How to prove that path of a charged particle in a uniform magnetic field is circular or helical?
Helical motion of charged particle in external magnetic field
Command-line to switch between profiles in gnome-terminal I created a new profile for gnome-terminal and I can switch between &quot;Implicit&quot; profile and the new created profile as you can see in the below image:  Now I wonder how can I switch between profiles using a command-line/script or maybe using a shortcut for this.  Note: Solutions like: gnome-terminal --tab-with-profile=Implicit &amp;&amp; exit   are excluded because I prefer not to close and open another terminal or get another terminal window in this process of changing the profile. Does this seemingly elementary question require König's theorem? How can I get a significant overall ANOVA but no significant pairwise differences with Tukey's procedure? I performed with R an ANOVA and I got significant differences. However when checking which pairs were significantly different using the Tukey's procedure I did not get any of them. How can this be possible?  Here is the code:  fit5_snow&lt;- lm(Response ~ Stimulus, data=audio_snow) anova(fit5_snow)  &gt; anova(fit5_snow) Analysis of Variance Table  Response: Response           Df Sum Sq Mean Sq F value  Pr(&gt;F)   Stimulus   5  73.79 14.7578  2.6308 0.02929 * Residuals 84 471.20  5.6095                   --- Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1   df&lt;-df.residual(fit5_snow) MSerror&lt;-deviance(fit5_snow)/df  comparison &lt;-  HSD.test(audio_snow$Response, audio_snow$Stimulus, df, MSerror, group=FALSE)  &gt; comparison &lt;-  HSD.test(audio_snow$Response, audio_snow$Stimulus, df, MSerror, group=FALSE)  Study:  HSD Test for audio_snow$Response   Mean Square Error:  5.609524   audio_snow$Stimulus,  means                  audio_snow.Response   std.err replication snow_dry_leaves            4.933333 0.6208034          15 snow_gravel                6.866667 0.5679258          15 snow_metal                 6.333333 0.5662463          15 snow_sand                  6.733333 0.5114561          15 snow_snow                  7.333333 0.5989409          15 snow_wood                  5.066667 0.7713110          15  alpha: 0.05 ; Df Error: 84  Critical Value of Studentized Range: 4.124617   Comparison between treatments means                                Difference   pvalue sig        LCL      UCL snow_gravel - snow_dry_leaves  1.9333333 0.232848     -0.5889913 4.455658 snow_metal - snow_dry_leaves   1.4000000 0.588616     -1.1223246 3.922325 snow_sand - snow_dry_leaves    1.8000000 0.307012     -0.7223246 4.322325 snow_snow - snow_dry_leaves    2.4000000 0.071587   . -0.1223246 4.922325 snow_wood - snow_dry_leaves    0.1333333 0.999987     -2.3889913 2.655658 snow_gravel - snow_metal       0.5333333 0.989528     -1.9889913 3.055658 snow_gravel - snow_sand        0.1333333 0.999987     -2.3889913 2.655658 snow_snow - snow_gravel        0.4666667 0.994348     -2.0556579 2.988991 snow_gravel - snow_wood        1.8000000 0.307012     -0.7223246 4.322325 snow_sand - snow_metal         0.4000000 0.997266     -2.1223246 2.922325 snow_snow - snow_metal         1.0000000 0.855987     -1.5223246 3.522325 snow_metal - snow_wood         1.2666667 0.687424     -1.2556579 3.788991 snow_snow - snow_sand          0.6000000 0.982179     -1.9223246 3.122325 snow_sand - snow_wood          1.6666667 0.393171     -0.8556579 4.188991 snow_snow - snow_wood          2.2666667 0.103505     -0.2556579 4.788991 prove that a connected graph with $n$ vertices has at least $n-1$ edges Show that every connected graph with $n$ vertices has at least $n − 1$ edges.  How can I prove this?  Conceptually, I understand that the following graph has 3 vertices, and two edges:  a-----b-----c  with $a$, $b$ and $c$ being vertices, and $\{a,b\}$, $\{b,c\}$ being edges.  Is there some way to prove this logically?  --UPDATE--  Does this look correct?  Any advice on how to improve this proof would be appreciated.  Thank you. Rainbow Six Siege Insane Load times? Computer or Internet? I have a high-end computer with decent ethernet yet whenever I join a game in Siege everyone finishes loading and it takes me 1-5 minutes. Is this because of my internet or computer?   Specs:  CPU - i7-7700k GPU - GTX 1080 16GG RAM 5900 RPM 4TB Hard drive Define the content of a polynomial (over an arbitrary commutative ring $A$) to be the ideal generated by its coefficients, denoted $c(f)$. I want to show that   $$c(fg) = c(f)c(g).$$      (I was told this is true.)    What I was able to show was that $c(fg) \subseteq c(f)c(g)$ (this is obvious), and that their radicals are the same. My reasoning for the latter was as follows: let $f = a_0 + \dotsb + a_nx^n,$ $g = b_0 + \cdots + a_mx^m,$ and consider the matrix  $$ \begin{pmatrix}  a_0 b_0 &amp; \cdots &amp; a_0b_m\\ \vdots &amp; \ddots &amp; \vdots\\ a_nb_0 &amp; \cdots &amp; a_nb_m \end{pmatrix} .$$  Then $c(f)c(g)$ is generated by the entries of this matrix, while $c(fg)$ is generated by the sums along the diagonals. Let $\mathfrak p$ be a prime ideal of $A$ not containing $c(f)$ or $c(g)$, and let $i,\,j$ be minimal such that $a_i,\, b_j \notin \mathfrak p$.  Then all the terms in the generator of $c(fg)$ corresponding to the coefficient of $x^{i+j}$ are in $\mathfrak p$ except for $a_ib_j$, so that $c(fg) \not\subset\mathfrak p$. It follows that $c(fg) \subseteq \mathfrak p \Longleftrightarrow c(f)c(g) \subseteq\mathfrak p$, and hence $\sqrt{c(fg)} = \sqrt{c(f)c(g)}$.     Can I go all the way and show that in fact $c(fg) = c(f) c(g)$? Can we get by as English speakers visiting Iceland? We're going to Iceland for the first time in September. Will we be able to get around ok just speaking English, or should we try to learn some rudimentary Icelandic? And is that even what their language is called? Are the decimal places in a CSS width respected? A question on limsup Defaults to true.  If set to true replaces all instances of known swear words with replacements (comical or * based). If false renders the content as originally presented.  Benefits:    Reduces moderation requirements necessary to ensure that people behind corporate firewalls are not blocked by an errent swear word.   Stops idiots like me complaining about it. How can I change the margins for only part of the text? Sometimes the margins need to be changed for a particular page, paragraph, or other section of text.  For example, if I'm writing a letter and want the left margin to be almost at the right side of the page for the four lines of my address, or if I am combining prose with poetry and want the poems to be indented relative to the rest of the work.  What is the best way to change both left and right margins on the fly? Is it true that $0.999999999\ldots=1$? Title pretty much sums it up.  The external style sheet has the following code:  td.EvenRow a {   display: none !important; }   I have tried using:  element.style.display = "inline";   and  element.style.display = "inline !important";   but neither works. Is it possible to override an !important style using javascript.  This is for a greasemonkey extension, if that makes a difference. Problem with Refsection Invalid SSL certificate when pushing to Git server Our GM won't let us affect the story; the more I play the less fun I have. What do I do? I was wondering if anyone could give me some advice or anything on a GM situation.   My current GM is running a campaign and it is very video game/anime like. It started out super fun and the deeper we get the less fun I have. The reason is, it feels like GM is telling us a story but we cannot interact or change anything during his, "cut scenes". No matter what any player does he just says, "No it didn't work," or, "Yeah you do that. See if you roll a 20." We also can't damage any boss he has until we reach the point in the story that he has planned, so our interactions we have with villains don't matter.  I feel like he is making us sit there and listen to an anime he has come up with in his head. Since the players can't change or react to the key situations. I find my self sitting there thinking, why would I react, it doesn't matter.  It's not just me either. Another player feels the same way. I am going to talk to my other fellow players soon about it and see how they feel.  Side note: the GM as a person is very defensive and I am afraid to confront him about it because I know he will throw a fit. We have also caught him straight lying about abilities and stats on his own PC in other campaigns, and he doesn't take to kindly when you call him out.  He is new to GMing. This is only the second game I have played with him as a GM. The first one had one session the he just never wrote more for.  Is there anything I can do to maybe get him to more lenient with his story? I really enjoy the story for the most part and if he would just let our characters do things and give us a chance at stopping his villians or letting us change the story based on our actions, everything would be cool. How to convert light activated switch Comment hint not disappearing when I start to type comment If I click "add comment", the comment hint (placeholder; "Request clarification or add information.") is not a hint, but editable text. If I delete the text and click outside of the comment then the hint appears as it should and when I click back into the comment box the hint disappears as it should.   I am on Internet Explorer 10.0.7.  Am I just oblivious and this is how it has always been (having to delete the comment hint), or is this a bug? Random integer in VB.NET I need to generate a random integer between 1 and n (where n is a positive whole number) to use for a unit test. I don't need something overly complicated to ensure true randomness - just an old-fashioned random number.  How would I do that? Python 3 with QGIS on Mac? I need to start learning Python, and I want to learn with version 3.x. I understand that QGIS--at least some plug-ins--relies on Python and Tcl/Tk, and probably on specific versions. Assuming that QGIS is dependent on Python 2, can anyone tell me how to set up parallel installations of Python 3 and Tcl/Tk such that QGIS will have what it needs and I can use Python 3 for other purposes?