$\text{SL}(2,\mathbb{R})$ is homeomorphic to $S^1 \times \mathbb{R}^2$
$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$
"Contact us"... and then what? Can we have a perfect field of characteristic $p$ with a non-archimedean valuation? Complex towers: $i^{i^{i^{...}}}$ If I've decided to update my SE avatar (in any of the various ways well described in the answer to this ), am I able to return to either the original stock image or previous avatar images I have used? Let $A$ be a $n\times n$ matrix with entries $a_{ij}=i+j $ . Calculate rank of $A$.  My work : I noticed that A is symmetric . Hence all of its eigen vectors are real .. That is all i have got .   Your help will be highly appreciated .Thank you . i have a simple sort of a database file which consists only of entries in the following format     variable=value   i want to create a simple vim syntax highlighting for it and set it for specific file extension  for instance, variable part could be light blue, and value part light red  i googled it and came across things such as $vimruntime\syntax\, syntax set=, syntax match, and hi keywords, but couldn't set it up myself eventually  so i want a very simple vim code snippet that would realize it by matching the left and right hand sides and coloring them separately In "Seeing like a state" of James C.Scott there is a sentence at the beginning of a paragraph (Acknowledgements xi):     There are a good many scholars whose writings opened up new   perspectives for me or provided outstanding analyses of issues that I   could not have hoped to study so comprehensively on my own.   What worries me is a good many scholars. Is this some type of inversion or ancient English? Is this grammatically right? Why is a here? I thought that appropriate version would be There are many good scholars. Am I wrong? What's the correct way to do division using a backslash? If I write $a/bc + d$, will that be equal to $(a/(bc))+d$?   Basically, if I place a slash, will I divide by what's directly behind the slash $(b)$, the term that's behind the slash $(bc)$, or everything after the slash $(bc+d)$? How to get git-completion.bash to work on Mac OS X? I have followed  to  add git-completion.bash to my /opt/local/etc/bash_completion.d/git-completion  and I put PS1='\h:\W$(__git_ps1 "(%s)") \u\$ ' in my .bashrc_profile  But now I am getting this -bash: __git_ps1: command not found everything I do a cd.  Can you please tell me what am I missing? After update, Ubuntu has now become PopOS I'm running Ubuntu 20.04.1 LTS. I was recently prompted to install an update from Ubuntu's GUI software updater, but I'm not sure which update. Now, I seem to have... a strange amalgamation of Ubuntu and PopOS all of a sudden? After booting up today, my desktop looks quite different:  It appears to be using Pop Shell, the tiling window manager that PopOS uses. The left-hand dock is gone. Looking at 'Appearance -&gt; Background' in Settings, the PopOS wallpapers are present. If I go to 'Appearance -&gt; Appearance', the light/dark theme chooser shows a visual with &quot;Pop!_OS User&quot; in it. The scrolling direction has been reversed. The icons are different. The Ubuntu Software Center is missing, but there's no &quot;Pop Shop&quot; in its place. The list goes on...  At the same time, a lot of my system is still very much how it was a few days ago when it was &quot;pure Ubuntu&quot; so to speak. My browser, IDE, files/folders, etc. are all the same... For context, I used to dual boot Windows and PopOS on a single SSD, but I've since wiped the drives I use (including resetting partitions). Now I use Windows on the SSD and Ubuntu on a new second drive... I can't imagine how anything from my old PopOS installation would carry over, let alone suddenly reappear inside Ubuntu after an update. EDIT: When running Software Updater again, it shows a full list of what it's trying to install: . When I try to install these packages, it fails midway through. That partially explains things... but why is this even happening? EDIT2: It is trying to install pop-gnome-initial-setup.deb which seems to conflict with gnome-initial-setup. Weird. What's the best way to use matching methods with multiple treatment groups? I'm assessing the impact of an intervention on an outcome. For my first analysis, I used the MatchIt package (see code below) to match treatment and control groups and to compare outcomes for the treated vs control. The Treat variable is 0 or 1.  nn_match &lt;- matchit(treat ~ cov1 + cov2+ cov3 ,                      method = "nearest", data = df)   In my second analysis, I'd like to assess the effect of different treatment levels (for example, I have Treatment A, Treatment B, and Treatment C) on the outcome, but I'm stuck on the best way to proceed. Does it make sense to run separate matching analysis for each treatment group while excluding the observations from the other treatments groups?   nn_treatA_match &lt;- matchit(treat_A ~ cov1 + cov2+ cov3 ,                      method = "nearest", data = df_minus_other_treatBC) nn_treatB_match &lt;- matchit(treat_B ~ cov1 + cov2+ cov3 ,                      method = "nearest", data = df_minus_other_treatAC)   Or is there a way to run the analysis with the entire dataset and match controls to each treatment at the same time? Other posts have mentioned the twang package, but I'm not sure if that's what I should go with as I'd like to compare outcomes for each treated group to the control and not just across treated groups. How do I disable search-as-you-type text in Ubuntu 17.10 and later? I cannot find enable-interactive-search in    gsettings set org.gnome.nautilus.preferences enable-interactive-search true.     I searched using    gsettings list-recursively | grep search     and    gsettings list-recursively | grep interactive     but no such option.  Can you help me disable this in Ubuntu 17.10 and later? Show that $a_{n+1}= 1 + \frac{1}{a_n}$ converges. I am a little at a loss.  A couple years before my mother-in-law passed away she signed her house over to my wife and her sister via a quitclaim deed.    After she passed we sold the house for $80,000. The money was used to pay off the mortgage, some other funeral expenses, and sent $15,000 to other family members.  All the money was dispersed at the closing of the house.  My wife and her sister each ended up with about $20,000.  How do we go about claiming this income?  Since there was a quitclaim deed I guess we technically owned the house for a few years prior to its sale.  How does that affect our tax and what is the basis?  Is the basis the selling price(fair market value) or something else?  How much does each sister claim for gains?  Sorry it this is a jumble of questions I am just trying to get this straightened out before tax is due. I am trying to follow the best practice design patterns for triggers but get the error Constructor not defined: [CaseTriggerHandler].(). What am I missing?  Trigger  trigger CaseTrigger on Case (before insert, before update, before delete, after insert, after update, after delete, after undelete) {      CaseTriggerHandler handler = new CaseTriggerHandler();      //Before Insert     if(Trigger.isInsert &amp;&amp; Trigger.isBefore){         handler.OnBeforeInsert(Trigger.new);     }     //After Insert     else if(Trigger.isInsert &amp;&amp; Trigger.isAfter){         handler.OnAfterInsert(Trigger.new);     }     //Before Update     else if(Trigger.isUpdate &amp;&amp; Trigger.isBefore){         handler.OnBeforeUpdate(Trigger.old, Trigger.new, Trigger.newMap);     }     //After Update     else if(Trigger.isUpdate &amp;&amp; Trigger.isAfter){         handler.OnAfterUpdate(Trigger.old, Trigger.new, Trigger.newMap);     }     //Before Delete     else if(Trigger.isDelete &amp;&amp; Trigger.isBefore){         handler.OnBeforeDelete(Trigger.old, Trigger.oldMap);     }     //After Delete     else if(Trigger.isDelete &amp;&amp; Trigger.isAfter){         handler.OnAfterDelete(Trigger.old, Trigger.oldMap);     }     //After Undelete     else if(Trigger.isUnDelete){         handler.OnUndelete(Trigger.new);     } }   Handler      public with sharing class CaseTriggerHandler {      private boolean isExecuting = false;      public CaseTriggerHandler(boolean isExecuting){         this.isExecuting = isExecuting;     }      public void OnBeforeInsert(List&lt;Case&gt; newCases){         //EXECUTE BEFORE INSERT LOGIC     }      public void OnAfterInsert(List&lt;Case&gt; newCases){         //EXECUTE AFTER INSERT LOGIC     }      public void OnBeforeUpdate(List&lt;Case&gt; oldCases, List&lt;Case&gt; updatedCases, Map&lt;Id, Case&gt; CaseMap){         //BEFORE UPDATE LOGIC     }      public void OnAfterUpdate(List&lt;Case&gt; oldCases, List&lt;Case&gt; updatedCases, Map&lt;Id, Case&gt; CaseMap){         //AFTER UPDATE LOGIC     }      public void OnBeforeDelete(List&lt;Case&gt; CasesToDelete, Map&lt;Id, Case&gt; CaseMap){         //BEFORE DELETE LOGIC     }      public void OnAfterDelete(List&lt;Case&gt; deletedCases, Map&lt;Id, Case&gt; CaseMap){         //AFTER DELETE LOGIC     }      public void OnUndelete(List&lt;Case&gt; restoredCases){         //AFTER UNDELETE LOGIC     }      public boolean IsTriggerContext{         get{ return isExecuting;}     } } When to use 'which' or 'in which' or 'that' (as relative pronouns)? I am confused about when to use:  'which' vs 'in which' vs or 'that'?    Would you please explain it as a writing rule? Here I have an example:       Authors reported shortened growth period and slower radial growth rate for low soil water potential which is directly attributed to the decrease in water potential, rather than to the constraints on carbon availability. SQL Server Dependencies Is there an easy way to chase down table/stored procedure/function dependencies in SQL Server 2005+?  I've inherited a giant application with lots of tables and even more stored procedures and functions that are long and interlinked.    At the end of the day is there a way to build a dependency tree?  Ideally what I'm looking for goes in both directions:  For a table/procedure - what depends ON it?:  Show me all the stored procedures that eventually reference it (ideally in a tree view such that sub procedures nest out to the bigger procedures that call them)  For a procedure - what does IT depend on?:  Show me all the procedures and tables that a given procedure will (or could) touch when running.  It seems this tool shouldn't be that hard to make and would be incredibly useful for DB maintenance generally.  Is anyone aware of such a thing?  If this doesn't exist, why the heck not?  The built-in functionality in Management Studio is nice but the information does not appear to be complete at all. Two functions $f:A\rightarrow B$ and $g:B\rightarrow A$. Can someone give me an example where $g\circ f=id_A$ but $f\circ g\neq id_B$? How to copy files via terminal? I have read about copying files with terminal but these examples will help me a lot. So here is what I want to do:  Examples:   I have a file in /home/levan/kdenlive untitelds.mpg and I want to copy this file to /media/sda3/SkyDrive and do not want to delete any thing in SkyDrive directory. I have a file in /media/sda3/SkyDrive untitelds.mpg and I want to copy this file to /home/levan/kdenlive  and do not want to delete any thing in kdenlive directory  I want to copy a folder from home directory to sda3 and do not want to delete any thing on sda3 directory and opposite I want to cut a folder/file and copy to other place without deleting files in that directory I cut it into. When I need to generate unique id's based on some information hashing is typical choice. However, sometimes that id needs to be of a particular size.  I've seen a lot of schemes (HMAC-MD5-96 in SSH, CGA in SeND for IPv6) that use a portion of a hash, so I'm thinking it might be alright to use it that way.    Which properties of a hash are preserved and which do not apply anymore?  Obviously, with less bits, the chance of collision goes up, but can I still rely on the partial hash for uniform character distribution?  What about the ?  I'm guessing that if a hash modifies the entire hash upon the smallest change to the input, that would imply that a portion of that hash would be equally well changed.  Am I thinking about this correctly?  Also, would these behaviors be different for different hashing algorithms?