a part vs. part
"As part of" versus "as a part of"
"Not possible" and "Impossible" When we say,      It is not fair.   or     It is unfair.   I'm not sure enough to say whether both of the sentences have the same meaning or not though superficially, there is no difference between them but if we say,     Something is not possible.   or     Something is impossible.   then there is really an observable difference in my first language. Hence, the question - is there a difference between the preceding two sentences in English? Theorem:   If $x$ is a positive integer and  $x^3$ is even, then $x$ is even.   My Proof by Contrapositive:  I. Assuming that $x$ is odd, then I will show that $x^3$ is odd.  II. $x$ is odd, so $x$ can be expressed as $2k+1$  III. $(2k+1)^3$  IV. $2k(4k^2 + 6k + 3) +1$  V. Since $2k(4k^2 + 6k + 3) + 1$ is an integer, we can conclude $x^3$ is odd.  Did I go about this correctly? Should step V say "$2$ multiplied by an integer and added by $1$ is odd therefore we can conclude $x^3$ is odd"? How to store and preserve lots of data? I have many DVDs with movies and games, they are getting old and dusty, some of them are damaged. And I have more than 300 GB of general files in my PC. I want to store and preserve all of my data and keep them for a long long time.  I want to know what is the best and more reliable storage media that I should use and... I can buy. I will copy all of my DVDs to a new media and throw all of my DVDs away. It will take a very smaller physical space.  I think that everything that I need to store is currently 800 GB.  I do not like optical discs, because they are fragile and I have lost many CDs and DVDs. Hard drives would be good, but they can suddenly stop working, and it is very expensive to recover data from a broken hard drive.  Other questions: How often will I have to copy the files to a new storage media? Is it good idea to have another copy of each file stored in another storage media?  Edit: If I put all of the movies and games in a HDD, if the HDD stops working, I will lose everything. This is how valuable they are for me: If I lose 10% of them, I will not cry, because I can get new ones, but if I lose 90%, it will be a disaster.  Edit 2: I do not think that that online storage is reliable even if I pay for it, because they may be closed or may get bankrupt. They are not from my country and it would be difficult to get the files back. Color of questions with accepted answers on user profile page Related to these questions:    When viewing the Meta user profile, the difference between questions with accepted answers and questions without is pretty subtle. At first glance, I couldn't tell the difference at all. Can the contrast between the styles be increased a bit to make it easier to see?     This screenshot doesn't look too bad, but when looking at a list going down a whole page, it gets a bit difficult to pick one out at a glance. (I'm guessing even more so for some of the color-challenged users) As the title say, I want to prove that  $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$  i.e.  $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n)  \quad (\star)}$. In a book I found that in order to prove it, we need the following claim:  Claim: Let $\displaystyle{\zeta \in C^{\infty} ([0, \infty)) }$ such that $\zeta (t)  =1 $ for $ t \leq 1 $ and $\displaystyle{ \zeta (t) =0 }$ for $ t \geq 2$. For $R&gt;0$ and  $x \in \mathbb R^n$ define $\displaystyle{ \eta_R (x) = \zeta \left( \frac{|x|}{R} \right)}$. If $ \displaystyle{ u \in W^{k,p} ( \mathbb R^n) }$ with $1 \leq p &lt; \infty $ then, $\displaystyle{ \eta_R u \to u }$ in $W^{k,p}(\mathbb R^n) $ as $ R \to \infty$.  If I prove the claim then the $(\star)$ follows using mollifiers. But unfortuently I have stuck, with the proof of the claim.  Any ideas? Maintenance plan for creating transaction log backups I have a database that is set to the 'Full' recovery model, and I'm creating a maintenance plan to backup the Transaction Log every 15 minutes. A full backup is then taken once every 24 hours.  I was wondering: will the maintenance plan for the transaction logs create a lot of different files every 15 minutes, or will it by default append to the same file for 24 hours?  Also: what is preferred? I couldn't find anything on this subject (or maybe I was Googling the wrong words).  Thanks for any pointers anyone can give me.  EDIT: I also see that the checkmark "Verify backup integrity" is OFF by default. Is there a risk with turning it on? It seems to be that it should be on by default, so is there a reason for not using this? How to design partitions for SSHD? How to add number of days to today's date? Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$), are true, why aren't logarithms defined for negative numbers?  Why can't $b$ be negative? Take $(-2)^3 = -8$ for example. Turning that into a logarithm, we get $3 = \log_{(-2)}{(-8)}$ which is an invalid equation as $b$ and $x$ are not positive! Why is this? How to replace all occurrences of a string in JavaScript? Aligning equations within multline I have this really long equation and I have used multline to break it up. I also want to align the broken bits of the equation to the left. When I use  align, for some reasons it aligns to the right when I want it to be on the left. How can I align to the left?  \begin{multline} \begin{align} \mathcal{H}_{q1} \mathcal{H}_{q2} \mathcal{H}_{q3}...\mathcal{H}_{qn}0=\frac{1}{2}H_{q1}\{\mathcal{H}_{q2}\mathcal{H}_{q3}...\mathcal{H}_{qn}0\}+\frac{1}{2} h_{q1} \\ =\frac{1}{2}H_{q1}\{\frac{1}{2} H_{q2}[\mathcal{H}_{q3}...\mathcal{H}_{qn} 0]+\frac{1}{2}h_{q2}\} +\frac{1}{2}h_{q1}=\\ \frac{1}{2}H_{q1} \frac{1}{2} H_{q2} [\frac{1}{2}H_{q3}(\mathcal{H}_{q4}..\mathcal{H}_{qn}0)+\frac{1}{2}h_{q3}]+ \frac{1}{2}h_{q2}+\frac{1}{2}h_{q1}=...\\          =(\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn}0+ (\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn-1}h_{q1}\\+(\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn-2}h_{qn-1}+.. \\+ (\frac{1}{2})h_{q1}=\sum^n_{j=1} (\frac{1}{2^j}) H_{qo}H_{q1} H_{q2}H_{q3}..H_{qj-1} h_{qj} \end{align} \end{multline} Do I need a visa to visit the UK with my EU partner? I am not from EU. I have Spanish partner and we live in Spain and, because of our official relationship, I have a Spanish family resident permit. We are going to travel to the UK together for tourism. I am confused whether I need an EEA family permit or not.  I will not live there and I'm going to travel with my partner. I was told that I don't need a visa, that showing our civil union document will be sufficent. Is this correct, or do I need a visa to enter the UK?    Thank you so much for your answer. Let me give you more detail.  I hold Turkish citizenship.  My resident card exactly says that Residence Card of a Family Member of a Union Citizen " tarjeta de familiar de la union.   my case is like For example, a non-EEA spouse of a German national living in Germany will usually hold a residence permit issued under German domestic law. Therefore, a United Kingdom EEA family permit is required for travel and entry to the UK , therefore I should consider that I need to EEA Family Permit even if for touristic trip with my partner. I can't play Pokemon GO! What's wrong? which of the following is NOT a possible value of $(e^{f})''(0)$?? Let $f$ be an analytic function on $\bar{𝐷} = \{z \in \mathbb{C}: |z| \le 1\}$. Assume that $|𝑓(𝑧)| ≤ 1$ for each $z\in \bar{D}$. Then, which of the following is NOT a possible value of $(e^{f})''(0)$??  $(A) 2$  $(B) 6$  $(C) \frac{7}{9}e^{\frac{1}{9}}$  $(D) √2 + 𝑖√2$  So I take a look at Cauchy Integral Formula and then $$(e^{f})''(0)=\frac{1}{\pi}\int_{|z|=1}\frac{e^{f(z)}}{z^3}$$  Taking modulus on both sides we have $$|(e^{f})''(0)| \le \frac{e}{\pi}2\pi=2e$$  Hence (b) cannot be the choice.   Is this alright!!  Thanks for the help! Joint distribution of a Normal and Truncated Normal I have a random variable $X\sim \text{Normal}(\mu,\sigma)$ and have the transformation $Y=\max\{0,X\}$.  Is the distribution of $Y$ a truncated normal where it is truncated to live on the positive half of the real line?  Secondly, is there a closed form solution for the joint distribution of a Normal random variable and a truncated Normal random variable? Regression in spacing before lstlisting environment with version 2018/09/02 after paragraph break Is a particular sequence required for detailed replicator requests?    To elaborate, is there a reason Captain Picard orders his tea, "Tea. Earl Grey. Hot." aka "Noun. Descriptor. Descriptor." and not "Descriptor. Descriptor. Noun." or "Descriptor. Noun. Descriptor." (in the event he forgot a descriptor)?  Would ordering it as "Hot. Earl Grey. Tea." or even the non-statement "Hot Earl Grey tea." produce a warm (hot), smoky (gray-ish) gas with the odor of tea?  I assume his ordering sequence is an acquired background affectation, but I'd like confirmation.  Note: I'm using "descriptor" instead of "adjective" to avoid minutia/debate about classification. A continuous mapping is determined by its values on a dense set Ubuntu system is broken after accidentally uninstalling many packages I accidentally removed all the packages starting with the name jpeg with:  sudo apt-get remove jpeg*   This screwed up a lot of things, removed Unity, Firefox, Flash player etc.  I somehow managed to recover the desktop and unity but then a lot of things are still buggy. For example .rpm files and .tag.gz just won't open up at all.  What I want is a way to fix everything that's amiss. Something that is the equivalent of a 'repair installation' with a windows disc.  Any tips, anyone? Can I travel after visa's intended travel date? I just finished applying for my visa and had originally planned to travel on the 30th of September, so I put that as my intended date of travel. But now I will not be leaving until close to the second week of October.  How does this work, will I still be able to travel with this visa even though it says I intended to arrive on September 30th?