I am stuck in one step of an application of Borel Cantelli Lemma.  I am trying to show the following : if $a_n$ is a sequence such that $0 &lt; a_n &lt; 1$ for all $n$ then $\prod_{n=1}^\infty 1 - a_n = 0 \Rightarrow \sum_{n=1}^\infty a_n = +\infty$. By using the log I can show that $\prod_{n=1}^\infty 1 - a_n = 0 \Leftrightarrow \sum_{n=1}^\infty \log(1-a_n) = - \infty $ (since $1-a_n &gt; 0, \forall n$) but I cannot conclude about the divergence of $\sum_{n=1}^\infty a_n$. Any hint ?
Prove that if $0\le p_n&lt; 1$ and $S=\sum p_n&lt; \infty$ then  $\prod (1-p_n)&gt;0$. Hint given: First show that if $S&lt;1$, then $\prod (1-p_n)\ge 1-S$.   Attempt: I was able to show the hint by using recursion setting $A_n=\prod_{i=1}^{n}(1-p_i)$ re expression the $\prod (1-p_n)$ as $1-S+\sum_{n_1, n_2=1,n_1&lt;n_2 }^{\infty} p_{n_1}p_{n_2}-..... $ and observed that every subsequent term is less than the previous term hence  $\prod (1-p_n)\ge 1-S$ which is $&gt;0$ if $S&lt;1$ but am unsure about how to extend it to $S\ge1$.
How to Drag and Drop files to open application? Since Ubuntu 18.04 I can't drag a file e.g. image file to an open application such as Inkscape or  email message window. In Ubuntu 17.10 I was able to select the image file by clicking on it, and while holding down left mouse button I would hit the supper key displaying all open windows where for there I could then drop the file in any open app or folders. This is no longer possible/default behaviour since Ubuntu 18.04. Is there a workaround that would allow me to achieve this or even a way to recreate the same behaviour like in 17.10? Fill cylinder cap with quads Is there any way of converting the triangle fan found on a cylinder cap into quads. Is there a way to take a screenshot using Java and save it to some sort of image? If I multiclass into 2 or more spellcasting classes, how do I determine my known/prepared spells? If I multiclass into 2 or more classes/subclasses that can cast spells, how do I determine what spells I know or can prepare?  For instance, if I am playing a multiclassed character with 1 level in warlock, 2 levels in bard, and 3 levels in paladin, how do I determine which spells I know/have prepared?    Related: Set multicolumn width to default table width What is the most efficient way to convert a MySQL query to CSV in PHP please?  It would be best to avoid temp files as this reduces portability (dir paths and setting file-system permissions required).  The CSV should also include one top line of field names. $S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective? $S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is  a continuous function.  I've already shown that $$T_{\mathrm{max}} := \mathrm{sup}\{ T(x) : x \in S^2\}  \: \: \mathrm{ and }\: \: T_{\mathrm{min}}:= \mathrm{inf}\{ T(x) : x \in S^2\}$$ are reached by $T$ with the argument that $S^2$ is closed and bounded and therefore after the Heine-Borel-Theorem compact, and therefore since T is continuous it assumes its max and min in $S^2$.   Now i have answer the following:     1) Is there a point $x_0 \in S^2$ with $T(x_0)=T(-x_0)$ ?      2) Is there a value $T \in(T_{\mathrm{min}}, T_{\mathrm{max}})$ which is only assumed at exactly one point of $S^2$ ?   I wanted to show that $T$ is injective to get that 1) is false and 2) is true but i didn't get anywhere. Any ideas? Thanks in advance! Does Power Word Kill kill druids in wild-shape? This question has been asked before, at least twice by people who were trying to cheat on the PROMYS Europe 2020 Application, and once in a  because the OP showed no effort. I haven't been able to solve it, so I'm going to to ask it again.  PROMYS applications have closed, and I hope the effort shown below is enough to keep this question from being closed.     The Mathematical Forest is grown in a two-dimensional plane, where   trees can only grow on points with integer coordinates. To start with,   there are no trees at all. The foresters plant the first tree at   $(0,0)$. Each year, they carry out tree planting according to the   following rule. If there is a tree on the point (𝑚,𝑛) but there are   no trees on the points $(m+1,n)$ and $(m,n+1)$, then they can choose   to remove the tree on $(m,n)$ and plant new trees on the points   $(m,n+1)$ and $(m+1,n)$. For an integer $k\geq1$, the $k$th diagonal   consists of all points $(m,n)$ with $m+n=k−1$. Is it possible for the   foresters to arrange their planting so that eventually there are no   trees on the first $2$ diagonals? What about the first $3$ diagonals?    $4$ diagonals? Can you generalize?   It's easy to get to a position with no trees on the first two diagonals.  (It only takes $4$ plantings.)  I can prove it's impossible to reach a position with no trees on the first $4$ diagonals, and I believe it's impossible to reach a position with no trees on the first $3$ diagonals, but I can't prove it.  For the $k=4$ case, I used a pagoda function, as in a peg solitaire problem.  For $k=1,2,3,\dots$ define the potential of a tree on the $k$th diagonal as $\frac1{2^{k-1}}$, and the potential of the forest as the sum of the potentials of all the trees in it.  When a tree of potential $\frac1{2^{k-1}}$ is removed, it is replaced by $2$ trees of potential $\frac1{2^k}$ so the potential of the forest never changes.  Initially, the potential of the forest is $1$.  Since there are $k$ trees on diagonal $k$, the potential of all points beyond the fourth diagonal is $$\sum_{k=5}^\infty\frac k{2^{k-1}}=\frac34&lt;1,$$ so it is impossible that there are no trees on the first $4$ diagonals.  The potential of the fourth diagonal is $\frac12$, so this argument doesn't show that it's impossible to have no trees on the first $3$ diagonals.  I've done a lot of experimenting, though, and I believe that the statement is true.  (I even wrote a little computer game so I could experiment quickly.)  The best I've been able to do is to get down to one man on the third diagonal.   The black circles are trees eligible for removal, and the gray circles are other trees.  You can see that there is a "traffic jam" in front of the lone tree on diagonal $3$, and there seems to be no way to clear it.  I've been trying to come up with an argument that reflects this traffic jam, but I haven't been at all successful. We know that after $n$ removals there are $n+1$ trees, and I've been trying to prove somehow that they can't all be sufficiently far from the origin to allow all trees to get past diagonal $3$, but I haven't come close.   In the diagram above, if the tree at $(1,1)$ were moved to $(0,2)$, and the tree at $(1,2)$ were moved to $(2,0)$, then it would be possible to remove the tree at $(0,2)$, emptying the second diagonal.  In the fictitious position, we still have one tree on diagonal $2$ and two on diagonal $3$.   So, I think an argument along the lines I was trying cannot just be concerned with the distance from the origin; it must somehow take into account how the trees interfere with one another.  I haven't been able to do this.    I'd be grateful for solutions, hints, or counterexamples.  I wasn't really sure what tags to attach.  Please add any that seem appropriate. Securing SSH server against bruteforcing I have a little SVN server, old dell optiplex running debian. I don't have that high demands on my server, because its just a little SVN server... but do want it to be secure.  I just renewed my server to a newer and better optiplex, and started looking a bit into the old server. I took it down after experiencing problems. When I check the logs, its full of brute-force attempts and somehow someone has succeeded to enter my machine. This person created some extra volume called "knarkgosse" with two dirs "root" and "swap1" or something.  Don't really know why and what they do, but sure do want to prevent this from happening again. I find this a bit strange though because I change my password ever few months or so, and the passwords are always random letters and numbers put together... not easy to brute-force.  I know I can prevent root from logging in, and use sudoers... and change the SSH port, but what more can I do?  So I have a few questions:   How can I prevent logging in for 5 minutes after X amount of incorrect tries. Or slow tries down after each incorrect try? Is there some kind of central blacklist which a server can connect to? A blacklist that keeps track of IP addresses that are "unsafe" and should never be granted access? What more can I do to apply safety to my server?   Like I said earlier, I am running Debian 5 with Apache (www-data user problem?), svn, mysql, php, phpmyadmin, hudson. It is on a home network with port forwarding on 80, 443, 8080, 8180, 23 and 22. How to tell Google to ignore URL parameters and forget about any already crawled? Difference between "in time" and "on time" I have an appointment at 8 and I arrive there at 7:55, is it "on time" or "in time"? What about "the nick of time"? Why is it safe to use 15 A receptacles on a 20 A circuit? Is it possible to create 3D red/cyan anaglyphs in a hand drawing? There are plenty tutorials online on how to create 3D red/cyan anaglyphs from existing photos in Photoshop or similar software. However, I'm wondering if it is possible to draw a 3D image by hand on for example a whiteboard using markers.  My plan is to tape together a red and blue marker and draw with it using the same rotation/inclination of the marker across the drawing. Will this give a 3D effect when viewed through red/cyan glasses? Or is the 3D anaglyph method more complex than just offsetting red and cyan lines? Why are moderators now required to report underage users? Just today (December 2, 2020), a  was enacted, which requires moderators to report users that they find to be underage:  If you find a user claiming to be under 16 escalate it to us through the &quot;Contact Community Team&quot; button and select the &quot;underage user&quot; template - this will flag the on-duty CM for action so we can investigate and, if deemed necessary, delete that account and all personal data associated with it. [...] This policy only applies to sites where you are a moderator. If you see things on a site where you’re not, there is no obligation to escalate.  Being part of the , moderators are required to follow the instructions there, and not following them is considered a violation of the moderator agreement. However, this contradicts a , which states (bold added):  Users and moderators are not legally responsible for underage users and are under no obligation to report them to us. If out of concern you would like to report it, [...]  As part of the re-enaction of the policy on December 2, the text &quot;and moderators&quot; in the above quote was edited out of that statement. Why are moderators now required as per policy to report underage users that they come across? Is there a legal explanation as to why SE's stance on this changed? I'm aware of  from a then-employee, which stated that moderators are expected, but not required as a condition of remaining a moderator, to report underage users. However, the new policy requires reporting them. Why? An example of a case where a moderator may not want to report an underage user is if the user claims to be from an EU country that has reduced the required age from 16. While SE's Terms of Use specify a blanket limit of 16 to comply with regulations in all EU countries, individual member states do have the ability to reduce the limit, so long as it's not below 13. There is no law violation if, say, a 15-year-old user from an EU country that has reduced the age limit to 14 registers, so a moderator may not feel like reporting them (since in that case they're merely enforcing the Terms of Use, which they're not otherwise expected to do).  The policy was recanted a few days after it was originally enacted on October 15, 2020, but was enacted again on December 2 after minor revisions following a review among moderators. The same question still applies to the revised policy, however. Triggered by a  I was wondering what actually would happen to a ship (e.g. the Enterprise-D NCC-1701) that collides with a stationary object such as a) another ship of a similar configuration or b) a gigantic structure like a moon or even a planet.  We know the deflector is supposed to keep interstellar dust (not to be confused with ordinary dust, which is a different thing) from ripping large holes in the ship.  At first, one thinks "Obvious. ." But does it really? Would the deflector manage to actually deflect both masses? If not, where is the threshold of what a deflector can handle? Would the ship maybe tunnel right through the object, because of wibbly-wobbly space wobblification by the warp field? It does do some weird things to the space right in front and behind the vessel.  Important remark: We cannot approach this with what we know about physics at all. For all intents and purposes, both ships must measure the other ship travelling faster than light, resulting in it having infinite (or rather undefined) mass and by extension momentum. If we start with that reasoning any ship jumping to warp would suck in the entire universe (it takes a moment, but think about it). So, obviously we cannot apply our intuition (even our relativistic intuition) here. How do I make my rim brakes stop squealing? My rear rim brakes are squeaking loudly. I don't know what the problem is, but I am sure it has nothing to do with rain because it happens all the time even if it is dry.  Anybody have an idea what could be problem and how to solve it? It makes the rear brakes kind of useless, because if I use them I will scare the people or drivers around me. Validity of maximising log-likelihood for maximum likelihood estimation For reasons owing to mathematical convenience, when finding MLEs (maximum likelihood estimates), it is often the log-likelihood function---as opposed to the standard likelihood function---which is maximised.  From what I've gathered, this approach is deemed valid as a result of the monotonically increasing nature of the (natural) logarithm function.  My understanding of a monotonically increasing function is: for all $x$ and $y$ (defined on a subset of the reals), if $x \leq y$ then $f(x) \leq f(y)$.  This, however, does not appear to be the case for all log-likelihood functions; for example: for the log-likelihood $\text{Gamma}(3, 5)$ function, if $x = 0.15$ and $y = 0.46$, then $f(x) = -0.59$ and $f(y) = -1.19$.  Clearly, I've misunderstood this concept. Fundamentally, I guess I'm asking can somebody (preferably mathematically) demonstrate why:  $$ \hat{\theta} = \text{argmax} \text{ } L(\theta) = \text{argmax} \text{ } \text{log} \text{ } L(\theta) $$  where $\hat{\theta}$ is the MLE for a given likelihood function. Organic yet accurate modeling with the golden spiral Foreword  First off, let me say that I am profoundly amazed at the orderliness in nature. If you haven't heard about the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, ...), you should check it out.  This mathematical concept of the Fibonacci numbers can be shaped into a set of squares, where the lengths of the sides of the squares follow the Fibonacci sequence:     This is sometimes called a golden ratio. It can be formed into a (golden) spiral. (a.k.a. logarithmic spiral):      Application  I am looking for ways on how to utilize this concept in Blender. Primarily in organic modeling. The golden spiral is found all throughout nature. From flower seed heads, sea shells, and ram horns to hurricanes and spiral galaxies.  My question is: how can I model based off of one or multiple golden spirals? Specifically, I'm interested in modeling a sunflower head like the one in the following image. I'm mostly interested in the point of origin of the individual seeds (not worried about the shape). Does Blender have any features related to this? Notify-osd notifications appear unthemed in top-left corner