Id not in set gives no result
Query to find GUID's "not in" null containing subquery returns no results
The set of integers $n$ expressible as $n=x^2+xy+y^2$ Let $S$ be the set of integers $n$, such there exist integers $x,y$ with $$n=x^2+xy+y^2$$   Is the implication $$a,b\in S\implies ab\in S$$ true? If yes, how can I prove it?  I worked out $$n\in S\iff 4n\in S$$ and $$n\in S\iff 3n\in S$$ I tried two approaches. The first is to express $$(a^2+ab+b^2)(c^2+cd+d^2)$$ in the form $$f^2+fg+g^2$$ with polynomials $f,g$ with integer coefficients. I however could not find suitable $f$ and $g$. The second approach is based on $$x^2+xy+y^2=\frac{(2x+y)^2+3y^2}{4}$$ If we have $n=x^2+xy+y^2$ , we have $u^2+3v^2=4n$ for some integers $u,v$ with equal parity. The main problem of this approach is to consider the equal parity. Any ideas ? How to find regression coefficients $\beta$ in ridge regression? In ridge regression, the objective function to be minimized is: $$\text{RSS}+\lambda \sum\beta_j^2.$$   Can this be optimized using the Lagrange multiplier method? Or is it straight differentiation? Is there a graceful way to handle passing a list of ids as a parameter to a stored procedure?   For instance, I want departments 1, 2, 5, 7, 20 returned by my stored procedure. In the past, I have passed in a comma delimited list of ids, like the below code, but feel really dirty doing it.   SQL Server 2005 is my only applicable limitation I think.  create procedure getDepartments   @DepartmentIds varchar(max) as   declare @Sql varchar(max)        select @Sql = 'select [Name] from Department where DepartmentId in (' + @DepartmentIds + ')'   exec(@Sql) Guaranteed lifetime of temporary in C++? Does C++ provide a guarantee for the lifetime of a temporary variable that is created within a function call but not used as a parameter?  Here's an example class:  class StringBuffer { public:     StringBuffer(std::string &amp; str) : m_str(str)     {         m_buffer.push_back(0);     }     ~StringBuffer()     {         m_str = &amp;m_buffer[0];     }     char * Size(int maxlength)     {         m_buffer.resize(maxlength + 1, 0);         return &amp;m_buffer[0];     } private:     std::string &amp; m_str;     std::vector&lt;char&gt; m_buffer; };   And here's how you would use it:  // this is from a crusty old API that can't be changed void GetString(char * str, int maxlength);  std::string mystring; GetString(StringBuffer(mystring).Size(MAXLEN), MAXLEN);   When will the destructor for the temporary StringBuffer object get called?  Is it:   Before the call to GetString? After GetString returns? Compiler dependent?   I know that C++ guarantees that a local temporary variable will be valid as long as there's a reference to it - does this apply to parent objects when there's a reference to a member variable?  Thanks. Please help me to prove this trigonometric equation.  $\cos \left( 36^\circ \right)-\cos \left( 72^\circ \right) = \frac{1}{2}$  Thank you. Prove that every prime larger than $3$ gives a remainder of $1$ or $5$ if divided by $6$ Can we prove that every prime larger than $3$ gives a remainder of $1$ or $5$ if divided by $6$ and if so, which formulas can be used while proving? $\varepsilon$-number countability without choice Let $\alpha\mapsto\varepsilon_\alpha$ be the enumeration of the $\varepsilon$-numbers--that is, those $\alpha$ such that $\omega^\alpha=\alpha$--by the ordinals.  If we know that countable unions of countable sets are countable, (slightly weaker than countable choice), then we can show fairly easily that $\varepsilon_\alpha$ is countable iff $\alpha$ is countable.  I think I saw a result at some point that $\varepsilon_0$ is countable without relying on any choice principle (though I can't recall where I saw that). Is this true? More generally, for which $\alpha$ can we conclude that $\varepsilon_\alpha$ is countable in ZF?  Update: I have been able to prove (in ZF) that if $\varepsilon_\alpha$ is countable, then $|\alpha|&lt;\aleph_1$, so that's one direction. I'm still a bit uncertain about the answers I've got so far. Is it possible to duplicate eyes? This question contains spoilers for anime viewers and early manga viewers.      When Nagato was revived with Edo Tensei, he had Madara's eyes. When Madara was revived, he had his original eyes too (Rinnegan, much like Nagato).   Is this exploitable? Can I repeat this cycle of transplant and revive to duplicate strong eyes? How does that work really? Producing doc/docx from LaTeX Is there any way to produce doc/docx files from LaTeX? Reverse / invert a dictionary mapping Union of uncountable cardinals I have to prove the following:  Let $\omega_1$ be the least uncountable cardinal and, for each $n \in \omega$ (here, $\omega$ is the set of natural numbers denoted as an ordinal/cardinal), $n\ge1$, let $\omega_{n+1}$ be the least cardinal that is greater than $\omega_n$. Show that $\bigcup_{n\in\omega}\omega_n$ is a cardinal.  I suppose the proof has something to do with transfinite induction/recursion, but I don't know exactly how I'm supposed to use it. Any thoughts? How to prevent system applications (like the Software Center) from asking for password? If $n = 51! +1$, then find number of primes among $n+1,n+2,\ldots, n+50$ If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$  Really speaking, I don't have any clue ... Rigorous proof that $dx dy=r\ dr\ d\theta$ I get the graphic explanation, i.e. that the area $dA$ of the sector's increment can be looked upon as a polar "rectangle" as $dr$ and $d\theta$ are infinitesimal, but how do you prove this rigorously?    1. Geometrically, the exact area would be $$\frac{(r+dr)^2d\theta}{2} - \frac{r^2d\theta}{2}$$$$= (r + \frac{dr}{2}) dr d\theta  $$$$= r dr d\theta + \frac{dr^2 d\theta}{2}.$$  How do we get rid of $\frac{dr^2 d\theta}{2}$? Is it too insignificant in value compared to $r dr d\theta$ so that it vanishes? If so, is it the reason it gets ignored? And what's more, geometrically speaking, why would the two areas - $dx dy$ and $r dr d\theta$, be equal?  2. Approaching it algebraically, setting $$x = r\sin\theta$$$$y = r \cos\theta$$  gives $$\frac{dx}{d\theta} = r\cos\theta,  dx = r\cos{\theta} d\theta$$$$ \frac{dy}{dr} = \cos\theta,  dy = \cos\theta dr. $$  Multiplying both equations, side by side, gives $$dxdy = r\cos^2\theta dr d\theta.$$  Again I get an extra term, which is $\cos^2 \theta$. In both cases I am unable to derive that $dxdy = rdrd\theta$.  What do I do wrong in my reasoning? It all makes me think that I'm getting something essential terribly wrong. Where can I find the status of the released versions of Ubuntu? Error Message:      "A reboot is pending. Please restart before starting Visual Studio".    I installed Visual Studio (Visual Basic 2015 and SQL 2015) and after what looked like a successful install I then got the "Reboot" window.   Not expecting it I had a bunch of things open. So I closed a bunch a bunch of things, and did the reboot myself.   Now I cannot run Visual Basic because it ONLY gives me the above error message in the pop-up window. It won't even let me do a re-install.   All it does is keep giving me the above error message.   If you have any advice that's better than a complete uninstall/reinstall I'm sure everyone in my situation would benefit from it.  How can I resolve this issue? How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer? $\forall m,n\in\mathbb Z$ , $m\ge1$ and $n\ge1$ how to prove that  $$\frac{(mn)!}{m!(n!)^m}$$ is an integer?  Thanks in advance. I am having a lot of trouble with these series questions. Up until this point, I had relatively little trouble with all the questions in the book. These seem to require knowledge about approximations of functions and other external experience-based knowledge, which I just don't have yet.  Determine convergence or divergence of the given series. In the case of convergence, determine whether the series converges absolutely or conditionally.  $$\sum_{n=1}^\infty (-1)^n\left[e-\left(1+\frac 1 n \right)^n\right]$$  It's easy to see that  $$\lim_{n\to\infty}\left[e-\left(1+\frac 1 n\right)^n\right]=0$$  however, in order to apply Leibniz's Rule and show conditional convergence I need to show that the sequence is monotonically decreasing. This doesn't seem doable with straight inequalities, so I tried taking the derivative, which just resulted in an uninterpretable mess. This doesn't even begin to address the question of absolute convergence/divergence.  There are 54 of these questions... I must be missing something really fundamental if they all take this long. How to make IntelliJ IDEA recognise code created by macros? Has there been any news from the maintainers of Ubuntu packaging of Ruby as to whether they'll be releasing a package for Ruby 2.0? If so, have they indicated whether they'll be releasing any packages for Ruby 2.0 on 12.04?