Let $ V(d,n) $ be the vector space over $ \mathbb{C} $, of forms of degree : $d$ , in $ \mathbb{C} [X_1 , \dots , X_n ] $.  How to show that : $ \mathrm{dim}_{ \mathbb{C} } V(d,n) = \binom{d+n-1}{n-1} $ ?  I think $ \mathrm{dim}_{ \mathbb{C} } V(d,n) = \mathrm{Card} \big( \{ (i_1 , \dots , i_n ) \in \{ 1 , \dots , n \}^n \ | \ i_1 + \dots + i_n = d \} \big) $ but, why is : $ \mathrm{Card} \big( \ \{ (i_1 , \dots , i_n ) \in \{ 1 , \dots , n \}^n \ | \ i_1 + \dots + i_n = d \ \} \big) = \binom{d+n-1}{n-1} $ ? This is in reality a combinatoric question, so we need to take in account repetition, without order. but, i still not understand why it's equal to : $ \binom{d+n-1}{n-1} $.  Thank you in advance.
Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ if $m &lt; 0$. This statement came up in an algebraic geometry text with no explanation provided, and I'm trying to understand why it's true.   Thoughts so far:   Since $H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m))$ is the vector space of regular sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ over $\mathbb{P}^n$, we can try to find a basis. Since the regular sections are the degree $m$ polynomials in $x_0, \dots, x_n$, it seems like the collection of degree $m$ monomials in $x_0, \dots, x_n$ should work. But I don't see how there are ${n + m \choose n}$ of these. Is it just a basic counting argument that I'm struggling with?  On a side note, I think I understand what global sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ look like, but how can we describe the regular sections over a distinguished open set $D(f)$? Is it something like degree $m$ polynomials with denominator $f$?
Are there regional distinctions in how hiccup/hiccough is spelled? Intuitive Explanation of Tippe Top Effect? Express solutions of equation $ \tan x= x $ in closed form I know that the equation $$ \tan(x) = x $$ can be solved using numerical methods, but I’m looking for a closed form of the solutions. In my opinion, having only numerical solutions means that we don’t know the problem, and sooner or later, we’ll be able to find a closed-form solution or at least a power-series solution. I’m looking for an explicit form of a sequence $ (x_{n})_{n \in \mathbb{Z}} $ in $ \mathbb{R} $ such that  $ 0 &lt; x_{0} &lt; \dfrac{\pi}{2} $, $ \tan(x_{n}) = x_{n} $ for all $ n \in \mathbb{Z} $, and $ (2 n - 1) \dfrac{\pi}{2} &lt; x_{n} &lt; (2 n + 1) \dfrac{\pi}{2} $ for all $ n \in \mathbb{Z} \setminus \{ 0 \} $.  The existence of such a sequence follows from the continuity of $ x \mapsto \tan(x) - x $ over its domain. Its uniqueness follows from the monotonicity of $ \tan(x) - x $ over the intervals $$ \left( (2 n - 1) \frac{\pi}{2},(2 n + 1) \frac{\pi}{2} \right), \quad n \in \mathbb{Z}. $$ Mismatch Reputation in the thumbnail and the actual Reputation Actually this was quite surprising. A casual troll in the Users Page revealed few users with negative vote. One example is here    Then trying to see what could have causes such a low reputation, I peeked inside the Profile Page of this particular User and was surprised to see a decent positive score.    Can someone explain to me how this can be possible? Difference between focusin/focusout and focus/blur, with example Maximum number of edges in a simple graph? I found that the maximum number of edges in a simple graph is equal to $$\sum\limits_{i=1}^{n-1} i$$  Where $n =$ number of vertices.  For example in a simple graph with $6$ vertices, there can be at most $15$ edges.  If there were any more edges then $2$ edges would connect the same pair of vertices and thus would not be a simple graph.  Is this a known theorem? Automate fixes for issues found by Sonar Taylor series $\frac{z^2+z}{(z-1)^2}$ at $z=-1$ split file lines based on first field I have file with content like below and want to convert my output like below   Input  1,a,b,c 2,b,c 3,e,f 4,l  Required output  1,a 1,b 1,c 2,b 2,c 3,e 3,f 4,l    Values on first field is unique and no duplicate lines for 1st field in the input.  I am new to scripting and not sure how can we do this. Top Down RPG Movement w/ Correction? I would hope that we have all played Zelda: A Link to the Past, please correct me if I'm wrong, but I want to emulate that kind of 2D, top-down character movement with a touch of correction. It has been done in other games, but I feel this reference would be the easiest to relate to.  More specifically the kind of movement and correction I'm talking about is:   Floating movement not restricted to tile based movement like Pokemon and other games where one tap of the movement pad moves you one square in that cardinal direction. This floating movement should be able to achieve diagonal motion. If you're walking West and you come to a wall that is diagonal in a North East/South West fashion, you are corrected into a South West movement even if you continue holding left (West) on the controller. This should work for both diagonals correcting in both directions. If you're a few pixels off from walking squarely into a door or hallway, you are corrected into walking through the hall or down the hallway, i.e. bumping into the corner causes you to be pushed into the hall/door.   I've hunted for efficient ways to achieve this and have had no luck. To be clear I'm talking about the human character's movement, not an NPC's movement. Are their resources available on this kind of movement? Equations or algorithms explained on a wiki or something?  I'm using the XNA Framework, is there anything in it to help with this? Several responses to a question  suggested I try Opera kiosk mode, so I decided to download it and give it a try. Once the download had completed and the disk image was being verified, Safari hung for over a minute and there were periods where Safari used a very high %CPU:    In Mac OS X 10.5, I was accustomed to seeing Safari freeze, but it never used up more than 200% CPU (2 cores x 100%). Is this a bug in Activity Monitor? Or does this mean that %CPU is now more akin to ? stop Ipod from asking for old Apple ID password Recently, I've changed my Apple ID because my old E-mail has been hacked. Now that I set the new one as my apple ID, the old Apple account still appears when I open almost all my apps on my Ipod Touch (4th Gen.), asking for the password to my old Apple ID. When I put in the old password to my old Apple ID, it'll keep asking for it no matter how many times I type it in. How do I fix this while keeping my account secure? Any suggestions would help! Replace a string in a text file using a shell script not working properly I have a text file containing this:  0 0 -1 0 1 0 0 0 0 -1 0 0 1.5 0.0 1.0 1   and I want to replace the 1.0 to 2.0 so that it becomes:  0 0 -1 0 1 0 0 0 0 -1 0 0 1.5 0.0 2.0 1   so I use this command:  sed -i 's/'1.0'/'2.0'/g' /home/user1/file1.txt   or this:  sed -i 's/1.0/2.0/g' /home/user1/file1.txt   but this is the result that appears and I don't know how to fix this:  0 0 -2.0 2.0 0 0 0 -2.0 0 1.5 0.0 2.0 1 Initializing K-means centers by the means of random subsamples of the dataset? If I have a certain dataset, how smart would it be to initialize cluster centers using means of random samples of that dataset?  For example, suppose I want 5 clusters. I take 5 random samples of say, size=20% of the original dataset. Could I then take the mean of each of these 5 random samples and use those means as my 5 initial cluster centers? I don't know where I read this but I wanted to know what you guys think about the idea.    UPDATE: Please see this thread  for the general discussion about the various initialization methods. How can I save the world from the spread of Hallow and Corruption/Crimson? After the Wall of Flesh, the Hallow appears and Corruption/Crimson spreads even faster.   Corruption/Crimson spreads through everything   Except Hallow  Hallow spreads through everything   Except mud (jungle) Except synthetic blocks like gray-brick  The results of Hallow and Corruption meeting are unpredictable   How can I stop them both permanently and effectively from taking over the world, considering both the top layer and underground?  (Eg, simply placing gray brick on top of the surface won't stop the underground spread?)  And, is it worth the effort? I've got some lenses of brand X, but I've just got a new camera of brand Y. How can I tell if it's possible to use my lenses on my camera? I accept that I may lose a lot of automatic functionality like autofocus, aperture control and metering when using an adapter to connect the two. Can someone give a simple explanation as to why the       $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$   doesn't converge, on the other hand it grows very slowly?   I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks. What is the average number of deaths per episode? While watching Game of Thrones and seeing people being killed left and right, I began to wonder:  Just how many people die per episode (not just major characters but the minor characters as well)?   I'm just wondering about an average, but a list that mentions the number of deaths for each episode would be good too. Weighing a flying bird How to close ports / disable services? I just portscanned my system with nmap and got the following result:  PORT    STATE SERVICE 25/tcp  open  smtp 631/tcp open  ipp   what are these services and how i do disable them? probably i don't need them.