I am solving the following : Vector space  $\mathbb{R}$ over rational number   $\mathbb{Q}$ is infinite dimensional.  I proved this by using that $\mathbb{R}$  is uncountable, but my professor suggested a following slightly different method : Suppose that this vector space is finite-dimensional and let {$r_1,...,r_n$} be a basis for it. Then find an element in $\mathbb{R}$ which doesn't belong to $\operatorname{Span}${$r_1,...,r_n$}. I've tried several methods, but it vain. How can I prove this?
It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
14 months ago, an  supplement was published that discussed how the ranger described in the Player's Handbook was under powered. These rules reimagined the ranger's abilities, and suggested that an official version would be released later.     Finally, we come to implementation. If this iteration of the ranger, or a future revision of it, grades high enough, our plan is to present it as a revised ranger in a future D&amp;D sourcebook.   Does that Sourcebook exist yet? If not, have the devs given an update or told us their intentions? I need to add a coloured line before my caption; this is what I used:  \usepackage[labelfont=bf]{caption}  \DeclareCaptionFormat{myformat}{\hrulefill #1  #2 #3} \captionsetup[figure]{format=myformat}   but the figure turns out to be line this:    and not a horinzontal line which what I want  2) I want the line to be coloured RGB [152 204 204] with thickness about the same as \top rule. Can anyone show me how to do this? How do I handle collision detection so fast objects are not allowed to pass through walls? I'm creating a 2d sidescrolling shooter, and I'm having a little trouble with collision detection for the bullets. Everything including the bullets are objects with their own polygons / update methods.  The trouble is that bullets go fast, and at 60 frames per second (what the game runs at) a bullet will often skip right through a wall -- since it moves more than the width of the wall during in the update interval -- and continue happily on its way since the polygons will never actually overlap.  What can I do about this? The only thing I've been able to come up with is to draw a line from the old position to the new position and do collision detection on that, but line drawing for collision detection is recommended against by the slick2d documentation. How can I solve this? I was given a laptop with a totally dead display. I do not want to pay to replace the broken display, but I would like to use the laptop as a server and connect to it via SSH to control it.  I have pulled out the internal hard drive from the broken laptop, and I want to install Debian GNU/Linux and the SSH server onto this HDD using a working Debian laptop.  However, the hardware in the two laptops is very different, and I am wondering what kind of problems I would run into when I plugged the drive back in to the laptop with the broken display because of this?  How would I make this work? What is the proper way for me to install a bootable copy of Linux onto a HDD that will be used by a computer with different hardware? Emboss text onto curved surface I am fairly new to blender and I am trying to do something which I am assuming is fairly straight forward with no success. All I'm looking to do is emboss some text onto the curved part of a yogurt pot. Image attached below. What is the quickest and easiest way of doing this?  Thanks in advance If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$. Was revising for a Linear Algebra when I came across this question.     "Given a Matrix $A$ over $\mathbb{R}^{n \times n}$, and $A^2 = 0$,   show that rank($A$) $\leq n/2$"   My attempt:  I was thinking of the theorem from Cayley-Hamilton but the matrix is a $n \times n$ matrix hence that's probably invalid.  Is there perhaps a significance of $A^2 = 0$? apart from a special case of the nilpotent matrix? Is there any way to check whether a file is locked without using a try/catch block?    Right now, the only way I know of is to just open the file and catch any System.IO.IOException. Java date format conversion - getting wrong month High p-values for logistic regression variable that perfectly separates? I'm using R to run some logistic regression. My variables were continuous, but I used cut to bucket the data. Some particular buckets for these variables always result in dependent variable being equal to 1. As expcted, the coefficient estimate for this bucket is very high, but the p-value is also high. There are about ~90 observations in either these buckets, and around 800 total observations, so I don't think it's a problem of sample size. Also, this variable should not be related to other variables, which would naturally reduce their p-values.  Are there any other plausible explanations for the high p-value?  Example:  myData &lt;- read.csv("application.csv", header = TRUE) myData$FICO &lt;- cut(myData$FICO, c(0, 660, 680, 700, 720, 740, 780, Inf), right = FALSE) myData$CLTV &lt;- cut(myData$CLTV, c(0, 70, 80, 90, 95, 100, 125, Inf), right = FALSE) fit &lt;- glm(Denied ~ CLTV + FICO, data = myData, family=binomial())   Results are something like this:  Deviance Residuals:       Min        1Q    Median        3Q       Max   -1.53831  -0.77944  -0.62487   0.00027   2.09771    Coefficients:                 Estimate Std. Error z value Pr(&gt;|z|)     (Intercept)     -1.33630    0.23250  -5.747 9.06e-09 *** CLTV(70,80]     -0.54961    0.34864  -1.576 0.114930     CLTV(80,90]     -0.51413    0.31230  -1.646 0.099715 .   CLTV(90,95]     -0.74648    0.37221  -2.006 0.044904 *   CLTV(95,100]     0.38370    0.37709   1.018 0.308906     CLTV(100,125]   -0.01554    0.25187  -0.062 0.950792     CLTV(125,Inf]   18.49557  443.55550   0.042 0.966739     FICO[0,660)     19.64884 3956.18034   0.005 0.996037     FICO[660,680)    1.77008    0.47653   3.715 0.000204 *** FICO[680,700)    0.98575    0.30859   3.194 0.001402 **  FICO[700,720)    1.31767    0.27166   4.850 1.23e-06 *** FICO[720,740)    0.62720    0.29819   2.103 0.035434 *   FICO[740,780)    0.31605    0.23369   1.352 0.176236     --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  (Dispersion parameter for binomial family taken to be 1)      Null deviance: 1037.43  on 810  degrees of freedom Residual deviance:  803.88  on 798  degrees of freedom AIC: 829.88  Number of Fisher Scoring iterations: 16   FICO in the range [0, 660) and CLTV in the range (125, Inf] indeed always results in Denial = 1, so their coefficients are very large, but why are they also "insignificant"? I'm looking for a site, similar to StackOverflow, but with possibility to ask opinion based questions Since opinion based questions are not allowed neither here nor on stackoverflow, I would like to ask if any of you, know some other web page (outside Stack* ecosystem), where we can ask questions like:  "Which Java Web framework would you recommend to learn"   or any other, which by nature are opinion based.  EDIT1:  Allow me to explain a bit more - I know there are webgroups, forums and mailing lists BUT most of those are for single programming language or framework and I'm looking for something more generic. Stackoverflow is great because there are experts from all sort of programming languages and frameworks, so if it would be allowed to ask:   "Best web frameworks similar to ASP.NET MVC4"  I would get a lot of opinions, from different languages and frameworks, and that exactly what I'm looking for. Why can a 1 element set be a member of another set but not a subset of it? So here is the problem      Solve $$4^x + 6^x = 9^x$$ for $x$.   I am trying to find its real solution.  I was trying in this way!! $$6^x\left( \frac {4^x}{6^x} + 1\right) = 9^x$$ $$\left({\frac 23}\right)^x +1=\left({\frac 32}\right)^x$$ but I'm stuck . . . Will anyone help ? Expected Value Of Number of Removals From Urn Here's the question: "An urn contains b blue balls and r red balls. You repeatedly and independently remove balls from urn (without returning them) until the first blue ball is drawn. All balls currently in the urn have an equal probability of being selected each time a ball is removed. Define the random variable X as the number of balls that are drawn (number of red balls that are removed plus the first blue ball that is removed). Find E[X], the expected value of random variable X."  I'm having trouble finding a closed form expression for the expected value. Here's the work I've done: block all but a few ips with firewalld On a linux  networked machine,  i would like to restrict the set of addresses on the "public" zone  (firewalld concept), that are allowed to reach it. So the end result would be no other machine can access  any port or protocol, except those explicitly allowed, sort of a mix of     --add-rich-rule='rule family="ipv4" source not  address="192.168.56.120" drop'    --add-rich-rule='rule family="ipv4" source not  address="192.168.56.105" drop'   The problem above is that  this is not a real list, it will block everything since if its one address its blocked by not being the same as the other, generating an accidental "drop all" effect,  how would i  "unblock" a specific non contiguous set? does source accept a list of addresses? i have not see anything in my look at the docs or google result so far.     EDIT: I  just  created this:  # firewall-cmd  --zone=encrypt --list-all encrypt (active)   interfaces: eth1   sources: 192.168.56.120   services: ssh   ports: 6000/tcp   masquerade: no   forward-ports:    icmp-blocks:    rich rules:    But i can still reach port 6000 from  .123   my intention was that   if a source is not listed, it should not be able to reach any service or port If $M$ isn't Noetherian, $M$ has a submodule maximal with respect to being not finitely generated. I'm answering this question:     Let $M$  be a module. Show that if  $M$ is not Noetherian then $M$ has a submodule $N$ such that $N$ is not finitely generated but $A$ is finitely generated whenever $N &lt; A ≤ M$.   The solution starts as follows (I'm following a given solution): Let $Y$ be the set of non-finitely-generated submodules of $M$. If $(N_α)$ is a chain in $Y$ then $Q= \bigcup_αN_α $ is in $Y$, else $Q$ is f.g., which implies $Q=N_α$ for some $α$, whence $N_α$  is f.g., contradiction.  I don't understand why if $Q$ is f.g., then $Q=N_α$ for some $α$? Am I missing some fact?  Thanks! Handling Calls to Remove a Moderator This discussion informed our process for seven years, but eventually required updates and augmentation. For discussion regarding the successors to this process, please see:    Stack Exchange has some of the best Moderators around — seriously — and that is due in no small part to the communities who scrutinize and vote in our Moderator elections.  We have a rather formal process for electing moderators, but up until now  has been quite informal. On those rare, rare occasion when things just aren't working out, you can see why it becomes such a sticky business to remove someone from a democratically elected position. Our current process is one of &quot;we've got your back.&quot; The Community Team routinely monitors moderator activities, so if a moderator violates the  or fails to uphold the , we can and do intervene. If the disputes and problems persist without resolution, eventually we can ask someone to step down. That's not good enough. We need a much more transparent and prescribed process for when — and more importantly how — to intervene when on-going problems are going unresolved. In urgent situations, we'll step in to protect a site, but ideally the process should be impartial and not subject to accusations of Stack Exchange simply applying undue influence over a community when the whim strikes. The Question: Before we start bouncing around ideas internally, I'd like to hear what you think. What process should be in place to add some checks and balances to deal with higher issues like possible impeachment of a moderator? When and how would such a process come into play? How should such situations be resolved? How do we keep it from being abused? Have at it. This is a round table vetting of ideas. I'd like to know if there is any alternative to Deep Freeze for Linux that still under development, because I've read some things on the internet and looks like the same company which developed Deep Freeze had a Linux version of it, but the project was discontinued. There have been several discussions regarding code-only answers (answers that contain only code blocks with no textual explanation whatsoever):         Personally, I can't see why the quality filter should let such answers through at all. The main problem I have with code-only answers, especially once they're flagged as not an answer or very low quality, is that we can't determine at a glance whether these answers are:   Genuinely attempting to answer the question  answers Trying to correct an error in the question without actually answering it Random code dumps that may or may not relate to the question even by programming language or keyword(s) in the question   In particular, what exactly is an "informative code-only answer"? Does it mean the code is self-documenting? Does it mean the code substitutes code comments for a proper write-up? This is all very fuzzy. From  to the last question above:     I have never seen a code-only answer that couldn't be made better with the addition of appropriate text. They are of lower-quality than the same answer with some text.   This text makes all the difference, IMO, between an answer and a non-answer. Without it, it's often needlessly difficult to tell.  Existing code-only answers that are good-faith attempts to answer the question should be edited or downvoted, yes. But if the quality filter already tries to block code-only questions, why not block code-only answers too? These answers are often low quality, and (almost) always have ample room for improvement, just like questions. Is something similar to Robin's theorem known for possible exceptions to Lagarias' inequality? Robin's theorem says that if $$\sigma(n)&lt;e^\gamma n\log\log n$$ holds for all $n&gt;5040$, where $\sigma(n)$ is the sum of divisors of $n$, then the Riemann hypothesis is true, but if there are any counterexamples, then they are colossally abundant numbers, and there are infinitely many counterexamples.  Lagarias' theorem says $$\sigma(n)\leq H_n+e^{H_n}\log(H_n)$$ holding for all natural numbers $n$ is equivalent to the Riemann hypothesis, where $H_n$ is the $n$th harmonic number, the sum of the reciprocals of the first $n$ positive integers. It looks like Lagarias' inequality is sharper than Robin's, so any counterexamples to Robin's inequality must also be counterexamples to Lagarias'. Is known to be impossible for integers that aren't colossally abundant numbers or of some similar more general type to be exceptions to Lagarias' inequality? Call to a member function on a non-object So I'm refactoring my code to implement more OOP.  I set up a class to hold page attributes.  class PageAtrributes  {   private $db_connection;   private $page_title;      public function __construct($db_connection)      {         $this-&gt;db_connection = $db_connection;         $this-&gt;page_title = '';     }      public function get_page_title()     {         return $this-&gt;page_title;     }      public function set_page_title($page_title)     {         $this-&gt;page_title = $page_title;     } }   Later on I call the set_page_title() function like so  function page_properties($objPortal) {         $objPage-&gt;set_page_title($myrow['title']); }   When I do I receive the error message:     Call to a member function set_page_title() on a non-object   So what am I missing?