Let $n,m\in\mathbb{N}_0$ and $\sqrt{n}\not\in\mathbb{Q}$. Conclude that $\sqrt{n}+\sqrt{m}\not\in\mathbb{Q}$   What would you recommend me to read to be able to solve this?  Because apart from the case that if $\sqrt{m}\in\mathbb{Q}$ I am kinda stuck.  If $\sqrt{m}\in\mathbb{Q}$ then by "rational+irrational=irrational" the statement holds. (still left to prove tho..)
Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}, \ \ a \in \mathbb{Q}$, which I don't think is possible, but how to show it?
Why does performance of a regular hard drive decrease over the duration of a benchmark while SSD doesn't? I use HD Tune to measure performance of the hard drive.  The tests typically take 2-3 minutes and the transfer rate of a regular hard drive decreases dramatically as the test comes to a close.    However, the SSD drive performance stays the same over (pic below) the life of the test.  This happens on all my computers.  Why is that? Is there an exact command for restarting GNOME Shell in Ubuntu 20.04 corresponding to Alt + F2 + 'r' + Enter? Is there a way to restart GNOME Shell other than by using Alt + F2, R + Enter? With respect to the question, I have gone through , but I didn't get what I was looking for. I am looking for a command which exactly does what Alt + F2, r + Enter do. Please note that I am not willing to do the gnome-shell -- replace command... I did check both the things Alt + F2,r Enter and gnome-shell --replace... There is a lot of difference, I found... Sort ArrayList of custom Objects by property Let $A,B$ be two sets. The  states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$.  I was wondering whether the following statements are true (maybe by using the AC if necessary):   Suppose $f \colon A\to B$ and $g\colon B\to A$ are both surjective, does this imply that there is a bijection between $A$ and $B$. Suppose either $f\colon A\to B$ or $g\colon  A\to B$ is surjective and the other one injective, does this imply that there is a bijection between $A$ and $B$. If $[\cdot]$ denotes greatest integer function, then what is the value of natural number $n$ satisfying the equation $$[\log_21]+[\log_22]+[\log_23]+\dots[\log_2n]=1538 ?$$   My try:  Note that  $$0+1\times2+2\times2^2+3\times2^3+4\times2^4+5\times2^5+6\times2^6+7\times2^7=1538$$  But the answer is $$n=\sum_{i=0}^72^i=255.$$  How is it derived? What are the rules for splitting words at the end of a line? What are the rules in English language to split words at the end of a line?   Where exactly must the hyphen split the word? Where are my Screenshots? I pressed F2 to take a few Minecraft screenshots. Later I couldn't find them. Where are they stored on my computer? I want to import graphics into my main input file using the macro \includegraphics. It does not work if the filename contains spaces.   also discusses this subject, but there is no solution there.  My compilation routine is latex->dvips->ps2pdf (because of PSTricks). Why should we typedef a struct so often in C? I have seen many programs consisting of structures like the one below  typedef struct  {     int i;     char k; } elem;  elem user;   Why is it needed so often? Any specific reason or applicable area? Account "last activity time" is not always updated It would appear that the account's "Last activity time" is not updated for every hit.  I've been active on SO all morning, and in the last five minutes have voted on some posts as well as posted an answer.  However,  says:     "Last activity: 2 hours ago from &lt;IP address&gt;"   Is there a certain amount of time that needs to elapse between hits before the activity time is updated again?  This issue could result in Enthusiast badges not being awarded when they should. How can I apply Newton's sums to solve this problem? How can text selection be disabled in Android 3.0 Honeycomb? I want the figures and tables to be numbered 1, 2, 3, ... until the end of the document instead of 1.1, 1.2, ... in chapter I, 2.1, 2.2, ... in chapter II and so on.  I know how to convert the "chapter" part of figure/table numbers from Roman to arabic:  \renewcommand \thefigure     {\ifnum \c@chapter&gt;\z@ \@arabic\c@chapter.\fi \@arabic\c@figure} \renewcommand \thetable     {\ifnum \c@chapter&gt;\z@ \@arabic\c@chapter.\fi \@arabic\c@table}   One could modify the above code in order to remove the "chapter" part, but the figure/table counters would still be reset to 0 at the start of new chapters. Meaning of "kernel" In analysis, there are at least three kinds of "" concepts:   In probability theory, there is a concept called , also called probability kernel, from one measure space $X$ to another $Y$. It is actually a family of measures on $Y$, indexed by members of $X$. It can transform a measure on $X$ to a measure on $Y$, in terms of integration.  A special case is  of probability density function in statistics, where the  is actually a probability kernel. In real/complex analysis, there is a concept called  from a product space $X \times Y$ to $\mathbb{R}$ or $\mathbb{C}$. It can transform some special function: $X \to \mathbb{R}$ or $\mathbb{C}$ to another one:$Y \to \mathbb{R}$ or $\mathbb{C}$. This is used for defining integral transformation In Hilbert space theory, there is concept called , which is a family of bounded mappings for a family of Hilbert spaces. For every two Hilbert spaces $X$ and $Y$ in the family, there are exactly two members of the kernel, one mapping from $X$ to $Y$ and the other from $Y$ to $X$.   I wonder    Are they related concepts, since they seem to share some indescribable similarity? Can they be unified?   In particular, the first two are similar to each other as the kernels induce transformations on measures and on functions in terms of integrals.  It is however not obvious to me how the third one, i.e. the Hilbert space one, is related to the second one. Are they related to the  concepts? Or even to the  concept?   I feel like some insights and references for catching the general ideas shared across as many kinds of "kernels" as possible.  Thanks and regards! Formula for completing the square? My math teacher said that this was the formula for completing the square.  Original function: $$ax^2 + bx + c$$ Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$  However, using this formula I'm not getting the same answers that I would get just by determining the stuff myself.  Is this correct? How to prove a matrix is nilpotent? Given an $n\times n$ upper triangular matrix $A$ with zero on main diagonal, show that $A^n = 0$.   I did some matrix operation and noticed that the diagonal moves up, ultimately all entries will be zero. Is there a nicer way to do it? $f$ be a smooth function on real line , $f(0)=0$ , $f(x)>0, \forall x \ne 0$ and any $f^{(n)}(0)=0$ ; is $\sqrt f$ smooth? Let $f: \mathbb R \to \mathbb R$ be an infinitely differentiable function such that $f(0)=0$ , $f(x)&gt;0 , \forall x \ne 0$ and   $f^{(n)}(0)=0$ ( the $n$-th derivative ) $,  \forall n \in \mathbb N$ , then is the function $g(x):=\sqrt{f(x)}$ infinitely   differentiable ? I am having difficulty checking whether any $n$-th derivative exist at $0$ . Please help . Thanks in advance $\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$ Last upgrade crashes network manager (no internet connection, no applet) While upgrading today, something caused my wireless network manager to disappear. Not even connecting via wired connection.  This error is being shown by files:  **(nm-applet:2716): WARNING **: Could not initialize NMClient /org/freedesktop/NetworkManager: The name org.freedesktop.NetworkManager was not provided by any .service files   (nm-applet:2716): nm-applet-WARNING **: Error connecting to ModemManager: Error calling StartServiceByName for org.freedesktop.ModemManager1: GDBus.Error:org.freedesktop.DBus.Error.Spawn.ExecFailed: Cannot launch daemon, file not found or permissions invalid   (nm-applet:2716): nm-applet-WARNING **: Could not find ShellVersion property on org.gnome.Shell after 5 tries   (nm-applet:2716): nm-applet-WARNING **: Failed to register as an agent: (2) The name org.freedesktop.NetworkManager was not provided by any .service files    Is there a patch I can download through windows and send through to my ubuntu, or is it a case of having to reload everything? I tried running two versions back but still no internet access. creating a csv file from two csv files I have two csv files:  sub_compiler.to_csv('sub_compiler.csv') sub_compiler.head()     and   sub_opt = pd.read_csv('sub_opt.csv') sub_opt.head()     and what I would like to do is to create a csv file where I have something of the form   compiler, opt   How could I do this? I need to do this to make a submission.  Thank's in advance.  [EDIT] Thank you for the answers. Now I have obtained :    don't understand why I have the columns called unnamed.   My objective is to get a csv file that contains only the columns compiler and opt. How could I do this? Thank's again.  [EDIT 2] I have solved my problem but if I open the csv file manually, so just click on the file, I have the following:    s it contains only 11 lines, but it contains 3000 lines in jupyter. Is it there something wrong?  [EDIT 3] I tried to to the following:  import pandas as pd  test = pd.read_csv('1495927.csv') test   and I have:    where 1495927.csv is the csv file I have created and that is in the image where I have 11 elements.