Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be holomorphic on the form $f(x+iy)=u(x)+iv(y)$ where $u,v$ are real functions. Prove that $$f(z)=\lambda z+c$$   for  $\lambda \in \mathbb{R}$ and $c \in \mathbb{C}$.    How do I prove this? I think I have to use the Cauchy-Riemann equations somehow but I can't quite figure it out. Thank you!
Finding all differentiable $f(z) = u(x) + iv(y)$ in $\mathbb{C}$ where $u(x),v(y)$ are real valued functions.   I’m not sure what to do. Would $f$ be differentiable simply if and only if both $u$ and $v$ were differentiable?   My thinking is that if the limit $$ \lim_{h \to 0}\frac{f(z+h)-f(z)}{h} $$  exists then it would be equal to  $$ \lim_{h \to 0}\frac{u(x+h)-u(x)}{h} + i\lim_{h \to 0}\frac{v(y+h)-v(y)}{h} $$  and therefore would exist if they existed. But that doesn’t seem quite right.  I also thought about using Cauchy-Riemann equations, but seeing that I’m trying to find ones that are differentiable rather than those that aren’t, I thought they wouldn’t be of much help.
Add "Delete with reason" button/dialog to existing moderator tools for flagged answers While this is an old problem, it's one that will only increase with time.  As Stack Exchange sites gain in popularity, more and more users are coming to the sites in search of answers to their specific problems.  Unfortunately, not all of them understand the basic operating principals behind Stack Exchange, and ask questions, or start discussions as separate answers.  Deleting these answers is fine, but more often (I would argue almost always) than not, a new user doesn't understand why.  Granted, comments are a way of doing this, but there are drawbacks to comments:   They require extra, replicated effort on the part of the moderator for something that they do repetitively The message will be inconsistent; messages like this would be consistent, cite the relevant resources that guide the user in the right direction The larger size would have a higher chance of grabbing the user's attention, as well as put more weight behind the action (as it's officially attached to the answer in a consistent manner). The banner will serve as an example of what not to do to other users who can see the deleted post.   Additionally, there are tools that can help with auto-commenting on posts, but these have their drawbacks as well:   Not all moderators use these tools The messages that are delivered by the tools are ultimately generated initially by the moderators, so the message will not be consistent These tools are not always available on other devices (e.g., mobile platforms)   It would be helpful to the user if they had a reason indicating why the answer was deleted.  Something along the lines of:    In order to accomplish this, I propose an additional (not a replacement) button in the moderation tools labeled "delete with reason".  Mods would still be able to delete answers outright, they just have the option now of stating why in an easy manner which will allows for more time to moderate harder.  Placement is not important (screen real estate is valuable, yes), but I envision it looking something like this in the moderator tools:    When clicking "delete with reason", a dialog, much like the "close" dialog will appear with options:    Initial options for the reasons could be:   Asking a new question Starting discussion   A mechanism like this has the potential for other benefits as well:   Users with an answer deleted with a reason might actually be directed to perform the right action now, and in the future (creating better site denizens) Users who wish to contest the deletion of an answer on meta can cite that specific reason, which mods can specifically address (which actually might reduce noise on meta about "why was my answer deleted?") Can reduce flags on answers asking "why was this deleted?" (admitted, I've not personally experienced this to be a pervading problem on Stack Overflow, but this might be the case for other mods on other Stack Exchange sites) Other Stack Exchange sites which don't have that many users will give a better experience to their users when their first action isn't ideal for a Stack Exchange site; this will enhance the chance that the site will retain the user and get good contributions from them (as per Farseeker's comments: , ).   Additionally, if this works well for the moderator tools, I imagine that it would not be such a leap to eventually have this graduated to users with the rep to cast delete votes, thus allowing non-mods to deliver a more clear message to new users about how to best participate on a Stack Exchange site. What to do with random effects correlation that equals 1 or -1? Not so uncommon occurrence when dealing with complex maximal mixed models (estimating all possible random effects for given data and model) is perfect (+1 or -1) or nearly perfect correlation among some random effects. For the purpose of the discussion, let's observe the following model and model summary   Model: Y ~ X*Cond + (X*Cond|subj)  # Y = logit variable   # X = continuous variable   # Condition = values A and B, dummy coded; the design is repeated  #             so all participants go through both Conditions   # subject = random effects for different subjects    Random effects:  Groups  Name             Variance Std.Dev. Corr               subject (Intercept)      0.85052  0.9222                              X                0.08427  0.2903   -1.00                      CondB            0.54367  0.7373   -0.37  0.37                X:CondB          0.14812  0.3849    0.26 -0.26 -0.56 Number of obs: 39401, groups:  subject, 219  Fixed effects:                  Estimate Std. Error z value Pr(&gt;|z|)     (Intercept)       2.49686    0.06909   36.14  &lt; 2e-16 *** X                -1.03854    0.03812  -27.24  &lt; 2e-16 *** CondB            -0.19707    0.06382   -3.09  0.00202 **  X:CondB           0.22809    0.05356    4.26 2.06e-05 ***   The supposed reason behind these perfect correlations is that we have created a model which is too complex for the data that we have. The common advice that is given in these situations is (e.g., Matuschek et al., 2017; ) to fix the overparameterized coefficients to 0, because such degenerate models tend to lower the power. If we observe a marked change in fixed effects in a reduced model, we should accept that one; if there is no change, then there is no problem in accepting the original one.  However, let's presume that we are not interested only in fixed effects controlled for RE (random effects), but also in the RE structure. In the given case, it would be theoretically sound to assume that Intercept and the slope X have non-zero negative correlation. Several questions follow:   What to do in such situations? Should we report the perfect correlation and say that our data is not "good enough" to estimate the "real" correlation? Or should we report the 0 correlation model? Or should we maybe try to set some other correlation to 0 in hope that the "important" one will not be perfect anymore? I don't think there are any 100% correct answers here, I would mostly like to hear your opinions. How to write the code which fixes the correlation of 2 specific random effects to 0, without influencing the correlations between other parameters? How can I verify the following statement?     The image of a morphism $f : A \longrightarrow B$ is defined as $\text{Im}(f) = \text{Ke}r(\text{coke}r f)$ whenever it exists (e.g., in every abelian category). The morphism $f : A \longrightarrow B$ factors   uniquely through $\text{Im} f \longrightarrow B$ whenever $\text{Im} f$ exists, and $A \longrightarrow \text{Im} f$ is an epimorphism and a cokernel of ker $f \longrightarrow A$ in every abelian category. I am attempting to search jobs by tag on StackOverflow Careers. On SO, this is done using square brackets, but that is simply doing a keyword search, and I still get results that don't have the tag I am looking for.  For example, when I search for the keyword "android", I get this:        which is the same as when I search for the keyword "[android]":        Is there a format I can use to search jobs by tag? If not, I believe this would be a nice feature (seems intuitive for the Stack Exchange invite-only job site). Difference of Gamma random variables I know that new arXiv papers usually appear in Google Scholar within a few days or weeks. My question is: if someone submits a replacement of his/her arXiv paper, does the new version of the paper also appear in Google Scholar? How to make crontab email me with output? How can I make crontab email me with the output of its jobs? I have MAILTO=redacted@yahoo.com.au above the jobs, but it doesn't work.  I know I need to use something like Postfix or Sendmail but for the life of me I cannot find out how to achieve this. I've searched and searched for a simple guide on setting up Postfix but they're all insanely complicated and expect you to be setting up a whole server to send and receive email and to know all the terms (like, what is a domain name for outgoing emails?).  All I want to do is have crontab email me. Why is this so difficult??  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