Given the following related to an ML-inequality for $R &gt; 1$:  Estimation of the numerator from the function $f(z)$ is supposed to develop as follows:  I'm wondering why and how exactly the power $3ix$ cancels out (why does it vanish/equal to zero) in the equality $|e^{-3y+3ix}| = e^{-3y}$ ?
Let $z \in \Gamma (R)$ where this is the upper semi circle centred at the origin with radius $R&gt;1$. Let $z=x+iy$ with $x \in \mathbb{R}$ and $y \geq 0$. So   $$|e^{i \lambda z}|=|e^{i \lambda z}| |e^{- \lambda y}|= |e^{- \lambda y}|$$  where $\lambda \in \mathbb{R}^{+}$.  How can the last equality possibly be true?
Deleting Individual SE Accounts unsigned int vs. size_t (Before anyone jumps to the conclusion that this is the same question as , please read it through. Also I am looking for more comprehensive and detailed data than the poster in .)  Many years ago I used to use a tool called Line Speed Meter (or Monitor) by TCPIQ. This was a brilliant piece of software that would sit quietly in the background and several times a day (according to a selectable frequency) would reach out and test internet line performance against servers all over the world. In the course of a week (or more), thousands of servers would be tested and enough data would accumulate that one could analyze performance according to time of day, day of week, location of server, etc. Drill-down was possible to the point where you could see every server that was tested.  This data was invaluable in determining if one really was getting the performance being paid for. It was also infinitely more valuable than spot checks you perform at sites like  or even the better-regarded ones like  or  (both of which, incidentally, are biased toward USA users - which I am not one of anymore).  Unfortunately Line Speed Meter is no longer supported and the company that made it now has instead a tool called isposure that is nowhere near as flexible or transparent - it is geared to supplying data to ISPs rather than to the consumers. For example, according to their FAQ, it usually runs only once a day automatically - I'd like to start with running it every couple of hours for a week or two - and scale back once I have a good basis so that I can notice when performance changes. Also, the details of the tests are no longer visible to the consumer-user.  My question: is there a (preferably free) tool/service available today to automatically collect comprehensive line speed metrics so I can map out and analyze the performance I am getting on my Broadband internet connection?  EDIT 2012-02-13: I left something out of the above.  In my country, each ISP sells a package that implies performance beyond the borders of my small country. They own (or lease) the lines going across the ocean. They have full control of the bandwidth leading from their servers to the backbones in the USA and EU. Peformance through the cloud is very much relevant and although not entirely, certainly largely under the local ISP's control, since traditionally, the bottleneck has been traffic out of the country.  As an example, at the moment, my ISP is advertising widely their new transatlantic cable and how their performance now outstrips everyone else's. I haven't "felt" it - so I want a tool to quantify it.  Performance within my country is usually good. The problem is that almost all my work involves access to the Internet in North America, Europe, Pacific Rim and others. So I care about and want to document the end-to-end performance I am getting from my ISP - and if I switch, compare to what I get at a new ISP.  EDIT 2013-03-07: As @harrymc pointed out I neglected to mention OS. All the machines on my home LAN are (sadly) Windows (XP or 7). If A tool requires *nix, I have a couple ancient machines in storage I could probably re-purpose - but my preference is for a Windows-based tool. Thanks.  EDIT 2013-03-10: As a further clarification - a method to monitor speeds achieved during regular work provides half of what I'm looking for. What was nice about TCPIQ is that it would ferret out thousands of sites around the world and do brief d/l &amp; u/l tests against them to check speed and then log the results. This gave a valuable statistical database for analysis of speed overall, as well as during times of day, days of week and according to geographical targets. That's the sort of functionality I'd like to achieve (again). What causes this corkscrew-shaped object to rise up in a stream of water falling down? Here's a corollary of AC stating that: if there exists an onto mapping $f:A\to B$, then there exists a 1-1 mapping $g:B\to A$. What I'm confused about is that whether or not  AC can be derived from this proposition, in return? I have tried but haven't found a proof. Any help would be appreciated. Hilbert space of harmonic oscillator: Countable vs uncountable? 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Using:  $ ps -o'pid,ppid,args' `pgrep pulse`   I see that the pulseaudio daemon is being started by systemd (pid=1)   PID  PPID COMMAND 2738     1 /usr/bin/pulseaudio --start   However, I was unable to find any unit-file on my system containing pulseaudio or even just pulse.  My specific questions are:  A) Is there a way to determine the systemd unit that caused the creation of a specific process (in my example output, process 2738, the PA daemon)?  B) Are there alternative approaches to find out which unit-dependency chain or other settings of systemd resulted in the invocation of /usr/bin/pulseaudio --start? Some lands have an ability to become a creature, but they usually say they are still a land. If a land is a 3/3 creature and a land, can you target and kill it with a  to send it to the graveyard, or does it remain in play as a land? Distorted Cabin & Oily Sheen: What is going on in this paragraph? Is the energy density of vacuum uniform in the whole universe? Does it make sense to say, "The speed of light varies?"  Some may say right off the bat "Yes, it changes as a wave passes through a different medium."  However, I'd like to say no, because when I hear someone say the speed of light, I always think of the constant $c$ (unless the medium is specified to not be a vacuum, but then it isn't $c$ anymore), not the speed of a particular wave.  To me, it makes more sense to say something like, "The speed of a particular wave varies." What is the correct way to state this in the professional world? And in general, when professional physicists say "the speed of light," are they referring to the constant or the actual speed of the wave? So I have these two wheels parented to the leg of a robot. I wanna make them spin a few times but they're spinning around the wrong axis. With other objects I could just press R, Y, Y to make them spin right when they're already at an angle, but for some reason it doesn't work with these. 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As a result, I'm a bit confused in relation to the verify_data size and the finished message structure. Can anyone help me to understand this and/or maybe put me on the right track somehow? What is the source of help? help  displays information about builtin commands. What is the source of help? Does it maintain a database for builtin commands, or does it read some files of each builtin commands (similar to the manpage of each utilities)?  Sometimes I find its information seems to expand that by --help  $ cd --help bash: cd: --: invalid option cd: usage: cd [-L|[-P [-e]]] [dir]  $ help cd cd: cd [-L|[-P [-e]]] [dir]     Change the shell working directory.      Change the current directory to DIR.  The default DIR is the value of the     HOME shell variable.      The variable CDPATH defines the search path for the directory containing     DIR.  Alternative directory names in CDPATH are separated by a colon (:).     A null directory name is the same as the current directory.  If DIR begins     with a slash (/), then CDPATH is not used.      If the directory is not found, and the shell option `cdable_vars' is set,     the word is assumed to be  a variable name.  If that variable has a value,     its value is used for DIR.      Options:         -L  force symbolic links to be followed         -P  use the physical directory structure without following symbolic         links         -e  if the -P option is supplied, and the current working directory         cannot be determined successfully, exit with a non-zero status      The default is to follow symbolic links, as if `-L' were specified.      Exit Status:     Returns 0 if the directory is changed, and if $PWD is set successfully when     -P is used; non-zero otherwise.   I thought help extract the help part of the executable, but for a  python script , it isn't  $ help ./pdf-merge.py bash: help: no help topics match `./pdf-merge.py'.  Try `help help' or `man -k ./pdf-merge.py' or `info ./pdf-merge.py'.  $ ./pdf-merge.py --help usage: pdf-merge.py [-h] [-v] [--ask] [--output OUTPUT] [--title TITLE]                     [--author AUTHOR] [--keyword KEYWORD] [--pdftk PDFTK]                     [--gs GS] [--pdfmarks PDFMARKS] [--unicode]                     PDF [PDF ...]  Merge PDFs preserving bookmarks. Thanks to Larry Cai for suggesting that Unicode be supported and for discussion about the `--pdfmarks` option.  positional arguments:   PDF                  an input PDF to merge  optional arguments:   -h, --help           show this help message and exit   -v, --version        show program's version number and exit   --ask                pause for manual pdfmark tweaking   --output OUTPUT      name of the output PDF   --title TITLE        title of output PDF   --author AUTHOR      author of output PDF   --keyword KEYWORD    keywords for the output PDF   --pdftk PDFTK        path to the pdftk executable   --gs GS              path to the gs (Ghostscript) executable   --pdfmarks PDFMARKS  path to pdfmarks file. If not given, a temporary file                        is used. If given and the file is missing, execution                        will stop after the file is created (before the                        Ghostscript run). 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I am trying to recall the name of a story I read when I was fairly young (maybe 20 years ago, but not sure how old the story is).    What I remember: A man acquires a set of creatures that he keeps in his house, I think in an aquarium.  They evolve over time, quickly, and think of him as their benevolent god. They build statues in his likeness.   Then, I believe some sort of accident happens, and the statues they build become a little less benevolent, which bothers him.  This pattern continues on with his cruelty increasing and their depiction of him becoming more cruel. Black screen after installation of Ubuntu 18.04