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A formula for the roots of a solvable polynomial Let $F$ be a field and $p(x)\in F[x]$ a separable polynomial, denote $K$ as the splitting field of $p$ and assume that $K/F$ is Galois with a solvable Galois group.  I don't understand if this imply of any formula (in radicals) for the roots of $p$ (however, I do understand how a formula would imply that $p$ is solvable by roots).  Is there some kind of a way to obtain the roots of a solvable polynomial ? Antisymmetric Relations Save Illustrator as PDF with transparent background In Java, how do I convert a byte array to a string of hex digits while keeping leading zeros? In Blender 2.8, how do I light an EEVEE scene with an HDR? Phase Potentials Looking to get a explanation of the following solution.   $x'=sinx$ I understand that you must integrate $sinx$ and then take the negative of it making it $v(x)=cos(x) + c = cos(x)$ with $c=0$. However I am having problems understanding the equilibrium points.   The equilibrium points of this system are,  $$x^*=kz$$ $x*=kz$ k is even, this is unstable $x^*=kz$ k is odd, are stable  Why is it unstable when even and stable when even? I am Peruvian, and will study in the Turkish Republic of Northern Cyprus for two years.  As a South American, what problems may I have if I decide to study in the TRNC, as it is recognized only by Turkey?  What if, having been in the TRNC, I decide to travel to the EU (Amsterdam) for summer vacations? Would I encounter any problem entering The Netherlands, a member of EU, after having been in the TRNC?  I have read that many tourists (Russians and Asians for now) are deported mostly because they go from Northern Cyprus to the Republic of Cyprus. Cyprus, a member of the EU, believes that those entering the country from the north are illegal immigrants. Does this rule apply to Cypriot internal issues only, or does it extend throughout the EU? Will it cause problems for my intended travel to Amsterdam? Unparsed HTML shows on search result page When you search on any of the SE network sites, with a tag (so that a description is shown); it shows the tag info content box, with the highlight-search-term span elements.  For eg. I searched ; ; and the results are as shown:                  Notice the &lt;span class='search-highlight'&gt; boxes around searched tag terms. When you pass a collection like list, array to another function in python, does it make a copy of it, or is it just a pointer? Closure of the set of degree $n$ polynomials with $n$ real roots A very beautiful (but maybe classic) exercise:   Consider the set $\Omega_n$ of polynomials of degree $n$ with real coefficients and $n$ distinct real roots.   What is the closure of $\Omega_n$ in $R_n[x]$?  ($R_n[x]$ denotes the set of polynomials with real coefficients of degree at most $n$)    The answer is the set of polynomials in $R_n[x]$ all of whose roots are real, plus the constant polynomials. Showing that this set is contained in the closure is not too hard, however the reverse implication is more difficult and I'm wondering what is the most elementary method to prove it (without talking about holomorphic functions for example). Switch statement fallthrough in C#? Problem with long chapter title and arsclassica $(S^1 \times [0,1])/\sim$ homeomorphic to unit disk $ D^2$ Let $\sim$ be the equivalence relation on $S_1 \times I$ given by $(x,t)\sim(y,s)$ if and only if $xt=ys$ where $I=[0,1]$  How do I prove that $(S_1 \times I)/\sim$ is homeomorphic to the unit disk $D^2$ with the induced topology? Finding files by only knowing extension? Solving a six-degree polynomial of the form $ax^6+bx^3+g$. I read that it is not always possible to solve but from Wikipedia:      Some sixth degree equations, such as $ax^6 + dx^3 + g = 0$, can be solved   by factorizing into radicals, but other sextics cannot.   So how do you solve this? I've stumbled upon this equation where; $a = n^2$, $d=m^2n-2p^2$, and $g=m^4$. Where the $a,d,g$ corresponds to the above mentioned sextic-equation and $m,n,p$ are input variables. I apologize for that ton of variables in use, might cause some confusion. Is this possible using Galois Theory or by any other means? Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$ Grid search vs. random search for hyperparameter tuning I am interested in comparing the computational cost of grid search vs. random search algorithm for hyperparameter tuning. Is random search computationally less expensive then grid search? Is there an article/paper that talks about this topic? Thank you, Movie about gene therapy and genetics-based jobs I saw a movie several years ago and I have not been able to find it since.  It was about a dystopian society where parents could modify the genes of their children before birth.  Some individuals, including the protagonist, had not had this done, and so had a somewhat inferior genetic makeup.  Jobs are based on the results of genetic tests, and the protagonist, a male, was slated for a less than ideal job. In the movie, he fakes his identity and has major surgery to make him a little taller to match his ID. He is trying to get his dream job (astronaut?).  The only other parts I remember are   the protagonist, to keep up the ruse, managing to run on a treadmill faster than his  genetic limit a very suspicious character pulling the protagonist's DNA from a hair left in a comb Can you make an opportunity attack against a moving object? Why do employers require you to spread your 401(k) contributions throughout the year to get the maximum match? I think the title says it all. I can't see any reason employers have to incentivize employees to contribute to their 401(k) evenly throughout the year. Instead couldn't they use a relatively simple formula that is based on how much of their income the employee has contributed to their 401(k) thus far for the year? If that or their current contribution is over the threshold, they get the match this pay period, even if they didn't contribute anything.  EDIT: To clarify, my employer matches 100% of the first 3% of my contribution and 50% of the next 2%. The match is added every pay period along with my contribution. I am pretty sure there is no "true-up" feature, so if I were to hit the $17.5K limit early, and not be able to contribute at least 5% of my salary, I would not get the full match. So I have to be careful to make my contributions evenly throughout the year to get the maximum match. Just trying to figure out my employer's motivation for requiring this.