickly. My question is, how do I use large bitmaps with .NET?  Is there anyway to stream them, or non-memory load them? Access images inside jar