Here are two quotes from :     According to the first quote a dual is obtained by reversing all the arrows. Accrording to the second quote, the second pair of functors is the dual of the first. But if so, why aren't the arrows (in the category of categories and functors) in the second pair reversed (they still go from left to right)? And why aren't the arrows in $\mathbf{Set}$ reversed (i.e., why isn't $\mathbf{Set}$ replaced with $\mathbf{Set}^{op}$)? As far as I can see, only the arrows in $\mathscr A$ are reversed. Why so? If it's easier to answer what exactly &quot;dual&quot; means (in the framework of Leinster up to p.90), that would be great too.
I have long been confused about this notion. I know that for a statement within a single category, forming the dual statement is just reversing every arrows. But what about a statement concerning several categories and functors between them?   Mac Lane suggests reversing all arrows in all categories and leaving the functors invariant in his book "Categories for the working mathematician" (page 32), but it seems that Mac Lane himself doesn't always follow this discipline (when he states "the dual of Yoneda Theorem", he doesn't reverse the arrows in $\mathbf{Set}$ !) Can anyone give me a formal definition of "dualize a statement in category theory"?
Solve $x^2+5x+6 \equiv 0 \pmod{\!11\cdot 17}$ Solve $x^2+5x+6 \equiv 187 \mod 187$ Solution $$x^2+5x+6 \equiv 187 \mod 187$$ $$ (x+\frac{5}{2})^2 \equiv \frac{1}{4}$$ $$ 4(x+\frac{5}{2})^2 \equiv 1$$ $$ y:= x+\frac{5}{2} $$ $$ 4y^2 \equiv 1 \mod 11 \wedge 4y^2 \equiv 1 \mod 17  $$ $$ ( 2y \equiv 1 \mod 11 \vee 2y \equiv 10 \mod 11 ) \wedge ( 2y \equiv 1 \mod 17  \vee 2y \equiv 13 \mod 17)  $$ $$ ( y \equiv 6 \mod 11 \vee y \equiv 5 \mod 11 ) \wedge ( y \equiv 9 \mod 17  \vee y \equiv 15 \mod 17)  $$ Combining that from CRT I got: $$ y \in \left\{49, 60,83,94 \right\} $$ and for example: $$ x+\frac{5}{2}  \equiv 94 \mod 187$$ $$ 2x  \equiv 183 \mod 187$$ some calculus and get... $$x \equiv 185 $$ And the same thing for each other case. Question Is there any faster (or smarter) way to solve equations like that? line segment joining two centers of circles is perpendicular to line segment joining two intersection point of the circles Regex to get string between curly braces Unfortunately, despite having tried to learn regex at least one time a year for as many years as I can remember, I always forget as I use them so infrequently. This year my new year's resolution is to not try and learn regex again - So this year to save me from tears I'll give it to Stack Overflow. (Last Christmas remix).  I want to pass in a string in this format {getThis}, and be returned the string getThis. Could anyone be of assistance in helping to stick to my new year's resolution?    Related questions on Stack Overflow: Multi armed bandit for general reward distribution I'm working on a multi-armed bandit problem where we do not have any information about the reward distribution.   I have found many papers which guarantee regret bounds for a distribution with known bound, and for general distributions with support in [0,1].   I would like to find out if there is a way to perform well in an environment where the reward distribution does not have any guarantees about its support.  I'm trying to compute a nonparametric tolerance limit and using that number to scale the reward distribution so I can use the algorithm 2 specified on this paper (). Does anyone think this approach will work?  If not, can anyone point me to the right spot?  Thanks a bunch! Sentence vs proposition How to automount NTFS partitions? I have two NTFS partitions, and I don't want to mount them manually every-time I start Ubuntu.   How can I do this?  Is there a tool or a code to use?   If so, is it safe to automount? specially when they are being used by another OS? Term for a person who can read but cannot write I'm looking for a term to accurately describe a person who can only read but cannot write. While I'm primarily concerned with people who have never learned to write, I would also be interested in any additional terms used to represent people who have lost the ability to write as a result of disuse.  Both illiterate and unlettered imply an inability to read as well as to write. System.FinalException: Record is read-only: Trigger.updateCompetitors: line 24 I am writing a trigger to get all the Opportunity competitors into a field on Opportunity, however, getting an error while testing it.  Error: updateCompetitors: execution of AfterUpdate caused by: System.FinalException: Record is read-only: Trigger.updateCompetitors: line 24  Can you please check the below code and suggest me what's wrong with this?  trigger updateCompetitors on Opportunity(after insert, after update){    List&lt;Opportunity&gt; OppIds = new List&lt;Opportunity&gt;();     for (Opportunity opp: trigger.new){         OppIds.add(opp);     }  List&lt;OpportunityCompetitor&gt; oppComps = new List&lt;OpportunityCompetitor&gt;();  oppComps= [select id, CompetitorName, Strengths, Weaknesses from OpportunityCompetitor where OpportunityId in: Trigger.newMap.keySet()];  List&lt;Opportunity&gt; OppstoUpdate = new List&lt;Opportunity&gt;();   for (Opportunity opp: OppIds){     String allcomps = null;      if( oppComps.size()&gt;0 ) {       for (integer i = 0; i &lt; oppComps.size(); i++){          allcomps = allcomps + oppComps[i].CompetitorName+',';       }     }    opp.OpportunityCompetitors__c = allcomps;    OppstoUpdate.add(opp);  }    update OppstoUpdate; } Quantum mechanical books for mathematicians I'm a mathematician. I have good knowledge of superior analysis, distribution theory, Hilbert spaces, Sobolev spaces, and applications to PDE theory. I also have good knowledge of differential geometry. I would like to study the Semiclassic Analysis, but perhaps I must first study the foundations of quantum mechanics. So I would like to know what book you recommend me to begin studying quantum mechanics. I'm primarily interested in a mathematical point of view. I have seen some books of this type, but I would like to have some other opinion.  Thanks for every reply iMac only show flashing folder with question mark after formatting OS X partition drive Today i format my "OS X El capitan" partition for installing "Yosemite". But due an error in bootable yosemite usb it didn't install on this partition. After some hours when i turned on iMac and hold down "option" key for boot into usb but just blank screen and mouse show. And every time when i turend on iMac it only show flashing folder with question mark. 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Obviously a solution for this would be to have an include that's an ISO-8859-15 file, but there must be a better way than to have another required include?  echo strtr(utf8_decode($input),             'ŠŒŽšœžŸ¥µÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖØÙÚÛÜÝßàáâãäåæçèéêëìíîïðñòóôõöøùúûüýÿ',            'SOZsozYYuAAAAAAACEEEEIIIIDNOOOOOOUUUUYsaaaaaaaceeeeiiiionoooooouuuuyy');   UPDATE: Maybe I was a bit inaccurate with what I try to do: I do not actually want to remove the umlauts, but to replace them with their closest "one character ASCII" equivalent. Determine which circuits are parallel and which are in series I need to find the Equivalent Resistance across A and B. The problem is, I don't know which one's are in series and which ones are in parallel. How do I identify which ones are parallel or series? Prove divisibility law $\,b\mid a,c\,\Rightarrow\, b\mid ka + lc$ for all $k,l\in\Bbb Z$ What is the difference between range and xrange functions in Python 2.X? 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Is easy to find the all four possible candidates but is cumberstone to show that they are automorphisms:  For multiplication is too long showing that  $f((a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6})(a'+b'\sqrt{2}+c'\sqrt{3}+d'\sqrt{6}))=f(a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6})f(a'+b'\sqrt{2}+c'\sqrt{3}+d'\sqrt{6})$  For some non trivial candidate since the left hand side expands in $16$ terms. What about $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}$? there are $7$ non trivial candidates and multiplication expands in $64$ terms!   How to do it in a tricker form?  Edit(Let me be more precise in my question):   I know that if $f\in\Gamma(\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q})$, then $f(\sqrt{2})=\pm\sqrt{2}$, $f(\sqrt{3})=\pm\sqrt{3}$ so there are $4$ candidates for $\mathbb{Q}$-Automorphisms. 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