If $0\leq A\leq B$ on Hilbert space and $A^{-1}$ exists, show that $A^{-1}\geq B^{-1}$
For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$?
I'm using R(3.1.1), and ARIMA models for forecasting. I would like to know what should be the &quot;frequency&quot; parameter, which is assigned in the ts() function, if im using time series data which is:  separated by minutes and is spread over 180 days (1440 minutes/day) separated by seconds and is spread over 180 days (86,400 seconds/day).  If I recall right the definition, a &quot;frequency&quot; in ts in R, is the number of observations per &quot;season&quot;. Question part 1: What is the &quot;season&quot; in my case? If the season is &quot;day&quot;, then is the &quot;frequency&quot; for minutes = 1440 and 86,400 for seconds? Question part 2: Could the &quot;frequency&quot;  also depend on what I am trying to achieve/forecast? for example, in my case, I'd like to have a very short-term forecast. One-step ahead of 10minutes each time. Would it then be possible to consider the season as an hour instead of a day? In that case frequency= 60 for minutes, frequency = 3600 for seconds? I've tried for example to use frequency = 60 for the minute data and got better results compared to frequency = 1440 (used fourier see link below by Hyndman)  (The comparison was made by using MAPE for the measure of forecast accuracy) In case the results are complete arbitrary, and the frequency cannot be changed. What would be actually the interpretation of using freq = 60 on my data? I also think it's worth mentioning that my data contains seasonality at every hour and every two hours (by observing the raw data and the Autocorrelation function) Query Feature and Usage based entitlements license Drush installed and running; No Drupal site found even with uri specified After much trial and tribulation, I thought I had drush all ready to roar. Much excitement. But then, no joy...  Drush is working. The sites operate fine via MAMP. Regardless, a "drush cc all" command returns:   No Drupal site found, only 'drush' cache was cleared.                                          [warning] 'all' cache was cleared.                                                                       [success]   This question has stumped drupalcon attendees and very experienced developers. Any ideas?  I am on Mac OS 10.9.3 using MAMP (free) Version 3.0.5  Here is my drush status:   Drupal version         :  7.29-dev                                 Site URI               :  http://default                           Database driver        :  mysql                                    Database hostname      :  localhost                                Database username      :  db_user                             Database name          :  db_name                        PHP executable         :  /usr/bin/php                             PHP configuration      :                                           PHP OS                 :  Darwin                                   Drush version          :  7.0-dev                                  Drush temp directory   :  /tmp                                     Drush configuration    :                                           Drush alias files      :                                           Drupal root            :  /Applications/MAMP/htdocs/mysite.dev   Site path              :  sites/default       Have tried executing drush from /Applications/MAMP/htdocs/mysite.dev; from /Applications/MAMP/htdocs/mysite.dev/sites/default; and using drush cc all --uri=http://localhost/mysite.dev  Tried running drush from sites/default. Same result. How to Disable new terms (free tagging) with Taxonomies? I have a "term reference" field, in a form that uses an auto-complete widget. I want to limit user input to terms in the vocabulary only, and output a message when a user tries to enter a non existing term.  I found the  module. However this does not work with forms that contain a field collection or a field group.  There is also a contributed patch . But I was wondering whether there was a simpler/cleaner solution like a module or maybe I missed an already existing way of doing it with latest drupal and taxonomy versions. Using special auto start servlet to initialize on startup and share application data Arrow keys, Home, End, tab-complete keys not working in shell I have installed ubuntu minimal(mini.iso) on my vm. I then used recovery mode to login as root and create an account with useradd -m admin and then set a password with passwd admin.  When I login on the new account, instead of the normal prompt I only see a $ sign. If I try to tab-complete a command or file name it prints a normal tab. If I try to use the arrow keys it prints ^[[A, ^[[B, ^[[C or ^[[D. Also, ls no longer adds colors.  None of these problems were in recovery mode. How can I fix this? How can I maximize the "blurry background, sharp subject" (bokeh) effect? I know that this effect occurs when there's a shallow depth of field.  My question is, what are the various ways I can increase this effect in my photos when I'm taking them?    Note that I'm not asking  how to use an editing program to achieve this effect after in post production or after the image is captured. How to close aberrant tooltips in Windows So, here I am, trying to watch The Cleveland Show (TM).      And, as happens from time to time, some program or another has left a modal tooltip.  I don't know which program; how can I lose the tooltip, short of logoff/reboot?  Currently happening on Windows 10, but I have seen it on other versions. How do I reset proxy in terminal to automatic if not connected via proxy I tried to reset proxy of the terminal by some commands but it doesn't happen and automatically switches back to this proxy 172.16.0.16 (which apparently was my college proxy).   I checked in my system settings.I don't understand why this is recurring. Please be comprehensive.Also I further would like to know how to bypass proxy server since I couldn't access any of the ubuntu repositories as they were blocked in my college's proxy settings as is Ubuntu's homepage.  Thanks for your time.    For sudo ls /etc/apt/apt.conf.d/        It displays a different set of options where proxy is not listed.I am on 12.10,if this should help any.I put a snap of the terminal after the above command has been entered. I updated MacOS Sierra's Apache and PHP, and now whenever I $ sudo apachectl restart I get this:  AH00557: httpd: apr_sockaddr_info_get() failed for My-MacBook-Pro.local AH00558: httpd: Could not reliably determine the server's fully qualified domain name, using 127.0.0.1. Set the 'ServerName' directive globally to suppress this message   Things I've tried so far that didn't work:   Setting ServerName localhost in /etc/httpd.conf Setting ServerName localhost:80 in /etc/httpd.conf Setting ServerName 127.0.0.1 in /etc/httpd.conf Setting ServerName 127.0.0.1:80 in /etc/httpd.conf Ensuring 127.0.0.1       localhost is set in /etc/hosts Repeatedly restarting Apache Can Conjure Animals beasts be used as mounts? Is it possible to use the animals that are summoned using Conjure Animals as mounts?  As an example: Summoning Giant Eagles to carry a group up the mountain or over a wall. Prevent Mac Terminal from reverting to default theme I can't find any way to change the Terminal theme and have that theme applied permanently.  I want to change to another darker theme other than the default 'basic' theme, and (most importantly) have it remain as the theme every time I open Terminal. Decode escaped Url without using HttpUtility.UrlDecode In the search results of stackexchange.com/search?q=*, allow people to click on a tag and go to that website with the tag page open.   For example,  shows me this:        But clicking on the hovered-on tag takes me to  and the panda.   Instead, it should take me to    The math.SE URL is already there in  /#question-summary-608993 &gt; div.summary &gt; div.started.fr &gt; a #question-summary-608993 &gt; div.hot-question-site-icon &gt; a Create a Symbolic link to the merged contents of 3 folders I have 3 folders with large amount of data in each which has to be maintained seperately. Can I create a sym link or shortcut.in windows which should show me the merged contents of the 3 folders. I am not looking for library conf in Win7.   Any suggestion is highly appreciated. When I compile with LuaLaTeX and TL2016 this MWE   \documentclass{article}  \usepackage[no-math]{fontspec} \usepackage{luatex85} \usepackage{shellesc}  \setmainfont{Minion Pro}  \begin{document}  \huge{huge} \textbf{\huge{huge}}  \Huge{Huge} \textbf{\Huge{Huge}}   \end{document}   I have this .pdf.     Any help? With TL2015 all is correct. Thanks How to detect when a variable changes value How can I easily detect when a variable changes value? I would like the execution of the program to break on the debugger whenever a specified variable changes value. Right now I'm using Eclipse's debugger. My disk already has 4 primary partitions, how can I install Ubuntu? I would like to install Ubuntu alongside Windows 7 on a HP G62 Notebook. Although I have installed  Ubuntu in a dual-boot many times before, I found out that this model has already four partitions.    Partitions:   SYSTEM (NTFS, 199MB, used 66.59MB) Partition without any tag: NTFS (579GM, used 129GB)  RECOVERY (NTFS 16.74 GB, used 2.42GB)  HP_TOOLS (FAT32, 103.34 MB, used 13.23MB)   Since I am not an expert with partitions I would like to get advice on how to do this.  My first idea is this one:   Free some space from /dev/sda2 (I don't know if could also free some other space) delete the HP_Tools partition (I have already created a backup) create an extended partition with the free space in #1 containing three parititions: swap (1gb); / (EXT4, 30GB);  /home (EXT4, 120GB)   Another option is to use wubi instead.  What do you think? Is there any other way to achieve this?  PS: I really think this HP policy of using 4 partitions is not a coincidence  PS: I tried using gparted from the live CD and I got a warning message saying that if I freed some space from /dev/sda2 I could create serious issues in the system Does this seemingly elementary question require König's theorem? Salesforce lightning get current users username How I would go about getting the current users username or first and last name?  I have been looking at this for the last couple of hours - I was just wondering how I would go about getting their name and then displaying it on the page.  For example the current user is 'John Smith' and in the lightning component I want to output John Smith's name like:     Welcome to our community John Smith!