My understanding is that a larger focal length corresponds to a larger working aperture diameter, since the numerator is larger.  Why doesn't a larger working diameter correspond to a brighter image? Shouldn't there be more light coming into the lens?  Wouldn't a subject framed with 42.5/1.2 lens be brighter than a 35/1.2 if the subject is framed the same?
Cheaper zoom lenses usually are faster at the wide end and slower at the long end (for example, the $150 Canon EF-S 18-55mm f/3.5-5.6). More expensive constant-aperture zoom lenses have the same aperture regardless (for example, $800 Canon EF 17-40mm f/4.0 L).  My question is: are these good lenses  at the wider settings, or do they have a different optic system that allows them to maintain the same aperture throughout the zoom range?
Best way to extract a subvector from a vector? mysql database no space available Can a package overwrite the original commands in Latex? An example:  Without the amsmath package:   The equation environment numbers the equation automatically. The align environment cannot be executed.   With the amsmath package:   align and equation work seamlessly in numbering equations: I can use equation to write my first two equations, and align to write my third equation, and Latex will still number them correctly.   My questions are:   Is the code for the command equation after loading the amsmath package still the same as it was before loading the package? Moreover, if this is the case, then, in general, can loading different packages cause conflicts? And how do we avoid it? On equations $m^2+1=5^n$ Is there a printf converter to print in binary format? I can print with printf as a hex or octal number.  Is there a format tag to print as binary, or arbitrary base? I am running gcc. printf(&quot;%d %x %o\n&quot;, 10, 10, 10); //prints &quot;10 A 12\n&quot; print(&quot;%b\n&quot;, 10); // prints &quot;%b\n&quot; How do I change my login provider(s)? I no longer use the service that I originally used to register, and would like to change the service I use to log in, or at least add a second account.  Is this possible? How do I change my login provider or associate a backup method to log in?     For more information, see "" in the . In how many ways can a number be expressed as a sum of consecutive numbers? All the positive numbers can be expressed as a sum of one, two or more consecutive positive integers. For example $9$ can be expressed in three such ways, $2+3+4$, $4+5$ or simply  $9$. In how many ways can a number be expressed as a sum of consecutive numbers?  In how many ways can this work for $65$?  Here, for $9$ answer is $3$, for $10$ answer is $3$, for $11$ answer is $2$. Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$ I was solving a few problems from a textbook and I came across this one:     Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through   the point $P=(12,3)$   I could find the tangency points $$ but my solution was completely messy (I had four equations that I had to go back-and-forth with.  Is there an elegant/simple solution to that?  Tangency points:  $$P_1 = (0,3)$$ $$P_2 = \left( \frac{24}{5} , 3-\frac{24}{5} \right)$$ Running an infinite loop on startup I'd like to run a command at startup which pings a certain address every 10 minutes and writes the result to a file. I've figured out now how to do the pinging and file writing and the 10 min intervals:  while true; do my-command-here; sleep 600; done   My question is, can I put this in /etc/init.d/rc.local or should I be putting it in /etc/rc.local or somewhere else entirely? I'm specifically concerned because it's an infinite loop so I'm not sure if I could put it in one of these startup scripts.  Some help would be appreciated. I'm using Ubuntu 12.04.5 Submitting a URL to Google using PHP/cUrl How to merge partitions? I have two partitions, one of them has a file system and the other is unallocated.  How to merge two partitions? Using subcaption package but subfigures end up stacked rather than side by side Is there an easy way to add a hole into an object? If I wanted to add a hole into an object e.g. Create a tube  Is there an easy way to do that?  I haven't seen anything regarding negative space so far in Blender, and it would be really helpful for my project  Thanks in advance! Let $\mathbb{R^{ n×n}}$ be the space of n × n real matrices. Show that if $ A(t) \in \mathbb{R ^{n×n}}$ is a differentiable function of $t$ then $\frac{d(det A(t))}{dt}$ is the sum of the determinants of $n$ matrices, in which the $i$th matrix is $A(t)$ except that the $i$th row is differentiated. Use this result to prove that the directional derivative of the determinant function at the matrix $A \in \mathbb{R ^{n×n}}$ along the direction $B \in \mathbb{R^{ n×n}}$ is given by $$(det)^{'} (A; B) = tr(Adj(A)B) = \langle Adj(A)^{T} , B \rangle $$ where $Adj(A)$ is the adjoint of $A$, and where the inner product on $\mathbb{R^{n×n}}$ is the trace inner product given by $\langle X, Y \rangle = tr(X^{T} Y )$ Conclude that $$D(det)(A) = Adj(A) ^T $$. ZFC + $\exists$ Standard model $\rightarrow$ Con(ZFC + $\exists \omega$-model) $ZFC + \exists V_\alpha$ model of $ZFC \vdash Con(ZFC + \exists$ transitive standard model of $ZFC)$  and then  $ZFC + \exists$ transitive standard model of $ZFC \vdash Con(ZFC + \exists \omega-model$ of $ZFC)$  For the first one :   We can always find a countable extentional $M \subset V_\alpha$ elementary equivalent to $V_\alpha$. Let $M&#39;$ be the mostowski collapse of $M$. $M&#39; \approx M$ so $M&#39;$ is model of ZFC. And because $M&#39;$ is countable and transitive then $M&#39; \in V_\alpha$ (since $H_{\omega_1} \subset V_{\omega_1}$ and $\alpha$ is surely far larger than $\omega_1$).   So $V_\alpha$ is the model of '$\exists$ a standard transitive model of ZFC'.  For the second one :   I don't really know how to do it... Does anyone have an idea ? Use central limit to solve a problem I am asked to solve the following question using central limit theorem. In an election between two candidates, A and B, one million individuals cast theirvote. Among these, 2000 know candidate A from her election campaign and vote unanimously for her.The remaining 998000 voters are undecided and make their decision independently of each other byflipping a fair coin.Approximate the probability pA that candidate A wins up to 3 significant figures.  It's easy to solve directly. PA = $\frac{0.5(10000-2000)+2000}{10000}$ = 0.501. However, I am quite confused about how to solve this problem by central limit theorem. Fixing "Page is not mobile friendly" report from Google test In search results google reports "this page is not mobile-friendly". I want to fix this.   when I test my website in the mobile-friendly test, it reports:   Content wider than screen. Clickable elements too close together.   How can I find the items/content that are creating these problems?  Any help would be appreciated. How to validate an email address in JavaScript How to detect if a variable is an array My question is about $CFT_1$. Page 18 of  says that $$L={\frac{\overset{.}{Q}^2}{2} - \frac{g}{2Q^2}}\tag{1.11}$$ is the most general Lagrangian that preserves time translation and scale invariance in $CFT_1$. How does one prove that?