What does ⟨ -,- ⟩ mean in the context of category theory?
UMP of product functor
Validate date format in php Why does the degree of freedom of SSR equals 1 in simple linear regression I just installed a fresh install of 32-bit 12.04 LTS Pangolin today. When I tried to install a few applications, it was really slow. But when I download a .deb or .tar.gz file from on my web browser, it is a lot faster.  Computer specs:  HP Pavilion Slimline s7700n PC  [Memory]         873.7 MiB  [Processor]      AMD Athlon(tm) 64 X2 Dual Core Provessor 3800+ x 2  [Graphics]       GeForce 6105 LE/integrated/SSE2/3DNOW!  [Disk Size] 638.5 GB  I normally get 200-400 kbps.  This is the comparison. Is sum of two orthogonal matrices singular? I am trying to solve following problem.      Let $A, B \in \mathbb{R}^{n\times n}$ be an orthogonal matrices and $\det(A) = -\det(B)$. How can it be proven that $A+B$ is singular?   I could start with implication:  $\det(A)=-\det(B) \Rightarrow B$ is created from $A$ by swapping two lines or columns.   But I am not sure if this implication is correct. How can Obito use his Mangekyo so often? How come everyone easily takes or say the name of "Voldemort" without any fear in Scorpius's world? A concise definition of a frame of reference in Newtonian mechanics? I've read  on frame of reference and also followed all of the references cited in the text (Salençon, Brillouin, Norton, etc) but I'm struggling to find any concise definition in all of that.   I would like a concise definition for a frame of reference in the context of Newtonian mechanics. This definition should not involve any additional qualifiers such as inertial and must be mature enough as to differentiate a frame of reference and a coordinate system. Is there one such definition? I've got a small piece of code that is parsing an index value to determine a cell input into Excel.  It's got me thinking...  What's the difference between   xlsSheet.Write("C" + rowIndex.ToString(), null, title);   and  xlsSheet.Write(string.Format("C{0}", rowIndex), null, title);   Is one "better" than the other? And why? Context Around this Code: I am making a schedulable class run each Sunday to track a couple of metrics important to the team.   Question: I am looking to consolidate DML operations and a big part of this is understanding pass by reference vs pass by value.  I want to pass the salesOps user object to the method SDREfficiencyLastWeek. I made the return type void, because I figured that I will be passing this by reference and I can make all the updates in the method without having to anything.   Why I am confused:    I read the above article and am thrown off by the following statement:  "Non-primitive data type arguments, such as sObjects, are also passed into methods by value. "   If the object is indeed passed by reference, then the query for the salesOps user in my code should be sufficient enough to edit all fields on that object without actually having to state each field in the query.  global class SDREfficientMetricsSchedulable implements Schedulable  {     global void execute(SchedulableContext sc)      {         User salesOps = [SELECT ID FROM USER WHERE id = '0056A000002Z3p9QAC'];         List&lt;String&gt; queries = new List&lt;String&gt;();          String contactsWorked_lastWeek  = 'select count(id) results from contact where LastActivityDate = last_week';         queries.add(contactsWorked_lastWeek);          String demosSet_lastWeek            = 'select count(id) results from form__c where CreatedDate = last_week';         queries.add(demosSet_lastWeek);          String demosCompleted_lastWeek  = 'select count(id) results from opportunity where (stagename = \'Unqualified Demo Completed\'  OR stagename = \'Qualified Demo Completed\' )  and RecordTypeId = \'0126A000000yudDQAQ\' and CloseDate = last_week';         queries.add(demosCompleted_lastWeek);          String qualifiedDemos_lastWeek  = 'select count(id) results  from opportunity where RecordTypeId = \'0126A0000004Ao6QAE\' and isclosed = false and  SDR_Owner__c != null and CreatedDate = last_week';         queries.add(qualifiedDemos_lastWeek);          String closedWon_lastWeek       = 'select count(id) results  from opportunity where RecordTypeId = \'0126A0000004Ao6QAE\' and iswon = true and  SDR_Owner__c != null and CloseDate = last_week';         queries.add(qualifiedDemos_lastWeek);            SDREfficiencyLastWeek(queries, salesOps);          update salesOps;     }       public void SDREfficiencyLastWeek(List&lt;String&gt; queries, User salesOps )     {        }  } Check ping statistics without stopping Is there a way to tell ping to show its usual termination statistics without stopping the execution?  For instance, I'd like to quickly view:  --- 8.8.8.8 ping statistics --- 2410 packets transmitted, 2274 received, +27 errors, 5% packet loss, time 2412839ms rtt min/avg/max/mdev = 26.103/48.917/639.493/52.093 ms, pipe 3   without having to stop the program, thus losing the accumulated data. A word for someone who you share a half-sibling with This is probably best explained with an example.  Alice and Bob have a child, Erin. Bob and Carol have a child, Frank. Carol and Dan have a child, Grace. Clearly, Erin and Frank are half-siblings, because of their father, Bob. Similarly, Grace and Frank share their mother, Carol.  Erin and Grace have no parents in common, and are thus not half-siblings. However, is there a term for their relation? Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$ SEO in Stack Overflow We've been out of private beta for only two days, and stackoverflow.com is already ranking high in .  Are there any specific techniques applied in the site? I've been researching the subject for some time and haven't found it to be so easy.   even returns site-links.  () How to delete one shape from another in Illustrator CS4? I have an image (below) of two hands over a black circle. All three are currently separate paths with a fill but no stroke, all on the same layer.  How do I delete the shape of the hands from the black circle, so that afterwards I can delete the hands themselves, and just have the circle, with the shape of the hands cut out of it? On average, how many times must I roll a dice until I get a $6$?  I got this question from a book called Fifty Challenging Problems in Probability.   The answer is $6$, and I understand the solution the book has given me. However, I want to know why the following logic does not work: The chance that we do not get a $6$ is $5/6$. In order to find the number of dice rolls needed, I want the probability of there being a $6$ in $n$ rolls being $1/2$ in order to find the average. So I solve the equation $(5/6)^n=1/2$, which gives me $n=3.8$-ish. That number makes sense to me intuitively, where the number $6$ does not make sense intuitively. I feel like on average, I would need to roll about $3$-$4$ times to get a $6$. Sometimes, I will have to roll less than $3$-$4$ times, and sometimes I will have to roll more than $3$-$4$ times.  Please note that I am not asking how to solve this question, but what is wrong with my logic above.  Thank you! How can I send an email using PHP? I've just happened to read this  on MO (that of course has been closed) and some of the answers to a similar  on MSE.   I know almost nothing of nonstandard analysis and was asking myself if something like the sentence « $1- 0.999 \dots$ is a nonzero positive infinitesimal» could be easily expressed and proved in nonstandard analysis.  First of all, what is 0.999... ? If we take the usual definition as a series or as a limit of a sequence of rationals, then it will still be a real number and equal to $1$ (I guess by "transfer principle", but please correct me if I'm wrong).  Instead, let's define  $$0.9_N:=\sum_{i=1}^N 9\cdot 10^{-i} $$  where $N\in{}^*\mathbb{N}\setminus\mathbb{N}$ is an infinite nonstandard natural number. This $0.9_N$ is a legitimate element of ${}^*\mathbb{R}$, expressed as $0.$ followed by an infinite number of "$9$" digits.  What can be said about $\epsilon_N:=1-0.9_N$ ? Is there an elementary proof that $\epsilon_N$ is a positive infinitesimal of ${}^*\mathbb{R}$ ? (by "elementary" I mean just order and field axioms and the intuitive facts about infinitesimals, like that for $x$ infinite $1/x$ is infinitesimal etc.; no nonprincipal ultrafilters &amp; C). Show that $n^3-n$ is divisible by $6$ using induction As homework, I have to prove that     $\forall n \in \mathbb{N}: n^3-n$ is divisible by 6   I used induction  1) basis: $A(0): 0^3-0 = 6x$ , $x \in \mathbb{N}_0$ // the 6x states that the result is a multiple of 6, right?  2) requirement: $A(n):n^3-n=6x$  3) statement: $A(n+1): (n+1)^3-(n+1)=6x$  4) step: $n^3-n+(n+1)^3-n=6x+(n+1)^3-n$  So when I resolve that I do get the equation: $n^3-n=6x$ so the statement is true for $\forall n \in \mathbb{N}$  Did I do something wrong or is it that simple? How discontinuous can a derivative be? There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property.  To my knowledge, not many "highly" discontinuous Darboux functions are known--the only one I am aware of being the Conway base 13 function--and few (none?) of these are derivatives of differentiable functions.  In fact they generally cannot be since an application of Baire's theorem gives that the set of continuity points of the derivative is dense $G_\delta$.  Is it known how sharp that last result is?  Are there known Darboux functions which are derivatives and are discontinuous on "large" sets in some appropriate sense? "quite" vs "pretty" What is the difference between quite and pretty in the following context:     The differences between these concepts are quite complicated.   and     The differences between these concepts are pretty complicated.