Approximation of measurable function by simple functions 2
Approximation of measurable function by simple functions
Darth Vader threw Emperor Palpatine to a pit. Well, , but unlike other times, Palpatine wasn't injured or unconscious when he was thrown.  Why didn't he simply prevent his fall using his lightsaber (I have seen such action in Star Wars: The Clone Wars TV series lots of time) and make his way out? For an ecology project, my lab group added vinegar to 4 tanks containing equal volumes of pond water, 1 control with no elodea (an aquatic plant) and 3 treatments with the same amount of elodea in each. The purpose of adding the vinegar was to reduce the pH. The hypothesis was that the tanks with elodea would go back to their normal pH quicker. This was indeed the case. We measured the pH of each tank daily for about two weeks. All the tanks eventually returned to their natural pH, but the length of time that this took was much shorter for the tanks with elodea.  When we told our professor about our experimental design, he said that there exists no statistical test that can be performed on the data to compare the control to the treatment. That because there was no replicate for the control (we only used one control tank) we cannot calculate variance and so we can't compare the sample means of the control and the treatment. So my question is, is this true? I definitely understand what he means. For example, if you took the height of one man and one woman, you can't draw conclusions about their respective populations. But we did 3 treatments, and the variance was small. It seems reasonable to assume that the variance would be similar in the control?  Update:  Thank you for the excellent answer. We got more water and elodea from the wetland and decided we would run the experiment again with smaller tanks but this time with 5 controls and 5 treatments. We were going to combine this with our original data but the starting pH of the tanks was different enough that it doesn't seem valid to consider the new experiment to be sampled from the same population as the original experiment.  We considered adding different amounts of elodea and trying to correlate speed of pH remediation (measured as time elapsed until pH returned to its original value) with amount of elodea, but we decided that wasn't necessary. Our objective is only to show that the elodea makes a positive difference, not to construct some kind of predictive model for exactly how the pH responds to differing amounts of elodea. It would be interesting to determine the optimal amount of elodea, but that's probably just the maximum amount that can survive. Trying to fit a regression curve to the data wouldn't be especially illuminating because of the various complicated changes that occur to the community when adding a large amount. The elodea dies, decomposes, new organisms start to dominate, and so on. I'm using 17.04 ( Clean Install ), and as far as I remember, every try to install any sofware- it fails with that error code.  I tried sudo apt-get update &amp;&amp; apt-get upgrade and problem persists.  any idea why?   edit: Using software Updater gave this result :   edit : output of update  guy@LG-LAPTOP:~$ sudo apt-get update &amp;&amp; apt-get upgrade [sudo] password for guy:  Ign:1 http://il.archive.ubuntu.com/ubuntu zesty InRelease Ign:2 http://il.archive.ubuntu.com/ubuntu zesty-updates InRelease               Ign:3 http://il.archive.ubuntu.com/ubuntu zesty-backports InRelease             Err:4 http://il.archive.ubuntu.com/ubuntu zesty Release                           503  Service Unavailable Get:5 http://security.ubuntu.com/ubuntu zesty-security InRelease [89.2 kB]      Err:6 http://il.archive.ubuntu.com/ubuntu zesty-updates Release                   503  Service Unavailable Get:7 http://archive.canonical.com/ubuntu zesty InRelease [10.2 kB]             Err:8 http://il.archive.ubuntu.com/ubuntu zesty-backports Release                 503  Service Unavailable Get:9 http://archive.canonical.com/ubuntu zesty/partner Sources [1,416 B]       Get:10 http://archive.canonical.com/ubuntu zesty/partner amd64 Packages [1,616 B] Get:11 http://archive.canonical.com/ubuntu zesty/partner i386 Packages [1,612 B] Get:12 http://archive.canonical.com/ubuntu zesty/partner Translation-en [708 B] Get:13 http://security.ubuntu.com/ubuntu zesty-security/main amd64 DEP-11 Metadata [11.7 kB] Get:14 http://security.ubuntu.com/ubuntu zesty-security/main DEP-11 64x64 Icons [9,219 B] Get:15 http://security.ubuntu.com/ubuntu zesty-security/universe amd64 DEP-11 Metadata [14.4 kB] Get:16 http://security.ubuntu.com/ubuntu zesty-security/universe DEP-11 64x64 Icons [31.0 kB] Reading package lists... Done                                          E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-updates Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. E: The repository 'http://il.archive.ubuntu.com/ubuntu zesty-backports Release' does no longer have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details. guy@LG-LAPTOP:~$    output of apt get  guy@LG-LAPTOP:~$ sudo apt-get install putty Reading package lists... Done Building dependency tree        Reading state information... Done The following additional packages will be installed:   putty-tools Suggested packages:   putty-doc The following NEW packages will be installed:   putty putty-tools 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 661 kB of archives. After this operation, 2,713 kB of additional disk space will be used. Do you want to continue? [Y/n]  Err:1 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty-tools amd64 0.67-3   503  Service Unavailable Err:2 http://il.archive.ubuntu.com/ubuntu zesty/universe amd64 putty amd64 0.67-3   503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty-tools_0.67-3_amd64.deb  503  Service Unavailable E: Failed to fetch http://il.archive.ubuntu.com/ubuntu/pool/universe/p/putty/putty_0.67-3_amd64.deb  503  Service Unavailable E: Unable to fetch some archives, maybe run apt-get update or try with --fix-missing? guy@LG-LAPTOP:~$ How to evaluate a math expression given in string form? Is $\mathbb{Z}[x]/(x^2+1)$  isomorphic to  $\mathbb{Z}[i]$?   My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. If it is possible then $\ker g$ is $(x^2+1)$? Am I on the right track? Please Help. What is the idiom, proverb for "Little problems often become big problems if no one takes the initiative to correct them" What is the idiom, proverb for "Little problems often become big problems if no one takes the initiative to correct them"  Which means in an example that If the employees don’t bother to report a malfunctioning machine or a slip-and-fall hazard, serious injuries could occur – injuries that could have been preventable. Ignoring any hazard can be detrimental to one's business and the safety of their employees. I've notice that the business to business site AliBaba and it's more consumer friendly AliExpress website have a number of bike components for sale. They tend to be complete component groupsets and the like but are some quite reasonable deals, especially if you are building a bike from scratch.  I've used AliExpress before to purchase electronic components, which have been good quality and are what they say on the tin.   Has anyone used AliExpress to buy bike components and have they had any trouble? (quality, counterfeit etc.)  One benefit is that your money is held in escrow until you receive the item and aprove the payment so some safety. Proving that $\frac{\phi^{400}+1}{\phi^{200}}$ is an integer. How do we prove that $\dfrac{\phi^{400}+1}{\phi^{200}}$ is an integer, where $\phi$ is the golden ratio?  This appeared in an answer to a  I asked previously, but I do not see how to prove this.. How can black hole increase its mass? why can't `_` be exported to the environment of bash? In bash, why can't _ be exported to the environment of the shell?  $ export _ $ export | grep _= $   output nothing. Or do I miss something?   How can I export _ into the environment?  See also . Thanks. Correct ssh config file settings to tunnel to a 3rd machine I'm trying to tunnel to a server via a bridge server. So far, I've been able to get it working from the command shell properly using the following command:  ssh -A -t bridge_userid@bridgemachine.com ssh -A remote_userid@remoteserver.com   But I've been trying to wrap this into my ~/.ssh/config file and I have troubles. I've tried:  Host axp   User          remote_userid   HostName      remoteserver.com   IdentityFile  ~/.ssh/id_rsa.eric   ProxyCommand ssh -A -t bridge_userid@bridge_userid.com ssh -A remote_userid@%h   But when I do, I get the following error message from remoteserver.com and I'm not sure what is causing it:     ksh: SSH-2.0-OpenSSH_6.8^M:  not found   I know that when I log into remoteserver.com , my shell is /usr/bin/ksh.  I've tried to add path arguments to the ssh commands in the config file, but it made no difference.  Any ideas what it can be? For a more comprehensive list of monitoring tools and their features, check out .   As the question states, what are the most commonly used tools used for this task and what are their strengths and weaknesses? Improving "demonstrate a minimal understanding" close reason 3D software like GeoGebra Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on. General relativity (gravitation) in time and one spatial dimension I don't have any idea of general relativity but intend to learn. Is it a good idea to study general relativity in two dimensions (time and single spatial dimension) in the begining to get good idea on the subject? If it is, then please give some references for such a treatment. We need a hard rate limit on suggested edits, to limit the damage someone can do with blasting the review queue with minor edit suggestions.  Case in point:  peppered the site with a whopping 133 tag-only edit suggestions in 2 hours and 15 minutes. That's one edit every minute on average, but towards the end he was entering a new suggestion every 20 seconds. He only stopped because a  drew enough rejections to have him banned from entering more.  If there was a limit to the number of suggested edits you can enter in a time period we could have stopped this guy earlier. No &lt;2k user needs to edit hundreds of tags across that many questions in 2 hours time.  Considerations   There are cases where a retag drive or tag burninate justifies editing a larger number of posts, but I'd be fine if such tasks were limited to people with 2k+ reputation. A retag or burninate job requires a minimal engagement with the community to know when they are justified anyway. The suggested edit review process, by itself, was clearly not enough to stop this barrage. It cannot be seen as a rate limiting measure, at least not as a sufficiently efficient one.  The vast majority of the edits made in the above example were too minor, leaving spelling, grammar and formatting mistakes in the post untouched. The user was going too fast to pay attention to such details anyway. Yet this user has 98 approved edit suggestions, but all but 6 or 7 I'd have rejected as too minor, so the review process itself is broken.   What should be limited   The number of review votes reviewers have is already limited; I burned through 20 reviews in short order just reviewing the suggestions from just this one user. Certainly the number of unreviewed suggestions should be limited to let reviewers catch up with a user like this. I'd say having 20 outstanding suggested edits pending review is plenty. Suggested edits should be made with some consideration; we don't want minor edits, so if you are suggesting more than 20 in an hour (that's 1 every 3 minutes) you are not improving posts anymore, you are gaming the system to gain reputation. Do people sometimes get blocked by a webservice for things someone else has done with their dynamic IP-Address It is common practice for operators of web services (e.g. web sites) to block IP addresses when there was malicious traffic originating from that address.  It is also common practice that internet users don't have a static IP address but get one assigned by their ISP via DHCP.  Now I would expect that it sometimes happens that a specific IP address gets banned from a service and later that address gets assigned to another user, leaving the new user (temporarily) banned from the service for no apparent reason. However, I never had this problem or heard of someone having that problem.  Does this sometimes happen? If not, what prevents this from happening? Stack Exchange doesn't allow me to answer protected questions because I gave away earned rep in bounties Render not showing UV texture I have added a texture to a cube in the UV editor and it shows up fine in textured view  But when I render it there is no texture  What have I done wrong? Showing $\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_0^1 \mid F^{-1}(u)-G^{-1}(u)\mid du$ with $F$, $G$ CDF functions Let's $X$ and $Y$ have CDF functions admitting moment of order $1$. Let's be $F$ cdf of $X$ and $G$ cdf of $Y$.  I want to show that $$\int_{\mathbb R} \mid F(x)-G(x)\mid dx = \int_{0}^{1} \mid F^{-1}(u)-G^{-1}(u)\mid du\,.$$