How does a 16-300mm f/3.5-6.3 lens differ from a 50mm f/1.8?
If I have a 18-55mm lens, is there a point in buying a 35mm prime lens?
Let $K$ be any field. Is it true that:   If $f_1,\dots, f_k \in K[x_1,\dots,x_n]$ are polynomials of degree $1$, then $I=(f_1,\dots, f_k)$ is all $K[x_1,\dots,x_n]$, or, if it is a proper ideal, then is it prime? In particular, is it true that $K[x_1,\dots,x_n]/I$ is isomorphic to a polynomial ring (in the case that $I$ is proper and hence prime by the statement above)?   If they are true, how can i prove these statements in an elementary but rigorous way?  P.S.: Birth of my question: I found on my book that "affine varieties given by linear equations are irreducibile", so I translated (I don't know if I have made it right) this geometric statement into the analogous algebraic version. What does echelon mean? When you solve a system of linear equations, you write down the augmented matrix and reduce this to reduced row echelon form. What is the meaning of the word echelon? Close Lightning Modal by clicking anywhere outside of it Just wondering whether this is possible? i.e. once a modal is open, close it by clicking anywhere on the grey backdrop?  I've tried invoking my custom close-modal function via: - Adding onclick to the slds-backdrop div - Adding onblur to the slds-modal__container div - Adding onclick to the slds-modal (just wanted to see whether this would work)  None of these seem to work. Covariance of Euler-Lagrange equations under change of generalized coordinates Developer Accounts for ExactTarget Cannot delete directory with Directory.Delete(path, true) I'm using .NET 3.5, trying to recursively delete a directory using:  Directory.Delete(myPath, true);   My understanding is that this should throw if files are in use or there is a permissions problem, but otherwise it should delete the directory and all of its contents.  However, I occasionally get this:  System.IO.IOException: The directory is not empty.     at System.IO.__Error.WinIOError(Int32 errorCode, String maybeFullPath)     at System.IO.Directory.DeleteHelper(String fullPath, String userPath, Boolean recursive)     at System.IO.Directory.Delete(String fullPath, String userPath, Boolean recursive)     ...   I'm not surprised that the method sometimes throws, but I'm surprised to get this particular message when recursive is true.  (I know the directory is not empty.)  Is there a reason I'd see this instead of AccessViolationException? Solving $\cos x=x$ I would like to know how can we solve the equation $\cos x = x$, without graphing. I know that  there would only be one solution, that is obvious, that too in between $0$  and $\frac{\pi}{2}$. Is there any real expression in finite terms [perhaps we call that closed form, I am not sure] that could give $x$ or $\cos x$. Although I have not studied Taylor series, I know that it only gives an infinite series, which is not what I want. I suspect that it could not be done, but can anyone explain me why?  Just for completeness, Wolfram Alpha, gives the approximate answer $x = 0.7390851332151606416553120876738734040134$, but fails to give a exact solution. Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2}  \right |\leq \left |b-c  \right |$. Give me a clue, I don't have any idea to work with that inequality. What is this “(arg: 1)” appearing in terminal/tty after I pressed a combination of keys? In a virtual terminal I (accidentally) pressed Win+1 and got a new line consisting of  (arg: 1)   As long as I keep Win pressed, all numbers I press get appended to the number in the parentheses, e.g. Win and 1+4 yields (arg:14).  When I (after releasing Win) press any key, that character is printed as many times as the number reads and the (arg: ...) expression is replaced by the standard prompt.  Does anybody know what happens here? Nick Fury told Captain America that he has been "asleep" for nearly 70 years. If Dominic Cooper was playing Howard Stark Senior (grandfather of Tony Stark) then it is fine, but if he was playing Howard Stark (father of Tony Stark) then after 70 years he should be around 90 or above.  Tony Stark, his son, should be 60 to 70 years.  Is that possible?  Tony doesn't look 70 years in The Avengers. I cannot make xmodmap run on login in Ubuntu 13.04. I have searched around, including in askubuntu and all answers I found do not work.  I have a file named .Xmodmap in my home folder which contains the desired mappings. If I run it manually it works fine, so there is no error there. In order to make it run on startup, I have tried:     Using various .xinitrc containing one of the following commands:      xmodmap .Xmodmap xmodmap ~/.Xmodmap   /usr/bin/xmodmap /home/NAME/.Xmodmap if [ -s ~/.Xmodmap ]; then xmodmap ~/.Xmodmap fi  Using .Xresources instead of .xinitrc with the same variations. Setting it as a startup application.   None of the above works in Ubuntu 13.04 (64-bit). I also saw a few ideas about globally changing the keyboard mappings, but most of them were not applicable in Ubuntu 13.04 (i.e. the corresponding files did not exist where specified) and I prefer not to touch the global settings anyway. Is it appropriate to ask questions about software licenses/licensing on ? Or is the appropriate place  ? Or is there no appropriate site currently and we should suggest such a site on ? I have a list of dictionaries and want each item to be sorted by a specific value. Take into consideration the list: [{'name':'Homer', 'age':39}, {'name':'Bart', 'age':10}]  When sorted by name, it should become: [{'name':'Bart', 'age':10}, {'name':'Homer', 'age':39}] Interpretation of R's lm() output The help pages in R assume I know what those numbers mean, but I don't. I'm trying to really intuitively understand every number here. I will just post the output and comment on what I found out. There might (will) be mistakes, as I'll just write what I assume. Mainly I'd like to know what the t-value in the coefficients mean, and why they print the residual standard error.   Call: lm(formula = iris$Sepal.Width ~ iris$Petal.Width)  Residuals:      Min       1Q   Median       3Q      Max  -1.09907 -0.23626 -0.01064  0.23345  1.17532    This is a 5-point-summary of the residuals (their mean is always 0, right?). The numbers can be used (I'm guessing here) to quickly see if there are any big outliers. Also you can already see it here if the residuals are far from normally distributed (they should be normally distributed).  Coefficients:                  Estimate Std. Error t value Pr(&gt;|t|)     (Intercept)       3.30843    0.06210  53.278  &lt; 2e-16 *** iris$Petal.Width -0.20936    0.04374  -4.786 4.07e-06 *** --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1    Estimates $\hat{\beta_i}$, computed by least squares regression. Also, the standard error is $\sigma_{\beta_i}$. I'd like to know how this is calculated.  I have no idea where the t-value and the corresponding p-value come from. I know $\hat{\beta}$ should be normal distributed, but how is the t-value calculated?  Residual standard error: 0.407 on 148 degrees of freedom   $\sqrt{ \frac{1}{n-p} \epsilon^T\epsilon }$, I guess. But why do we calculate that, and what does it tell us?  Multiple R-squared: 0.134,  Adjusted R-squared: 0.1282    $ R^2 = \frac{s_\hat{y}^2}{s_y^2} $, which is $ \frac{\sum_{i=1}^n (\hat{y_i}-\bar{y})^2}{\sum_{i=1}^n (y_i-\bar{y})^2} $. The ratio is close to 1 if the points lie on a straight line, and 0 if they are random.  What is the adjusted R-squared?  F-statistic: 22.91 on 1 and 148 DF,  p-value: 4.073e-06    F and p for the whole model, not only for single $\beta_i$s as previous. The F value is $ \frac{s^2_{\hat{y}}}{\sum\epsilon_i} $. The bigger it grows, the more unlikely it is that the $\beta$'s do not have any effect at all. I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day. Why are two different concepts both called "heap"? How to preserve the same parskip in minipage Upgrade to version newer than repositories I have  about how to upgrade ghostscript to a version that is available on 15.04 but not on 14.04LTS (which I have). On this site there are numerous almost identical question regarding different packages, a few examples include:         So I thought it would be very useful in general to know: is there  a general, automated way to upgrade a package, including all its dependencies, to a version that is newer than Ubuntu's repos? Is this more easily done if the newer version exists in more recent Ubuntu releases? As the title says, how can I make this shape without resorting to a Boolean operation?    This is obviously a very simple example, but I find myself needing to do something like this relatively often when modelling.  Despite using Blender for a couple of years, and considering myself an intermediate to possibly advanced user, I still know of no easy way to make this kind of indentation in an object.  Obviously, if you are looking to make an indentation in the middle of the surface, you can simply select a face and extrude down. As you will know, this does not work at the edge of the surface.  I can create this shape using loop cuts, deleting the faces and then using snap to edge to extrude the edges down to the same level (and then filling in the faces), but it feels cumbersome and a lot of work for what feels like it should be very simple. How do I set the default program?