This question is from my assignment in complex analysis and I was unable to solve it.  Prove that tanz doesn't assume the value i, -i. Does this contradict Picard's Theorem.  Attempt- tanz = $ i \frac{e^{iz} -e^{-iz}} { e^{iz}+ e^{-iz}}$  =&gt; $e^{iz}=i$ and z=π/2 and z=-π/2 is not in domain. But how does it tells which values will never be attained?  So, how to deduce it? I think picard theorem is not violated as tanz is not entire, due to cosz being in denominator. Am I right?
Wolfram alpha says there are no solutions. Is this because of the following?  $$\tan(z)=i \Rightarrow \frac{e^{iz}-e^{-iz}}{i(e^{iz}+e^{-iz})}=i \Rightarrow \frac{e^{iz}-e^{-iz}}{(e^{iz}+e^{-iz})}=-1$$ Let $x=e^{iz}$. Then,  $$\frac{x-1/x}{x+1/x}=-1 \Rightarrow x-1/x=-x-1/x \Rightarrow 2x=0$$  Then, $$x=e^{iz} \Rightarrow \ln(0)=iz$$ but $\ln(0)$ does not exist so we have no solutions?
Best way to transfer files from Windows to Windows via Internet I can never figure out whether I should use who and whom. Most people use who for both colloquially, but some people say this is not correct.  What’s the rule for using who and whom correctly? I have been trying to install a fresh copy of Ubuntu 14.04 (64 bit version) on my desktop PC from a USB memory stick, but I'm experiencing a problem.  Each time I attempt an install I received an  "[Errno 5] Input/output error", followed by the following message before the installer exits:  "The installer encountered an error copying files to the hard disk: [Errno 5] Input/output error This is often due to a faulty CD/DVD disk or drive, or a faulty hard disk. It may help to clean the CD/DVD, to burn the CD/DVD at a lower speed, to clean the CD/DVD drive lens (cleaning kits are often available from electronics suppliers), to check whether the hard disk is old and in need of replacement, or to move the system to a cooler environment."  In order to attempt to remedy this I have so far tried installing Ubuntu to two different hard disks, one of which is brand new, and each time I have received the same error "Errno 5" message. I have also tried installing from 3 different USB sticks using torrent and Ubuntu site iso images burned using both Unetbootin and Startup Disk Creator.  Again, these measures have proved fruitless.  The installation process varies depending of the iso I use, with the Ubuntu download page iso exiting at an earlier stage compared to the torrented iso.  The later torrent iso returns an error and exits when I am prompted for my name, computer's ID and password.  I've never experienced anything like this when installing previous versions of Ubuntu.  I'm absolutely stumped!  Does anyone have any more ideas? Noetherian property for exact sequence Let $0 \to M' \xrightarrow{\alpha} M \xrightarrow{\beta} M'' \to 0$ be an exact sequence of $A$-modules. Then $M$ is Noetherian is equivalent to $M'$ and $M''$ are Noetherian.  For the ''$\Leftarrow$'' case:  I guess if we let $(L_n)_{n\geq 1}$ be an ascending chain of submodules of $M$, then $(\alpha ^{-1}(L_n))_{n\geq 1}$ is a chain in $M'$, and $(\beta(L_n))_{n\geq 1}$ is a chain in  $M''$. For large $n$ both these chains are stationary. Then why we know the chain $(L_n)$ is stationary?   Thanks! $|f'(x)| \le g(x)$ implies $|f(b) - f(a)| \le \int_a^b g(x) dx$, without assuming $f'$ to be integrable. Explanation of unpredictable behaviour of tee I encountered a behaviour that I don't understand while testing a script that sums the outputs from repeated executions of a program. To reproduce it create the text files out, which represents the output of my program, and sum, the file that holds the sum of the values returned on previous executions and which starts out as a copy of out,  cat &gt; out &lt;&lt; EOF 2 20 5 50 EOF cp out sum   The strange thing happens on running  paste out sum | awk '{$1 += $3; $2 += $4; NF = 2; print}' | tee sum   several times (15-20 times might be needed). Each time it runs, this command should add to the values in sum the corresponding values in out and write the results back to sum. What I get is that it works an unpredictable number of times, then sum reverts back to  2 20 5 50   I have later learned that  and solved the issue using a temporary file, still, this behaviour baffles me:   why does … | tee sum work at all (even if only for a limited number of iterations), while … &gt; sum never overwrites sum? why doesn't it work a predictable number of times? At work we were discussing this as my boss has never heard of normalization. In Linear Algebra, Normalization seems to refer to the dividing of a vector by its length. And in statistics, Standardization seems to refer to the subtraction of a mean then dividing by its SD. But they seem interchangeable with other possibilities as well.   When creating some kind of universal score, that makes up $2$ different metrics, which have different means and different SD's, would you Normalize, Standardize, or something else? One person told me it's just a matter of taking each metric and dividing them by their SD, individually. Then summing the two. And that will result in a universal score that can be used to judge both metrics.  For instance, say you had the number of people who take the subway to work (in NYC) and the number of people who drove to work (in NYC).   $$\text{Train} \longrightarrow x$$ $$\text{Car} \longrightarrow y$$  If you wanted to create a universal score to quickly report traffic fluctuations, you can't just add $\text{mean}(x)$ and $\text{mean}(y)$ because there will be a LOT more people who ride the train. There's 8 million people living in NYC, plus tourists. That's millions of people taking the train everyday verse hundreds of thousands of people in cars. So they need to be transformed to a similar scale in order to be compared.  If $\text{mean}(x) = 8,000,000$  and $\text{mean}(y) = 800,000$  Would you normalize $x$ &amp; $y$ then sum? Would you standardize $x$ &amp; $y$ then sum? Or would you divide each by their respective SD then sum?  In order to get to a number that when fluctuates, represents total traffic fluctuations.  Any article or chapters of books for reference would be much appreciated. THANKS!  Also here's another example of what I'm trying to do.   Imagine you're a college dean, and you're discussing admission requirements. You may want students with at least a certain GPA and a certain test score. It'd be nice if they were both on the same scale because then you could just add the two together and say, "anyone with at least a 7.0 can get admitted." That way, if a prospective student has a 4.0 GPA, they could get as low as a 3.0 test score and still get admitted.  Inversely, if someone had a 3.0 GPA, they could still get admitted with a 4.0 test score.  But it's not like that. The ACT is on a 36 point scale and most GPA's are on 4.0 (some are 4.3, yes annoying). Since I can't just add an ACT and GPA to get some kind of universal score, how can I transform them so they can be added, thus creating a universal admission score.  And then as a Dean, I could just automatically accept anyone with a score above a certain threshold.  Or even automatically accept everyone whose score is within the top 95%.... those sorts of things.  Would that be normalization? standardization? or just dividing each by their SD then summing? Why does cycle rendering only use one core from an AMD GPU? How can I change the default date format? When I paste a date into Excel 2010 in the format "2012-12-07 00:00:00.000" I'd like it to be displayed by default as "2012-12-07", but it is actually displayed as "00:00.0".   I know I can work around this by prepending a ' or changing the cell format, but I'm interested in changing the default format.  An approach that almost works is going to Control Panel->Region and Language->Long Time format and changing it to something like "h". This results in Excel displaying the dates I paste in as "2012-12-07 00:00:00.000", but screws up times displayed outside of Excel.  The dates I'm pasting in are from SQL Server. How does Cooper get out of the black hole? In the movie Interstellar,  Cooper (played by ) enters a black hole and sends a message to Murph (his daughter). Later, he is found by rangers, but I don't understand how he gets out of the black hole?  As far as I know, nothing can get out of a black hole, as it has extremely high gravity. I created a thick glass bottle and duplicated the interior faces to make a new object to create the liquid inside.   I went in the edit mode, selected the faces I wanted to copy and duplicated them with ShiftD. Then I pressed P to  separate the duplicated faces.   Now I try to make a duplicate of that liquid to make it flow out of the bottle, and I simply need the duplicate of one face of it. So I go to edit mode and do as I did to make the liquid from the bottle faces... but as soon as I'm back to the object mode and I want to move the duplicated faces, every thing moves with it...   I played around and tried things, and while doing so I realized that if I dupe the liquid object in object mode using ShiftD and press 0 to place it where the original is, they move together if I duplicate it and place it somewhere else so that they are separated.   Also if I pick one of the two separated objects and make it intersect with the other one, release my left mouse button to place it there, and then try to move it again, they both move together, but not at the same speed.  I really don't understand what is going on and I'm sure it's really badly explained, so what I'll do is make a screen capture of my Outliner and record the behavior of the objects and upload those to help you understand. System settings won't open on Ubuntu 20.04 LTS after adding System76/pop PPA I want to prove that every real polynomial of odd degree has at least one real root, using  the intermediate value theorem.  Let $P(x) = x^{2n+1} + a_n x^{2n} + . . . + a_0$ for each $a_i \in \mathbb{R}$ and $n \in \mathbb{N}$.  By the fundamental theorem of algebra I know that $P(x)$ has exactly $2n+1$ complex roots, so  $P(x) = (x+r_1)(x+r_2) . . . (x+r_{2n+1})$ for each $r_i \in \mathbb{C}$  I do not know how to complete this but I do know that, at some point, I probably have to show that each root with imaginary part non zero has to come in conjugate pairs, and since $2n+1$ is odd there is at least $1$ root that is imaginary part $0$ and thus real. For about 2 days I'm trying to configure Ubuntu so that I can send an email, but when I type in:  echo "Subject: sendmail test" | sendmail -v my@email.com   I'll get:  postdrop: warning: unable to look up public/pickup: No such file or directory Mail Delivery Status Report will be mailed to &lt;andreigbr&gt;.   Even if I change my@email with mine, the inbox will be empty. Can somebody help me, please? I watched about 5 tutorial about how to send an email fom the command line but still doesn't work. Two logos in beamer after certain frame Converting a Uniform Distribution to a Normal Distribution How can I convert a uniform distribution (as most random number generators produce, e.g. between 0.0 and 1.0) into a normal distribution? What if I want a mean and standard deviation of my choosing? Render Glass like materials with Alpha Channel to composite in external app Hope you can help me with some issues I've been dealing these days.  As I'm rendering products, most of the time I need to render some transparent materials in order to do a post-pro in Photoshop, change the backgrounds, superpose images, etc.  I've been looking for an answer but yet I've couldn't get a solution for this problem.  I need to render materials with transparency and no background in Cycles.  I've found that there are two ways to do this:   Render with transparency in the materials.  Render and get two renders, the original and another one only with the alpha channel to edit in Photoshop.   For the first option, I've achieved to render a glass-like material, as PET plastic using this node tree:    For the second option, I haven't found a way to get the alpha channel in cycles that includes the object with transparency and, if possible, the shadows in the background| but without the background. I've posted a lot to StackOverflow, and I've noticed something kind of irritating with the formatting in writing questions and comments. If I need to put something inside a &lt; and a >, without spaces in the middle (like right there), it will think it's code and not show up, like this   (obviously it won't come up what I wrote there, or else this wouldn't be an issue, but if you open up the question to edit you can see what's written there.)  If I want it to show up, right now I have to add an extra set of brackets inside of the originals with a junk variable, like right &lt;here> (again, open in edit window to see).  This is really annoying when trying to show variables like lists that require those brackets, and I feel like the junk variables could confuse people looking at the edits. Is there a way to get around this formatting? Thanks! Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional? It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this? How to convert Bedrock texture pack to Java Edition Minecraft