Trouble printing a multiple-line String with PrintWriter into a File
How do I get a platform-dependent new line character?
This question aims to create an &quot;&quot;  of numerous questions that ask about determinants of specific matrices (I may have missed a few):              The general question of this type is  Let $A$ be a square matrix of rank$~1$, let $I$ the identity matrix of the same size, and $\lambda$ a scalar. What is the determinant of $A+\lambda I$?  A clearly very closely related question is  What is the characteristic polynomial of a matrix $A$ of rank$~1$? Is there any intuition or meaning regarding Cauchy-Riemann equations? How do I disable 'Activities' hot corner in GNOME shell? GNOME shell has the so-called feature "hot corner", that opens the dashboard  if the mouse touches the top-left corner of a monitor screen.  I want this disabled I used to have a GNOME shell extension, yet the repository is not found anymore:  git clone git@github.com/Ahrak/gnome-shell-extension-disable-hot-corners.git fatal: repository 'git@github.com/Ahrak/gnome-shell-extension-disable-hot-corners.git' does not exist How do I set my level, unlocked weapon equip slots and playthrough mode? I usually render my animation into image sequence format, It usually takes a lot of time, so I turn off my monitor and come back later to check the rendered result then find that blender crashed before rendering all the frames.  Since rendering can be done from the command prompt I thought about writing a script (running in my OS) to automatically restart rendering from the last successful rendered frame until all frames are finished*.  But I believe this problem is facing a lot of people out there, is there a way to deal with it? instead of reinventing the wheel?     for example if the total render time is 8 hours I don't want come back to check my computer every 20 minutes (some times this is not available I'm asleep or away from my PC) Prove $4(x + y + z)^3 > 27(x^2y + y^2z + z^2x)$ Prove that, for all positive real numbers $x$, $y$ and $z$, $$4(x + y + z)^3 &gt; 27(x^2y + y^2z + z^2x)$$  I've tried expressing it as a sum of squares, but haven't got anywhere.   Hints are also welcome. Why is the time limit for editing comments 5 minutes? When do I apply Significant figures in physics calculations? I'm a little confused as to when to use  for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.  Would I need to apply the rules of significant figures to this step of the problem? $$ C = 2\pi (1000~\mathrm{m}) = 6283.19 $$  Or do I just need to apply significant figures at this step? $$ \text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s} $$  Should I round $125.664~\mathrm{m}/\mathrm{s}$  to  $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two? Reputation misaligned in new topbar Sometimes I wonder if he is actually/really  as lavish as he   pretends to be.   I know that this sentence can work without either actually or really, but if we were to use one of them to make an impact on the listener, which one should it be? How did Walter White poison that character? I've seen Breaking Bad complete series 2-3 times, but I never found out how Walter White was able to poison Brock.  I mean did he himself directly feed the Valley of the Lilly berries to Brock or did he let Saul do the work? I can't figure it out. How to create a database directory structure for MySQL? How do I compare 2 rows from the same table (SQL Server)? I need to create a background job that processes a table looking for rows matching on a particular id with different statuses. It will store the row data in a string to compare the data against a row with a matching id. I know the syntax to get the row data, but I have never tried comparing 2 rows from the same table before. How is it done? Would I need to use variables to store the data from each? Or some other way? (Using SQL Server 2008) Find children of the process What is a word for someone without allegiance? Matrix doesn't shrink when put in fraction. Please enable low-rep users to easily breakout up and down votes on their own questions I know once you get a certain amount of reputation, you can see the total up and down votes for questions. Well, my proposal is to always be able to see this on your own questions and answers. You can see it by going to your user page anyway, so why not make it possible to do by simply clicking on the number of votes (like you can when you have the rep)? How do I grep for all non-ASCII characters? I've been reading forum posts and blogs who mention "my guy" syndrome as a specific type of difficult player, but I can't seem to find a solid definition for the term. Can someone explain this particular kind of problem player to me? How do you deal with this kind of problem? Isomorphism between $\Bbb R$ and $\Bbb R(X)$? My questions are:     $1.$ Is there a field morphism $\Bbb R(X) \hookrightarrow \Bbb R$ ?      $2.$ If the answer to $1.$ is "yes", are $\Bbb R$ and $\Bbb R(X)$ isomorphic as fields?   $ $    $ $  For $1.$, here are some comments:   This is true for $\Bbb C$, see . The key point here is the fact that $\Bbb C$ is algebraically closed, of cardinality $2^{\aleph_0}$, and of characteristic $0$. The same argument doesn't work for $\Bbb R$. We can actually find a subfield $F$ of $\Bbb R$ such that $F \cong F(X) \cong F(X,Y) \cong \cdots$, see .  that $\Bbb Q(X)$ does not embed in $\Bbb Q$ (as a field).  This is equivalent to: Is there an injective ring morphism $\Bbb R[X] \hookrightarrow \Bbb R$ ? (By taking the fraction fields).   As for 2. :   They are isomorphic as abelian groups, and actually as $\Bbb Q$-vector spaces. They are not isomorphic as $\Bbb R$-algebras, because they haven't the same transcendence degree over $\Bbb R$. Let's try to see how to construct such a field isomorphism. We could try$^{[1]}$ to find a subset $B$ of $\Bbb R$ such that $\Bbb R = \Bbb Q(B)$ and $C \subsetneq B \implies \Bbb R \neq \Bbb Q(C)$.  The set $B$ must contain transcendental elements, pick such an $x_0 \in B$.  Then $\Bbb R = \Bbb Q(B) = \Bbb Q(B \setminus \{x_0\})(x_0)$ and we could try$^{[2]}$ to find a field isomorphism $\Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$, so that $\Bbb Q(B \setminus \{x_0\})(x_0) \cong \Bbb Q(B)(x_0) \cong \Bbb R(X)$.  For ${[1]}$, I tried to use Zorn's lemma on $\mathscr B = \{B \subset \Bbb R \mid \Bbb R = \Bbb Q(B)\}$ and the partial order $B_1 ≤ B_2 \iff B_1 \supset B_2$. But this is not clear to me why this should be an inductive set.  For ${[2]}$, my idea was to define $f : \Bbb Q(B \setminus \{x_0\}) \cong \Bbb Q(B)$ by choosing a bijection $b : B \setminus \{x_0\} \to B$ (these sets should be uncountable), and then trying to extend $f\vert_{B \setminus \{x_0\}} = b$ to a field morphism.     Edit for $^{[1]}$: in  article (MINIMAL GENERATING SETS OF GROUPS, RINGS, AND FIELDS, by LORENZ HALBEISEN, MARTIN HAMILTON, AND PAVEL RUZICKA) the theorem 2.4. proves that $\Bbb R$ has no minimal generating set as field (or $\Bbb Q$-algebra, i.e. as ring). The argument uses Artin-Schreier theorem. Minimal generating sets of modules are   .  I think that $B$ is a minimal generating set of a subfield $K \subset \Bbb R$ over $\Bbb Q$ if and only if any element of $B$ cannot be written as a rational fraction of other elements of $B$. Under these conditions and using (transfinite) induction, we can probably prove (and this was my first idea) that any function $f: B \to \Bbb C$ such that $b$ and $f(b)$ are conjugate over $\Bbb Q$ (i.e. have the same minimal polynomial) extends to a (unique) field morphism $\sigma : K = \Bbb Q(B) \to \Bbb C$.  Actually, the collection $\mathscr B$ defined above doesn't have to be an inductive set, at least if we replace $\Bbb R$ by $K =\Bbb Q(\sqrt 2, \sqrt[4]{2},\sqrt[8]{2},\dots)$, because the descending chain $\left(E_n = \{\sqrt[2^m]{2} \mid m \geq n\}\right)_{n \geq 1}$ satisfy $K = \Bbb Q(E_n)$, but $K \neq \Bbb Q\left(\bigcap_{n≥1} E_n\right)=\Bbb Q$, so the chain has no upper bound w.r.t. $≤$ in $\mathscr B$.