What is the meaning of $C^{op}$ if $C$ is a category without inverses?
Do opposite categories always exist?
In which order should I play the Borderlands series? How do I know if food left at room temperature is still safe to eat? Finding an isomorphic subring of matrices I'm struggling a fair amount with this exercise:     Find a subring of $M(2,\mathbb{Q})$ which is isomorphic to a) $\mathbb{Q}$ x $ \mathbb{Q}$ b) $\mathbb{Q}$ c) $\mathbb{Q}[x]$/$x^2$   Now I know a subring must be a subgroup, must contain the elements $0,1$ and must be closed under multiplication. As we are looking for isomorphisms then they must also be ring homomorphisms and must be bijective.  I tried to come up with a random subring and attempt to prove it is an isomorphism. E.g $\left\{\left.\begin{matrix}  a &amp; b \\ c &amp; 0  \end{matrix}\right|a,b,c \in \mathbb{Q}\right\}$ is a subgroup of $M(2,\mathbb{Q})$ and then try and show it's an isomorphism. However I'm stuck here as I don't actually know how to go about finding these specific isomorphisms and I completely lose what I'm doing. Any help would be great. I found an explanation for my question here , which suggests that water washes out sebum (special oil, which covers our skin) and water gets into outer layer of now unprotected skin by osmosis. This article also suggests that fingers are not really shriveled - they are water-logged. But is this right? I think I observed many times my wrinkly fingers after swimming, but I never felt that they are water-logged. Any thought on this matter would be highly appreciated. Attempt to de-reference a null object: Trigger There were custom validation error(s) encountered while saving the affected record(s). The first validation error encountered was:      Apex trigger updatequotestatus caused an unexpected exception, contact your administrator: updatequotestatus: execution of AfterInsert caused by: System.NullPointerException: Attempt to de-reference a null object: Trigger.updatequotestatus line 5, column 1.   Error Occured at Line:     if((t.what.getsobjecttype() == quote__c.sobjecttype) &amp;&amp; (t.subject.substring(0,5) == 'Email')  trigger updatequotestatus on task (after insert, after update){     Set&lt;Id&gt; quoteIds = new Set&lt;Id&gt;();     List&lt;quote__c&gt; quotes = new List&lt;quote__c&gt;();      for(task t:trigger.new){         if(t.whatId!=null){             if((t.what.getsobjecttype() == quote__c.sobjecttype) &amp;&amp; (t.subject.substring(0,5) == 'Email')){                 quoteIds.add(t.whatId);             }         }     }      for(quote__c q:[select StageName__c from quote__c where Id IN : quoteIds]){         quote__c qt = new quote__c();         qt.Id = q.Id;         qt.StageName__c = 'Proposal Sent';         quotes.add(qt);     }      if(quotes.size()&gt;0){         update quotes;     } } I need to know the ip address of another machine in a lan network.  What is the Windows command for getting ip address from machine name and vice versa? Flair for Unicode usernames broken Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$? Is there a measurable set $A$ such that $m(A \cap B)= \frac12 m(B)$ for every open set $B$?  Edit: (t.b.) See also  for further answers. Words that mean ''to say something unintentional'' I am looking for words that mean ''to say something unintentionally'' Is it correct to change the common structure in these phrasal verbs? I just read a book to learn English. And the topic I read is about the phrasal verbs, but a big doubt has come up to my mind.  Is it correct to change the position of the preposition (putting it with the verb instead of after the direct object)?  As in:  Common structure:   My wife backs me up over my decision to quit my job. We have to blow 50 balloons up for the party.  I need to break these shoes in before we run next week. Please help me out to get this job done.   So, would it be correct to do the following:   My wife backs up me over my decision to quit my job. We have to blow up 50 ballons for the party.  I need to break in these shoes before we run next week. Please help out me to get this job done. I have to give a combinatorial proof of  $$\sum_{k=1}^nk{n\choose k}^2=n{2n-1\choose n}.$$  I find it difficult to solve such problems. I'm not a brilliant person and never will be so I need to have algorithms to solve most problems. Usually I won't just "see" solutions unless I have a thorough enough understanding of a theory. This is also the case here. I know what the symbols mean, but I don't see what I should count and how to obtain the identity.  I think the right-hand side counts $n$-element subsets of a $(2n-1)$-element set with one element chosen to be "special". I think I should find a partition of the set of such subsets into $n$ parts so that the left-hand side is the sum of the cardinalities of the parts. Is there $n$ of anything on the right-hand side? Well, there are $n$ choices for the "special" elements, but only after I've chosen the $n$-element subset. Otherwise I can choose the "special element" in $2n-1$ ways.  This is a homework problem so I probably shouldn't ask for a full solution. But I would like to have some guidelines for approaching such problems, perhaps based on this example. Is there a way to disable certain Windows 7 keyboard shortcuts?  Specifically, in this case, I want to disable Alt+Space and WinKey+Space. "As a beginner, I am happy that I learned a new thing."  You will recognize this as your DoReMi-s.( at 4:52)  What is the meaning of the "AS" there and when do we use that?  In depth answers would be appreciated.  Similarly:     Do I have your word as a lawman not to shoot me down like a dog in the street?    Can you explain this sentence?   Can I understand it as a lawman wouldn't shoot down people on the street or wouldn't have his word or what? Frequent Write to Non-volatile Memory I am designing a device that automatically adjusts it's physical position as temperature changes. If the device it shut off or power is disconnected the device needs to remember it's last temperature and position. I have the ability to store these values in EEPROM but the problem is the position and temperature could be changing very rapidly. If I were to write the temp and pos to EEPROM after every time they changed that would (1) slow down the firmware a bit, and (2) likely kill the EEPROM after a year or two. So as I see it my options are as follows...  1) use a capacitor/battery to keep the device powered for a short time after power is lost so that I can write the values to EEPROM at that time only. I don't like this because the board is kinda power hungry and this would require a big cap. And I don't have a ton of free space. And I don't want the added cost of a battery and battery holder / or a big cap.  2) use F-RAM instead of EEPROM so that I can write to it trillions of times without wearing it out. I don't like this option because FRAM is quite a bit more expensive than EEPROM and this is for a production product (not just one).  3) Only write the position and temperature every 5 minutes or so. That way I always have a fairly recent position/temp recorded but I am not writing every second so my program is not slowed down and the EEPROM won't die as fast. This seems like my best option.  Does anyone else have any suggestions that I'm not thinking of? What is aperture, and how does it affect my photographs? How to connect to Salesforce REST API without OAuth? How does Jitterbit do it? Is there a quick way to set an HTML text input (&lt;input type=text /&gt;) to only allow numeric keystrokes (plus '.')? How can a Photon have a "frequency"? I picture light ray as a composition of photons with an energy equal to the frequency of the light ray according to $E=hf$. Is this the good way to picture this? Although I can solve elementary problems with the formulas, I've never really been comfortable with the idea of an object having or being related to a "frequency". Do I need to learn quantum field theory to really understand this? Difference Between Sequential Workflow & State machine workflow? What is the difference between Sequential Workflow and Stater machine workflow? What context should I take account to when choosing one of them? What does the term "86'd" relate to? What does it mean when someone or something is referred to as being "86'd"?