Assuming you're not doing some early cheese build, I'd think this would be something you could just do some math on to figure out which is better. Has this problem been solved? Which is better, and what is the math behind it?
What is the advantage on saving the money for the overlord when you have 9 drones, and not filling your supply first and then getting the overlord?  In my theory, in the first case you get to wait the saving time with 9 drones, and get to wait almost the same time with 10 drones in the second case. Second case should be better, since 10 drones mine faster than 9? So why does everybody go with the first case?  All this is assuming you go 12 pool or something like that, where you will get more drones after the overlord. 6- and 8-pools are obviously different.
What's the use/meaning of the @ character in variable names in C#? I discovered that you can start your variable name with a '@' character in C#.  In my C# project I was using a web service (I added a web reference to my project) that was written in Java. One of the interface objects defined in the WSDL had a member variable with the name "params". Obviously this is a reserved word in C# so you can't have a class with a member variable with the name "params". The proxy object that was generated contained a property that looked like this:  public ArrayList @params {     get { return this.paramsField; }     set { this.paramsField = value; } }   I searched through the VS 2008 c# documentation but couldn't find anything about it. Also searching Google didn't give me any useful answers. So what is the exact meaning or use of the '@' character in a variable/property name? Optional argument after splat argument Creating shapefile from current dataframe extents in layout view of ArcMap? Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ? Problem:  Let $A_{n.n}$ be square complex matrix. Prove the following: $$\left \| A \right \|=\left \| A \right \|_{HS}\Leftrightarrow  rank(A)\leqslant 1$$. Where $\left \| . \right \|_{HS} $ is the Hilbert Schmidt Norm.  Please read my solution and tell me whether it is correct. If not, let me know where the mistake is.  Proof of the implication $\Leftarrow $:   If $rank(A)=0$, then in this case $A=0$. It follows that $\left \| A \right \|=\left \| A \right \|_{HS}=0$ If $rank(A)=1$, than: $ A=\begin{pmatrix} A_{1}\\  \alpha _{2}A_{1}\\  ...\\  \alpha _{n}A_{1} \end{pmatrix}$ where $A_{1}$ is the first row of $A$   and in this case: $\left \| A \right \|=\left \| A \right \|_{HS}=\left \| A_{1} \right \|\sqrt{1+\alpha _{1}^{2}+...+\alpha _{n}^{2}}$  Proof of the implication $\Rightarrow $   We know that $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|$  On the other hand: $ A=\begin{pmatrix} A_{1}\\  A_{2}\\  ... \\ A_{n}  \end{pmatrix}$. So, $$\left \| Ax \right \|=\left \| \begin{pmatrix} \left \langle A_{1},x \right \rangle\\  \left \langle A_{2},x \right \rangle\\  ...\\   \left \langle A_{n},x \right \rangle\end{pmatrix} \right \|=\sqrt{\left \langle A_{1},x \right \rangle^{2}+\left \langle A_{2},x \right \rangle^{2}+...+\left \langle A_{n},x \right \rangle^{2}} $$  Where $ \left \langle A_{i},x \right \rangle$ is the inner product of $A_{i}$ and $x$  Using the Cauchy-Schwarz inequality: $\left \langle A_{i},x \right \rangle^{2}\leq \left \| A_{i} \right \|^{2}\left \| x \right \|^{2}$, we get: $\left \| A \right \|=max_{\left \| x \right \|=1}\left \| Ax \right \|\leq max_{\left \| x \right \|=1}\left \| x \right \|\left \| A \right \|_{HS}=\left \| A \right \|_{HS}$. In order to have the equality: $\left \| A \right \|=\left \| A \right \|_{HS}$, we should have: $\left \langle A_{i},x \right \rangle^{2}=\left \| A_{i} \right \|^{2}\left \| x \right \|^{2} $ which occurs only if $A_{i}=\lambda _{i} x$. So, the rows of A are dependent, which implies that $rank(A)=1$. Note that the case $rank(A)=0$ happens when $A=0$  Please let me know if my solution makes sense. I'm going to install Linux on Lenovo B5400, which contains:   500GB Hybrid SSHD with a 8 GB SSD and 500 GB HDD   Designing partitions for my 2 Linux distros was quite easy before, when I had only HDD. I always had:   10GB for Linux distro (mounted as "/") 10GB for another distro (again "/"; for another distro; for testing purposes or to assure continuity when switching distros) 2GB for swap (used for both distros; no need to separate) 300GB+ for "/home" (used for both distros; different usernames eliminates possible conflicts)   It's simple, isn't it? I would like to keep this approach, but I know that SSD has its own specificity - it prefers to be read, continuous writing may affect its life span. I don't want to kill SSD in a year, so I would like to move some system partitions to HDD in order to protect SSD. As far as I know /tmp is one of often read&amp;written locations. How much space it needs? What else should be placed on HDD?  Or: how to design partitions to achieve exactly the same effect and to keep SSD as safe as possible?  I'd be grateful for your help. What space does the spell Cloud of Daggers fill? I am trying to workout what template  (PHB, p. 222) would be on a grid. The spell states:     You fill the air with spinning daggers in a cube 5 feet on each side, centered on a point you choose within range.   So does that mean it would look like this:     Or like this:    From what I've read online, there seems to be some suggestion that it is a single square (tile) though this does stack with the spell description of "a cube 5 feet on each side".  Clearing this up for my group would be great, and I am sure our DM will be pleased as his monster were nicely diced by this spell. Thanks. How can I write multiple lines in a subscript? Is there a way to force a line break inside a math mode text?  My example:  \[   \underbrace{....}_\text{Some long text that should be multiline} \]   Trying  \text{Some long text that\\ should be multiline}   Didn't do the trick. As a non-native speaker of English, it can often be hard to spot some of the errors that a native speaker would find. For a long time, I've been thinking about getting a native speaker to help me with proof-reading my texts (articles, theses, application, etc), but I don't quite know where to look. What are some good ways to get proofreading for your texts?  For example, are there any websites on the Internet where one can exchange proofreading services between different languages? (Say, if you help me proofread my text in language x, I'll help you with yours in language y.) Notation for infinite product in reverse order Cannot select vertices from different object while in edit mode Whenever I enter edit mode, with 2 separate meshes (cubes), I can only select vertices/edges/faces of 1 mesh.  Example (the screen shot did not include the mouse cursor):    Also, when I enter edit mode with both meshes selected, one of the meshes has a red border and cannot be selected (it also has this border in Object mode): Are there any programs that will let me dictate (speech-to-text) to any program? Or at least to a text editor and text fields in Firefox.  I’m not interested in anything that uses an online service, it has to be on-device. It may be a commercial product. Improving Review Queues - Design overview II: Changing review bans and other new features Is the space $\mathbb{N}$ is a baire spaces ? yes/No Is  the  space $\mathbb{N}$  is a  baire  spaces ?  My attempts :   No ,  It  will not form a  baires  spaces ,  let $C_n= \mathbb{N}\setminus\{n\}$. This is open (since $\{n\}$ is closed in $\mathbb{R}$, hence in the induced topology of $\mathbb{N}$) and dense, since every open ball with center in $\mathbb{n}$ intersects ${C_n}$. But  $$\bigcap_{n\in\mathbb{N}} {C}_n = \emptyset$$  So $\mathbb{N}$ is not a baires  spaces because for  Baire Space the intersection of a countable family of open dense sets must be dense.   Is  Its correct or not ? This was a question my friend found in a book.  A block attached to a spring pulled by a constant horizontal force is kept on a smooth horizontal surface. Initially the spring is in its natural state. Then the maximum work that the applied force F can do is:  (a) $F^2/k$ (b) $2F^2/k$ (c) $4F^2/k$ (d) $F^2/2k$  My friend thought that the answer is (a) since the maximum displacement is $x=F/k$ and $W=Fx=F^2/k$. But he saw that the book's answer is different and he asked this question to me. I at first thought that the answer will be (a), but then I realised that this situation is somewhat similar to a charging capacitor. The final energy of the spring is actually $\frac12kx^2=F^2/2k$. So probably the answer is (d)? But the work done is still (a)? But the book says neither. It's (b). This is the logic in the book:  Method 1: in the book:  $$Fx=\frac12kx^2$$  $$x=2F/k$$  $$W=\frac12kx^2=2F^2/k$$  (since the book is an objective questions book it did not have detailed explanations)  What is wrong in the following methods?  Method 2: by calculating the maximum displacement  $$x=F/k$$  $$W=Fx=F^2/k.$$ Method 3: by calculating the final energy  $$x=F/k$$  $$\frac12kx^2=F^2/2k$$  PS: We both know calculus so don't hesitate to use it in the answers. Using `M` instead of `k` when rep is >= 1M on https://chat.stackexchange.com/users I read on  ()    Shouldn't it be 1.03m or (if we really follow the conventions of the metric system) 1.03M instead? How do I set a variable to the output of a command in Bash? Where can I find people for hire with LaTeX skills and designer skills? Understanding a particular evaluation of $\prod\limits_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)$ I'm having a hard time understanding the following evaluation of the infinite product  $$ \prod_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right).$$ In particular, I don't understand how you go from line 2 to line 3. Here $\omega = -\frac{1}{2}+ i \frac{\sqrt{3}}{2}$, which is a primitive third root of unity.  $$\begin{align}\prod_{n=2}^{\infty}\left(1-\frac{1}{n^3}\right)  &amp;=\prod_{n=2}^{\infty}\frac{(n-1)(n^2+n+1)}{n^3} \\  &amp;=\lim_{m\to\infty}\frac{1}{m}\prod_{n=2}^m\frac{(n-\omega)(n-\omega^2)}{n^2} \\  &amp;=\lim_{m\to\infty}\frac{\Gamma(m+1-\omega)\Gamma(m+1-\omega^2)}{m(m!)^2\Gamma(-\omega)\Gamma(-\omega^2)(1-\omega)(1-\omega^2)(-\omega)(-\omega^2)} \\  &amp;=\frac{1}{3\Gamma(-\omega)\Gamma(-\omega^2)} \\ &amp;=\frac{\sin{\pi(-\omega)}}{3\pi} \\  &amp;=\frac{\cosh (\frac{\sqrt{3}\pi}{2})}{3\pi } \end{align}$$ EDIT: I think I'm starting to make sense out of this by writing out the terms. $$ \begin{align} \prod_{n=2}^{m} \frac{(n- \omega)(n-\omega^{2})}{n^{2}} &amp;= \frac{(2-\omega)(3-\omega) \cdots (m- \omega)(2-\omega^{2})(3-\omega^{2})\cdots (m- \omega^{2})}{2^{2} \cdot 3^{2} \cdot \cdots  \cdot m^{2}}  \\ &amp;= \frac{1}{(m!)^{2}} \frac{\Gamma(m+1-\omega)}{(1-\omega)(-\omega)\Gamma(-\omega)} \frac{\Gamma(m+1-\omega^{2})}{(1-\omega^{2})(-\omega^{2})\Gamma(-\omega^{2})} \end{align}$$ Now I have to figure out how to take the limit as $m \to \infty$. Custom fields not useable in Approve/Reject Approval Request I'm trying to get through my first SalesForce mod and it's definitely been a rocky path. I have one task I'm trying to do, but this section is holding me back.  I have two custom fields.  One a pick list and the other a text box. I have added them to the Approve/Reject Approval Request section in the Opportunity page. I added the fields through the Select Fields to Display on Approval Page Layout and they DO show up on the Approve/Reject Approval Request section. I added them in like ().  However, just the label for the fields show, the field itself is nowhere to be found and can't be used.    You can see and even use the fields when you open the Opportunity page itself, but when you go to use the Approve/Reject, you can't do anything with them or see any values.  I was never a SalesForce user and got this tossed in my lap and really don't know my way around yet. I appreciate any help on this!!  Thanks!!  Kurt