Installing 32 bit libraries on 64 system, libz.so.1 missing
Unable to correct problems, you have held broken packages
$C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ I understand that they force you to implement methods and such but what I cant understand is why you would want to use them. Can anybody give me a good example or explanation on why I would want to implement this. Can a US citizen leave and enter the US using a foreign passport? 32 bit or 64 bit OS on a 512 mb VPS? If one obtains results in one area of a field (say during an MSc thesis) before deciding that one is not interested in making a career in that area, should one take the time (as a PhD student) to get make those results publishable, or is it better to focus on the new area? Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$.   I want to prove that $$ \frac{N(t)}{t} \rightarrow \lambda, \quad t \rightarrow \infty \quad \text{ a.s.}. $$  To prove this I received a hint that I should use the martingale law of large numbers which states that:  Theorem: (Martingale LLN)  Let $S= (S_n = \sum_{k=1}^n X_k)_{n=1,2,\ldots}$ be a martingale with respect to the filtration $(\mathcal{F}_n)_{n=1,2,\ldots}$. Then $\sum_{k=1}^n X_k/k$ converges almost surely on the set $\left\{ \sum_{k=1}^\infty k^{-2} \mathbb{E}\left[ X_k^2 \mid \mathcal{F}_{k-1} \right] &lt; \infty \right\}$. Hence $S_n/n \rightarrow 0$ a.s. on the same set.  Now to use this theorem I think I need to use Doob-Meyer to decompose $N(t)$ as the sum of a martingale and an increasing process. Which would be the martingale $M(t) = N(t) - \lambda t$ and increasing process $A(t) = \lambda t$. But how is $M(t)$ the sum of a martingale? And I do not know how this exactly relates to the thing I have to prove. Thanks for any help. Terminal appears with much delay when shortcut is used When I press Ctrl+Alt+T the Terminal opens after around 30 seconds. That's pretty annoying. If I open it with Alt+F2 and enter gnome-terminal the window appears instantly as usual.  I'm using Ubuntu 16.04 with gnome3.  Thank you for your help in advance! Encode URL in JavaScript? What is the difference between String and string in C#? Example (note the case):  string s = "Hello world!"; String s = "Hello world!";   What are the guidelines for the use of each? And what are the differences? Consider the following MWE:  \documentclass{article} \usepackage{pstricks} \begin{document} A \end{document}   It produces the error  Runaway argument? ] {pgffor.code.tex}  ! Paragraph ended before \ProvidesPackageRCS@i was complete. &lt;to be read again&gt;                     \par  l.11   How do I fix this? Did not receive site association bonus (+100 rep) I want to export my objects to unity, but in blender I`m using Eevee engine. However the light in rendered more and material preview mode is completely different, which impacts the colors of objects. Which one can i trust to see the true final colors of the objects so that they will be the same when i export them to unity? Series as an integral (sophomore's dream) Prove that a continuous function on a closed interval attains a maximum After lethal first strike damage to an indestructible creature, will it still do damage? If I were to attack with a 4/4 indestructible creature and it's blocked by a 4/2 creature with first strike, would my indestructible creature still attack during the second combat damage step, or would it just tap, not assigning any damage? Prove that $\sum\limits_{k=0}^m\binom{m}k\binom{n}{r+k}=\binom{m+n}{m+r}$ using combinatoric arguments. Prove that $$\sum\limits_{k=0}^m\binom{m}k\binom{n}{r+k}=\binom{m+n}{m+r}$$ using combinatoric arguments.   I prefer you to answer using an analogy to a combinatoric problem or else I'll have trouble understanding.  I tried:  Let $m$ represent a group of boys and $n$ represent a group of girls. We want to form a committee with the condition that there must be $r$ more girls than boys on the committee. There's two ways of doing this. (I'm having trouble with the right-hand side.)  The left-hand side is one way of doing this. So we split the problem to different cases. One such case is where there are no boys on the committee, and so, there's $\binom{m}0\binom{n}{r+0}$ ways of forming a committee. Next, we can have 1 boy on the committee. There's $\binom{m}1\binom{n}{r+1}$ ways to do this. We repeat this process until we have $\binom{m}k\binom{n}{r+k}$ ways of forming a committee. Summing up the cases will result in the right-hand side. However, as stated before, I don't really know how to explain the right-hand side. Maybe I used the wrong analogy? If $(Tx \mathbin{|} x) = 0$ for all $x$ then $T = 0$ Let $T$ be a linear operator on an inner product vector space $V$. I'd like to prove that if $$(Tx|x)=0 \quad \forall x \in V$$ then $T$ is the null operator. I can't figure out this proof using contradiction, do I have to choose an appropriate vector? Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$? Given three matrices $A$, $B$ and $C$ such that $ABA^T C$ is a square matrix, the derivative of the trace with respect to $A$ is:  $$ \nabla_A \operatorname{trace}( ABA^{T}C )  = CAB + C^T AB^T $$  There is a proof . However, I was not able to understand the second line of the proof. it says something like this:  $$ \nabla_A \operatorname{trace}( ABA^T C )= \nabla_\bullet \operatorname{trace}(f(\bullet)A^{T}C) + \nabla_\bullet \operatorname{trace}(f(A) \bullet^T C) $$  Where  $f(A) = AB$  I tried searching the proof or any hint using google, but I couldn't fine anything. Could anyone please help me with this second line. I cannot sleep. I am obsessed with this proof. Distribution of linear combination of OLS regression coefficients I have a simple linear OLS regression $Y_i = \alpha+ \beta_1 X_{1i} + \beta_2 X_{2i} + e_i$ where $e_i \sim N(0,\sigma)$. I have estimated the regression from the data and obtained estimates for my coefficients as well as the corresponding covariance matrix. Assume my dataset is large (>500)  Now I would like to construct the 95% confidence interval for $Z = a_1 \beta_1 + a_2 \beta_2$. My first thought was that I could assume that Z follows a normal distribution because my regression coefficients are approximately normal and a linear combination of normal variables is also a normal variable. I do find some sources where they treat $Z$ as a normal random variable.  But now I am in doubt because I found that a linear combination of normal random variables is only normal if they are independent. So is it still save to assume $Z$ is random? How do I make Units efficiently? I just blew almost all my money on a really cool lookin' ship. How do I regain solvency without spending a lot of time fiddling around in my inventory?  At the moment I've found a sequence of steps that'll turn 50 carbon and 100 Plutonium into ~5000 Units (by creating Suspension Fluid and then turning it into Electron Vapor). Are there better recipes or sequences of recipes I should look for? Or should I just drop the whole thing and go mine a particular mineral?