I enjoy taking photos with my current telephoto (55-250) but I wonder if the 70-300 could be a worthwhile upgrade. As of November 2013, it would be around a $350 investment if I deal hunt and sell my current lens.  I believe that I would benefit from the USM, for achieving focus quikcly as I am usually shooting moving subjects and my current lens is a bit slow.  How can I assess if the upgrade would be worthwhile? What improvements I could expect (with regard to quality of the pictures, AF, quietness, build quality....) stepping up to the 70-300 lens? Are there any downsides to consider?
Should I buy a Canon 55-250mm lens or 70-300mm for my Rebel T3i? I have the basic 18-55mm lens that came with my camera but I need a next step up that's not too expensive.   What are the most common types of photos shot with either lenses? Or what is each lens best for?
EDIT:  The vector space for the $(\frac{1}{2},0)$ Representation is $\mathbb{C}^2$ as mentioned by Qmechanic in the comments to his answer below! The vector spaces for the other representations remain unanswered.  The definition of a representation is a map (a homomorphism) to the space of linear operators over a vector space. My question is: What are the corresponding vector spaces for the   $(0,0)$ Representation $(\frac{1}{2},0)$ Representation  $(0,\frac{1}{2})$ Representation $(\frac{1}{2},0) \oplus (0,\frac{1}{2}) $ Representation $(\frac{1}{2},\frac{1}{2})$ Representation infinite dimensional Representation? I'm an elected diamond moderator on one site, and a user of several others. I think it would be extremely helpful for our flagging process if users had the option to enter freeform text explaining their flag on types of flags other than the "in need of moderator attention" flag.  An example of when this is relevant: Sometimes posts have had rude/abusive flags, but the rude/abusive part might be difficult to notice, or it might be a single word or sentence buried somewhere in the middle of the post. Users have to choose between "flag it as rude with no explanation" or "flag it as misc and explain what's going on". Often they'll pick the first because of course it seems like what they should do (e.g. they know the system has special handling attached to rude/abusive flags) but it means they can't explain what's going on to us, so we have to reach out directly in private to find out.  Sometimes it also helps to be able to explain why a comment is no longer needed, because past experience tells me the "no longer needed" flag on its own might be declined on a particular site -- so I pick the custom reason instead of "no longer needed" just so I can provide explanation.  I'd prefer users be able to understand they can pick the appropriate reason and be able to attach any explanation they deem necessary to further direct moderators to the issue. If not for all kinds of flags, then at least for spam or rude/abusive flags. Link Google Form responses To Already Existing Google Spreadsheet Tab I have a Google Form. I had all of my Google Form responses linked to a specific tab in my Google Spreadsheet. Somehow my form responses were unlinked from my tab. When I go to re-link my form responses to my already existing tab it only gives me the option to create a new tab in my spreadsheet.   How do I simply re-link my Google Form responses to my already existing tab in my Google Spreadsheet? Two monitors when only having one in 13.10 I just did a fresh install of Ubuntu 13.10 64bit, and it says that I have 2 monitors, when I actually only have one    I have a Dell XPS 15 with Nvidia graphics card if that helps in anyway I have to find the limit of this sum: $$\sum_{r=1}^{\infty}\cot ^{-1}(r^2+\frac{3}{4})$$  I tried using sandwich theorem , observing:  $$\cot ^{-1}(r^3)\leq\cot ^{-1}(r^2+\frac{3}{4})\leq\cot ^{-1}(r^2)$$  Now when I was calculating the limit of left hand expression, I convert it to $\tan^{-1}$, by using:  $$\tan^{-1}\frac{1}{x} = \cot^{-1}x$$  but couldn't sum up the terms of arctan series.     How can I proceed?   Is there any better way ? Why do we have an elementary charge $e$ in physics but no elementary mass? Is an elementary mass ruled out by experiment or is an elementary mass forbidden by some theoretical reason? What do I if I used "Reset All" or "Erase All Content and Settings" on a jailbroken iOS Device? I recently received an iOS device and it was jailbroken. When I wanted to unjailbreak it, I used "Erase All Content and Settings." The loading screen came on and it has not changed. After rebooting the device, it is now stuck at the Apple logo. What should I do? A random variable that induces a $\sigma$-algebra the same as the one in the sample space I sometimes browse erotic media files but I would like to avoid having them appear in files &amp; folders / recent. How can I do that?  The old "making .recently-used.xbel a folder" trick from 10.10 and older doesn't work any more.   Plus I'd prefer to turn off tracking only temporarily.  I also prefer not to use a separate user for this, as switching users sometimes isn't fast enough. To what extent is WhatsApp's statement on secure messaging realistic? Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $ I found an interesting exercise of other amazing book (Jech's) that I'm not totally sure about how to do it.   Prove the following form of the distributive law:   $$\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $$  Assuming that $F_{a,b_1} \cap F_{a,b_2} = \emptyset $ for each $a\in A$ and $b_1,b_2\in B$ and $b_1\not=b_2$.  First of all, I'm not completely sure about what $ F_{a,b}$ really means. I suppose that is  the range of a family with a domain $A \times B$.    And second, I cannot figure out how to do the converse (assuming that the first part it is correct, I used as normal in this kind of proof element chasing)  Proof:  ($\Rightarrow$) ...  ($\Leftarrow$) Suppose that $z\in \bigcup_{f\in ^AB} \big(\, \bigcap_{a \in A} F_{a,f(a)}\,\big)$; then there is $f\in \,^AB$ such that $z\in  \bigcap_{a \in A} F_{a,f(a)}\, $. Let $a \in A$ be arbitrary.   Claim 1 $\, \bigcap_{a \in A} F_{a,f(a)}\subseteq F_{a,f(a)} \subseteq \bigcup_{b\in B} F_{a,b}$.  Proof of the Claim 1:   For the first inclusion: Suppose $z\in\bigcap_{a \in A} F_{a,f(a)}$; then for each $a\in A$ we have that $z\in F_{a,f(a)}$. Then $\bigcap_{a \in A} F_{a,f(a)}\subseteq F_{a,f(a)}$.  For the second inclusion: Now suppose $z\in F_{a,f(a)}$. Clearly $f(a)\in B$ because $f\in ^AB$. Then there is some $b\in B$ such that  $z\in F_{a,b}$ and hence that $z\in \bigcup_{b\in B} F_{a,b}$. $\square$  Since $a$ was arbitrary it follows that $z\in \bigcup_{b\in B} F_{a,b}$ (by claim 1) for each $a\in A$ and hence  that $z\in \bigcap_{a\in A} \big(\, \bigcup_{b\in B} F_{a,b} \, \big)$ as desired.  I really, really would very much appreciate some help with that. Thanks in advance as usual. Undefined Symbol ___gxx_personality_v0 on link Is it possible to switch from one area of graduate study to another in US universities? For example, suppose someone has enrolled in a computer science phd program. Can he switch over to math(or physics) phd program in the same school later?(or say from applied mathematics to pure mathematics?)  What are the steps for doing for doing that? Given a number, how do I discover in what table and column it could be found within? I don't care if it's fast, it just needs to work. Posting and answering questions you have already found the answer to Concatenate several columns into one in Google Sheets I'm planning on selling a USB external hard drive that currently contains an old Ubuntu installation with stored passwords and banking information.  How can I securely erase the drive before selling it? I am creating a controller for a 12 V system using an Arduino Uno microcrontroller. For the outputs I am using a relay shield to switch the 12 V components. I have a 12 V toggle switch that turns on some 12 V components in the system and I want to use a trigger signal off of this same switch to send to an Arduino digital input. I know that the Arduino can only handle 5 V max. What would be the best way to step down the 12 V coming off of the switch to the 5 V for the input?  EDIT: The system is for use in a car. Would the amperage of the car battery need to be lowered somehow as to not blow up the components? Trying to understand what happens with a brokerage firm's customers when it goes bankrupt (case of study: Lehman Brothers) I'm trying to understand how secure it is to own mutual funds/ETFs, not in terms of market volatility etc., but in terms of judicial security (i.e. what happens to your holdings if the firm goes bankrupt). To understand it, I have read this article about the bankruptcy of Lehman Brothers:  Note: From now on, I'm always referring to investors who own funds/ETFs managed by Lehman Brothers, not by investors who invested in Lehman Brothers or subprime mortgages (e.g. S&amp;P 500 ETF investors). From the above article, what I have understood is that the funds/ETFs managed by Lehman Bros. were shifted to other brokerage firms to be managed by them. Is this what actually happened? Still, what happens with the investors if, hypothetically, no brokerage firm in the world wants to manage these financial products? Proving $6^n - 1$ is always divisible by $5$ by induction I'm trying to prove the following, but can't seem to understand it. Can somebody help?     Prove $6^n - 1$ is always divisible by $5$ for $n \geq 1$.   What I've done:  Base Case: $n = 1$: $6^1 - 1 = 5$, which is divisible by $5$ so TRUE.  Assume true for $n = k$, where $k \geq 1$: $6^k - 1 = 5P$.  Should be true for $n = k + 1$  $6^{k + 1} - 1 = 5Q$  $= 6 \cdot 6^k - 1$  However, I am unsure on where to go from here.