We want to prove that $$\sum_{k=0}^n (-1)^k{n \choose k}^2=\ \Bigg\{                 \begin{array}                  \ \hspace{2mm}0 \hspace{20mm}\text{ if n is odd} \\                   \ (-1)^{\frac{n}{2}} {n \choose \frac{n}{2}}\hspace{6mm}\text{ if n is even}                 \end{array}$$ The odd proof is pretty easy by creating a bijection between odd and even subsets, but I cannot figure out the even proof. Any help would be significant!
I am trying to prove the following identities:  a. $$\sum_{k=0}^n(-1)^k{n\choose k}^2 = \bigg\{^{0 \ \text{if k is odd}}_{(-1)^m{2m\choose m} \ \text{if n = 2m}}$$ b. $$\sum^k_{i=0} {n+i \choose i}={n+k+1\choose k}$$  For part $b$ I know I can use induction. It is clearly true when $k=0$ and then assuming for $k-1$ we get $$\sum^k_{i=0}{n+i\choose i}={n+k\choose k-1}+{n+k \choose k}={n+k+1\choose k} \ ...$$
I am trying to prove that all vector spaces can be written as the direct sum of two subspaces.  My attempt: If each vector in vector space V can be uniquely written as the sum of two other vectors in V, the two vectors can be categorized into subspaces and thus V can be written as the direct sum of these subspaces. I am not sure how to show that this is true for all cases though. Edit Sorry for the confusion,to clarify I meantGiven a vector space V with a subspace W, prove that there exists a subspace W' such that the direct sum of W and W' is V.I am not sure about my proof method at all. Thank you. Determining size of grid cells created by vector grid tool of QGIS? I am working with QGIS 1.8.0 and Im making a vector grid on a map. Does someone know how big are the squares if the grid settings look like in the picture? Recommendations for good Newtonian mechanics and kinematics books What are some good books for learning the concepts of Kinematics, Newton laws, 2D Motion of Object etc.? Is there an (easy) way to create or personalize .bst files? When you prepare an article for a journal (in my case, Quantum Information and Computation, Rinton Press), often they ask you to use some format for your references; in my case, the following:     [1] R. Calderbank and P. Shor (1996), Good quantum error          correcting codes exist,  Phys. Rev. A, 54, pp. 1098-1106.      [2] M.A. Nielsen and J. Kempe (2001), Separable states   are  more disordered globally than locally, quant-ph/0105090.      [3] A.W. Marshall and I. Olkin (1979), Inequalities:   theory of majorization and its applications, Academic Press (New   York).   The issue is, sometimes (and this is the case) they do not provide a .bst style file for bibtex users. I (and I guess many other people) prefer not to copy all the references of a bibtex file one by one into a tex document. So here is the question I am trying to answer:     What is the easiest way to create a .bst file? (for Rinton or any other publisher)   At first sight, I thought this would be very easy to do: after all, a .bst file is just a mere script to give format to a list. Strangely to me, after a long search I have been unable to find a simple clean solution to this problem. These are the the options I have been looking at:   I have tried to adapt a previous .bst file: concretely, apsrev4-1.bst, which has a similar format. Other people have tried this solution before , . However, it is an ugly option. First, it requires going through the script of the .bst and understanding it. Second, the script format used in different .bst files differs (quite annoyingly), so it is not easy to modify them unless you do understand what's written. The above solution is quite hideous and you really lose time on it, so I was wondering: isn't there any software tool (hopefully with a GUI) that does this job? It should, after all, pretty easy to program for a .bst script expert some sort of program to generate .bst files. There are some related questions in this site asking for something similar , ; there I have seen a couple of proposed apps, but going through their manuals they seem not so easy to use. Isn't there some software tool available to generate a .bst in less than 1 hour of work?   Remarks: ideally I would like to use/personalize/create .bst styles that support eprint fields (e.g. aps4-1.bst). Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice? Are battery packs allowed in hand luggage? When flying, are you allowed to bring battery packs / power banks? I'd like to bring my ACC 10400mAh Power Bank on several flights. "Notice: Undefined variable", "Notice: Undefined index", and "Notice: Undefined offset" using PHP Using CLI, how do I change the user's profile picture? A similar task for changing the desktop background, can be achieved using the following script:   FILE="/backgrounds/background.png" gsettings set org.gnome.desktop.background picture-uri "$FILE"   How would I achieve the same result, but with the user's avatar/profile picture? Does changing your Compute Device, change all computes of Blender? So, I have recently moved from using Blender, on my family's Nettop PC, to my personal, gaming PC. And with this move, came to option to use CUDA instead of CPU.  Which leaves the question, if I move to CUDA, does that change all computing, for like physics baking, or only for rendering?  Just a nooby question that I wanted to know, and yes, I know you can specify GPU rendering separately. How can I move or force a program installation to another location, even when unsupported? I have an SSD as my main Windows drive, with a 640GB 2.5" HDD, partitioned to store programs and user settings, and also to act as backup (it's the only thing I had lying around at the time of building my PC).  The problem is that whenever I install a program, it installs to C:\Program Files [(x86 for the 32 bit programs]\, although I have changed the environment variables. This wouldn't normally be an issue, however every installation program points its shortcut to my 640GB HDD.  The root layout of both drives:   To clarify:   Program files get installed to C:\ Program shortcuts are always pointed to Z:\, my 640GB HDD   Modifying the relevant environment variables doesn't do anything, I looked at , but however it only talks about modifying the registry and environment variables, which I have already done so.  I install to the Z:\ drive if the installation program lets me change the installation path, but however the installation programs sometimes don't let me change this.  Is there a way that I can force every program to install to the relevant location on Z:\, but at the same time move any programs currently installed to the new location? Perhaps I'm missing something here?  My target is to have every program installed on Z:/, with only the core Windows operating system on C:/.  Found ; would it be appropriate to use in my case? I would be able to move the entire Program Files (and its x86 version) to Z:\, without impacting on the performance.* My father-in-law is a builder. He has put in 20 amp and 15 amp breakers, and ran 12-2 as the home runs to the switch's and the first plug to make them hot. He then ran 14-2 to feed the rest of the plugs, lights, and ceiling fans.   Is it okay that he used different gauge wires on the same circuit? Does an ability resolve if the source of the ability leaves the battlefield? Player A taps  targeting Player B; its "deals 1 damage to target player" ability is placed on the stack.  Player B gets priority and responds with , targeting Goblin Fireslinger.  Both players pass priority. Snap resolves, and Goblin Fireslinger is returned to its owner's hand.  Does Player B still take 1 damage from Goblin Fireslinger's activated ability?  I know that if the creature had been killed/sent to the graveyard, the ability would still resolve, but I was told recently that because a player's hand is a private zone it couldn't. If anyone could confirm or deny this for me, it would be most appreciated. Greyed-Out vs Invisible Compare two files with first column and remove duplicate row from 2nd file in shell script I will ask my question with an example. I have 2 files:  File #1:  118D FC300_R5_TP  FX.B      32775       2112   6       2038   6       2112   0 118E FC300_R5_TP  FX.B      32775       2136   7       2065   6       2136   0 118F FC300_R5_TP  FX.B      32775       2124   6       2064   6       2124   0 1190 FC300_R5_TP  FX.B     819210     814632  99     814609  99     814632   0 1191 FC300_R5_TP  FX.B     819210     104100  13     103714  13     104100   0 1192 FC300_R5_TP  FX.B    1638420    1609476  98    1609402  98    1609476   0 1196 FC300_R5_TP  FX.B    1638420    1638432 100    1638379 100    1638432   0 119A FC300_R5_TP  FX.B    3276840    3271776 100    3271698 100    3271776   0 119E FC300_R5_TP  FX.B    3276840    3264120 100    3264034 100    3264120   0 11A2 FC300_R5_TP  FX.B    3276840    2328648  71    2328546  71    2328648   0 11A6 FC300_R5_TP  FX.B    3276840    2328444  71    2328355  71    2328444   0 11AA FC300_R5_TP  FX.B    3276840    2328528  71    2328403  71    2328528   0 11AE FC300_R5_TP  FX.B    3276840    2328648  71    2328468  71    2328648   0 11B2 FC300_R5_TP  FX.B    3276840    2130000  65    2129766  65    2130000   0 173A FC300_R5_TP  FX.B    6553680    6478572  99    6477747  99    6478572   0   File #2:  11AA FC300_R5_TP  FX.B    3276840    2328528  71    2328403  71    2328528   0 11AE FC300_R5_TP  FX.B    3276840    2328648  71    2328468  71    2328648   0 11B2 FC300_R5_TP  FX.B    3276840    2130000  65    2129766  65    2130000   0 173A FC300_R5_TP  FX.B    6553680    6478572  99    6477747  99    6478572   0 0BDB FC600_R5_TP  FX.B   33554640    6044364  18    6033105  18    6044364   0 0BDC FC600_R5_TP  FX.B   33554640    6613536  20    6481974  19    6613536   0 0BDD FC600_R5_TP  FX.B   33554640    4435848  13    4057170  12    4435848   0 0BDE FC600_R5_TP  FX.B   33554640    6620868  20    6249518  19    6620868   0   Desired Output  File #3:  0BDB FC600_R5_TP  FX.B   33554640    6044364  18    6033105  18    6044364   0 0BDC FC600_R5_TP  FX.B   33554640    6613536  20    6481974  19    6613536   0 0BDD FC600_R5_TP  FX.B   33554640    4435848  13    4057170  12    4435848   0 0BDE FC600_R5_TP  FX.B   33554640    6620868  20    6249518  19    6620868   0   I'd like to compare file 1 and file 2 using their first columns and remove the entire line or row from file 2 where they match in file 1. I'd also like to save the results to a 3rd file, file #3. Meaning of Conditional sentences with 'should' and 'happen to' I'm a bit familarized with conditional sentences. But the following sentences with 'should' and 'happen to' are quite confusing too me. They are:   If it should rain, I will stay at home. 2.If it should happen to rain, I'll stay at home   If it happens to rain, I'll stay at home.    What are the meaings of those sentences ? Do they all mean the same ? Thank you a lot in Advance for your answer. The youngest person on Earth in a society of immortals The protagonist is the youngest person on Earth -- the only one to be born in many centuries. Terrestrial society is small but every individual is unique and multitalented, as they are all immortal and have lived for centuries. They live in flying habitats.  No one knows how to raise children anymore; the protagonist wears diapers into adulthood because he was never potty-trained, and his education was haphazard. Consequently, he is quite lost in this advanced society, and only tolerated by the sophisticates around him.   When the Earth is invaded, the immortals put up only a show of a fight, arriving at the scene of battle in a colorful flotilla of unarmed flying habitats. To the horror of the protagonist, they are wiped out by the invaders. He realizes that the immortals were weary of life and welcomed this end.   This is probably a novel, though it might be a novella. It was written in English and I read it probably in the 1970s or 1980s. Where is .bashrc? How to keep any Window "Always on Top"? Some software have an option for their window to be "Always on Top". Is there any way to get the same functionality for any window. Let me give an example where this would be useful.  Suppose I have a small browser window with real time stats from a game. The window is automatically updated at the rate of once every 10 seconds. I would like the small window to be always on top. so that I can just look at it whenever I want to, without actually going to it and at the same time I could continue working in my main application, which is at full screen.  Is there any Windows hack or App which can give me this functionality? How do I center text horizontally and vertically in a TextView? Review Count - Fix It or Lose It Another meta question identified a bug that was marked as designed. To summarize the bug, the little number that appears next the Review link in the header shows an incorrect value most of the time.         I would like to propose that the feature be dropped. Honestly, I would prefer it be fixed, but the response to the question above gives me little faith that it will be. Half of the time, I click that little brown box and find out that I was lied to. There's nothing to review.  The little number no longer does the job it was created to do.        Oh sweet! Three things I can review!        Just kidding...