How to prove $X^{\omega}$ with a product toplogy is metrizable with $\rho(x,y) =\sum_{n=1}^{\infty}2^{-n}\frac{d(x_n,y_n)}{1+d(x_n,y_n)}$?
Show that the countable product of metric spaces is metrizable
What is the difference between user and service account? I would like to know difference between user and service account.  I know that e.g. Jenkins installed to ubuntu is not a user, .    What is use of service account? When we need them? How can I create service account? How to find out who deleted some data SQL Server My boss had a query from a customer yesterday asking how they could find out who deleted some data in their SQL Server database (it is the express edition if that matters).  I thought this could be found from the transaction log (providing it hadn't been truncated) - is this correct? And if so how do you actually go about finding this information out? The question is (from Topology without tears) that: Let $(X,d)$ be a metric space and $\tau$ the corresponding topology on $X$. Fix $a \in X$. Prove that the map $f:(X,\tau) \rightarrow \mathbb{R}$ defined by $f(x) = d(a,x)$ is continuous.  My first attempt was that let an open ball, $B_{\epsilon}(a)\subset \mathbb{R}$ and show that $f^{-1}(B_{\epsilon}(a)) \subset \tau$. Since $f(x) = d(a,x)$, $f^{-1}(B_{\epsilon}(a))=B_{\epsilon}(a) \in \tau$.   I am pretty sure the last sentence is wrong. I am not sure if every open ball is an open set in topology induced by metric space.  Please give me some hint and direction. Thanks I know the minimum number of reference letters required at many U.S. PhD programs in the STEM fields is three -- and that it is also not uncommon for someone to submit a fourth letter.  What about a fifth letter?  Would a fifth letter of recommendation be perceived as "too many"? Books for a beginner I am a student of 11th grade and i have completed the syallabus of both 11th and 12th grade maths with complete understanding and it was possible coz of the love for this subject that i have. I don't want to sound "larger than life" here, but my curiousity has now increased to learn maths at a deeper level. As of now i am reading "Linear algebra done right" by Axler. Thats an amazing piece of work and now i truly understand what "matrix" is all about. I also want to extend my knowledge on the following topics :  1) Algebra (groups,subgroups,homomorphisms etc)   2) Analysis   3) Geometry/Topology  Why i am here is because i wanted you guys to recommend a book on each of the above topic that would be appropriate for a beginner like me (I must mention here I have been working on Apostol's volume 1 calculus, thats a great text, but for a beginner, its best if the concept is explained in a broad manner and in as simple and easy words as possible) . I just want a book which explains the concept broadly rather than coming to the conclusion directly (which is not a great sight for a beginner like me) .   Having said that, no book is complete i understand. Thats why i am asking your recommendation as you guys are aware which text would be the best to start with for a beginner. I have searched on the net about this, but there are dozens of works available, and out of them all , i want the one which is the best (approximation) for a beginner .   Please don't misunderstand me in any way. I am just confused which text to go for, and i understand the books which u may suggest may be for undergraduate level course as these topics are for undergraduate level courses, thats not a problem at all . I need a easy to learn (i mean easy in terms of "broad" explanation) book. I hope you guys don't mind me questioning such a question on this forum . Thanks for all the help you guys have been providing me on this forum . Maths Maths Maths... the world is beautiful coz of u! :)) When using find, how do I return the file name and the line number when searching for a string? I manage to return the file name in one command and the line numbers with another one, but I can't seem to combine them.  File names: find . -type f -exec grep -l 'string to search' {} \;  Line numbers: find . -type f -exec grep -n 'string to search' {} \; I'm talking about at least two juvenile sci-fi novels, read between second and fourth  public grades (U.S.A.) which for me places them around 1956-9.  The protagonists are a brother and sister, presumably my (then) age. They are friends  with a single man, a scientist who has built a rocket ship. The scientist is little fey,  physically; a (relatively) large, round head, balding, short in stature and striking long,  thin hands and fingers.  The rocket ship's final preparation is to slop a viscous, clear gloop over EVERYTHING, inside and out from a pail with a paintbrush. In flight, there is an encounter with the moon, and a very small planet with (non-threatening) inhabitants not physically unlike the fey rocket scientist. The planet orbits either the moon or the earth, not the sun, and strangely, some kind of mushroom(!) theme runs through the books. Really.  That's it. That's all I got (I was nine at the most, fer chrissake). My recollection tells me there was concealment from parents, a particular name for the planet and people  and flight through a spatial fog. I mention these things separately because I am less sure  of them than the previous two paragraphs. The author was blissfully unrepentant of the laws  of physics:  Earth gravity on a micro-planet, fog in space and super-gloop to ready the  ricketiest rocket. Worked for me.  I'm guessing even Richard, my previous mentor/savior might not be familiar with these.  Anybody? Author, titles? Or do you just recognize one of these? Does $\int_{0}^{\infty}\frac{dx}{1+(x\sin5x)^2}$ converge? Transform the NFSA with transition table Transform the NFSA with transition table  $$\begin{array}{ccc} X&amp;a&amp;b\\ A&amp;\{\}&amp;\{F\}\\ B&amp;\{\}&amp;\{\}\\ F&amp;\{\}&amp;\{A,S\}\\ S&amp;\{F\}&amp;\{A,F\} \end{array}$$  and starting state $S$ inta a DFSA and give the transition table of that DFSA. Place the letter $X$ (uppercase) in the top left corner of the matrix. Your states in the first column and inputs in the first row can be in the order of your choice.  The question then offers a $7\times 3$ matrix to be filled in.  Unsure how to transform the NFSA with transition table.  This is my answer but it is incorrect.  $$\begin{array}{ccc} X&amp;a&amp;b\\ \{S\}&amp;\{F\}&amp;\{A,F\}\\ \{F\}&amp;\{\}&amp;\{A,S\}\\ \{A\}&amp;\{\}&amp;\{F\}\\ \{A,F\}&amp;\{\}&amp;\{A,F,S\}\\ \{A,S\}&amp;\{F\}&amp;\{A,F\}\\ \{A,F,S\}&amp;\{F\}&amp;\{A,F,S\} \end{array}$$ top output: cpu usage > 100% Find the number of zeroes (counting multiplicity) of $p(z)=z^6+z^3+10z^2+4z+3$ inside the annulus $1&lt;|z|&lt;2$. I think this can be solved using Rouché’s theorem. First consider $|z|\leq 1$ on the boundary $|z|=1$ and $|z^6+z^3|&lt;|10z^2+4z+3|$ which can be shown using triangle inequalities, the left side is at most $2$ while right side is at least $3$. We could also consider $|z|\leq 1+ \epsilon$ and our proof still works. Thus $p(z)$ has 2 roots inside $|z|\leq 1$. Also for $|z|= 2$ we have $|z^6+z^3|&gt;|10z^2+4z+3|$. Thus we have 6 zeroes of $p(z)$ inside $|z|&lt;2$ and therefore 4 of them are in $1&lt;|z|&lt;2$. Is this correct? The only &quot;tricky&quot; part is ruling out that there are zeros on $|z|=1$. I am confused whether drift velocity depends upon area or not . I checked answers for this question but all of them were cancelling each other out . from the equation $$I = neAv$$ It is clear that drift velocity should depend upon area but at the same time don't you think that by increasing the area of cross section of a conductor should have resulted in increase in drift velocity please help me out with it in detailed answer of how should i go with all the possible cases and explanation ,thanks :) How to fade one material into the other? Hello blender community.  Could someone please let me know how to fix this issue? I have one material for the whole face and a different one for the lips.  there's a noticeable edge that I want to get rid of. I just want it to have an ease fade instead of a sharp edge.   what's the best way to go about this?  Thank you in advance!    Finished Update  I wanted to give an update on how it was fixed with @hexbob6 and @Lemon 's help.    I needed two textures. One is the actual colored version and the 2nd is the black and white masked version. Hexbob6 described very well on the process here:   A: The node process on how to do it.    The B/W masked texture is going to be connected to the Fac of the mixed shader along with its UV coordinates. Make sure the image is on clip instead of repeat. the 1st shader is going to be the black part of the B/W masked texture. An important thing to note is if you have a texture already for the object, be sure to connect the other UV coordinates! the 2nd shader will be the white part of the B/W masked texture.   B: This section is the UV layout called "mainUV" where all the color goes.  C: This section is the B/W masked texture to focus on the lips.  D: Just the render view  E: Best way to texture something is to be able to see it in 3D view and texture paint.  F: Just showing that there are two textures to keep track of. One is the main UV and the 2nd is to focus on the lip.   That was it. Once again thanks to those who replied and helped me figure this problem. In the process I learned other things that I didn't know about blender even though I've been using it for years. How to do a secure erase on a Macbook Pro? I have a Macbook Pro with SSD and I would like to completely erase every single bits of data on it. How would I do that? On a PC, I would just run Darik's Boot And Nuke, but can I run it on a Mac too, or is there a better alternative? ANOVA analysis with only mean and standard deviation Two of my friends are getting married. They're both huge Doctor Who geeks, and actually met through a DW RPG. Their wedding invitation contained a very subtle DW reference (you wouldn't have been able to tell unless you were very familiar with the show yourself.)   Doctor Who's been running long enough that there are quite a few nice quotes for basically every occasion in life. I was hoping someone would know a nice wedding-related one, or maybe one about love, that I could include in a wedding card? Can I answer my own questions, even if I knew the answer before asking? Windows Unable to Delete ._. File What is the "Wanna Cry" ransomware's possible impact on Linux users? It's just come to light that there's a $300 ransom you have to pay because ransomware targeting Microsoft Windows has encrypted your data. What steps do Linux users need to protect from this if for example they are using wine?  This ransomware is widely reported to be based on a tool developed by the NSA to hack into computers. The NSA tool was used by a hacker group called the Shadow Brokers. The code can be found in .  Microsoft released a patch (MS17-010) against this vulnerability on March 14, 2017. The mass infection is reported to have begun spreading on April 14th. This is discussed .  As I haven't booted Windows 8.1 in 6 to 8 weeks, can I apply this patch from Ubuntu without booting Windows first? (After research it may be possible ClamAV could report the vulnerability from the Linux side looking into Windows partition but it's unlikely it could apply the patch. The best method would be to reboot into Windows and apply patch MS17-010.)  Individuals and small companies who subscribe to Microsoft Automatic Updates are uninfected. Larger organizations who delay apply patches as they are tested against organization intranets are more likely to be infected.  On May 13, 2017, Microsoft took the extraordinary step of releasing a patch for Windows XP which has been unsupported for 3 years.  No word if wine is doing anything about a security update. It was reported in a comment below that Linux can be infected too when users run .  An  registered a domain name that acted as a kill-switch to the ransomware. I presume the non-existent domain was used by the hackers on their private intranet so they didn't infect themselves.  Next time they will be smarter so don't rely on this current kill-switch. Installing the Microsoft patch, which prevents exploiting a vulnerability in the SMBv1 protocol, is the best method.  On May 14, 2017 Red Hat Linux said they are not affected by "Wanna Cry" ransomware. This might mislead Ubuntu users along with Red Hat, CentOS, ArchLinux and Fedora users. Red Hat supports wine which answers below confirm can be effected. In essence Ubuntu and other Linux distro users googling this issue might be mislead by the Red Hat Linux Support answer .  May 15, 2017 Update. Over the last 48 hours Microsoft released patches called KB4012598 for  to protect against "Wanna Cry" ransomware. These Windows versions are no longer on automatic updates. Although I applied security update MS17-010 on my Windows 8.1 platform yesterday, my old Vista Laptop still needs patch KB4012598 downloaded and manually applied.       Moderator note: This question is not off topic - it asks about whether or not any Linux users need to do any steps for protecting against the risk.      It is perfectly on topic here, because it's relevant to Linux (which Ubuntu is), and it's also relevant for Ubuntu users running Wine or similar compatibility layers, or even VMs on their Ubuntu Linux machines. How to represent kWh usage by year against average temperature?