Watermark for Textbox
Watermark TextBox in WinForms
What is in the space between a nucleus of an atom and its electrons? Is it true that all numbers of the form $\sqrt[b_1]{a_1}+\sqrt[b_2]{a_2}+\cdots+\sqrt[b_k]{a_k}$ are not integers, where $a_i,b_i$ are positive integers such that none of the $k$ terms is an integer? I think this must be a theorem somewhere, probably a fundamental theorem in number theory, but not sure what it would be called. Proving that closed (and open) balls are convex I work with QGIS 2.10.1 on Windows 7 (x64). I have dissolved polygons from an attribute value and the polygons geometries being not perfectly joined together, I get some 'garbage' nodes I need to delete. I have tried to edit the layer and using nodes tool, selecting and deleting the nodes, but it seems impossible to do (nodes are not deleted at all!). Any idea on how to process this kind of clean-up?  EDIT : you can  for testing. Can someone use DC generator along with electric motor to make electricity along with mechanical energy? I know this might sound stupid and maybe stupid totally, but I need some clarification....  I came through electric motor principle.  and then I came across the following link on DC generators    Now my question is, can't we combine them both to provide DC current with little DC used in motor side? By "combine," I mean using a DC generator coil only at the other side of the coil of the motor in the motor itself. The condition is that the 2 coils are connected via some apparatus that causes the 2nd coil to move along with the 1st coil.   In this way we can use both the mechanical energy produced by the 1st coil and the electrical energy produced by the 2nd coil, right?   This is a great concept if it works, right? Please throw some light on this. Check if a user has a role How can I check if a user has a certain role?  I found  but it is for Drupal 6. Baking with oven without bottom heating element My mom wants to do baking but her oven doesn't have bottom heating element. It only has an element on top, with a fan at the back.  She told me that she can't make me a good pizza or cake without it.  Is there any way around? Forced to Quit PhD although the hard and proved work I am a first year Ph. D student, and I have been working on a new topic of research.  I worked day and night to prove something new and verified it. I began my Ph. D while my supervisor was on travel, but he returned before I had been here for long.  When he returned, he told me we have to stop and not continue without any reasonable reasons.  After a few months he began treating me poorly---always belittling and underestimating---although I got two grants from a top conference, made videos and podcasts, and constantly received praise for my good work and personality.    It seems that I have been treated as a competitor, not a collaborator.  Although agreeing with my approaches in private, my supervisor said false words to the committee to take the decision and stop my Ph. D work!  It is not surprising, as there was another former senior Ph. D student who I met on campus, who was not allowed to defend his Ph. D: this same supervisor claimed he was not able to finish his dissertation although he had published in a peer reviewed journal and having citation as well, it was scary and not familiar.  In front of committee, he lied, and said something without proving. I asked if he can make a detailed constructive criticism, but he could not.  All I can say that they are a gang literally, and he said words that show that he is jealous and doesn't want to see me having self confidence in my self. I know no one will believe me, but this is the truth.  Honestly, I am distressed and I'm losing sleep.  I had this dream where there were many people who saw me fail.  All things considered, I don't know whether what happened is good or bad, and I definitely don't know what I should do next!  I would be grateful for your advice and whether there is someone has encountered any similar situation. Correct usage of "lest" RAR password cracker Does the  file format use a serious encryption (such as )?  Will it take unfeasibly long to brute force a password?  If not, what tools do you suggest that do so automatically? Difference between "commit suicide" and "suicide" How do I prove if $\sup\{ (n!)^{1/n} | n \in\Bbb N\} = +\infty$?  I don't know how to properly explain it Roman numerals with over- and underline This question led to a new package:      (using a variation of egreg's code, and stating this in the manual; the package isn't really new now, sorry for the delayed announcement)  I have read about "How to get Roman numerals" but asked myself how to get Roman numerals formatted as in     i.e. with fitted over- and underline (without manually using boxes and such)? I'm curious for your answer(s)! Do the -ing and to-infinitive "verbs" that follow catenative verbs always take the grammatical function of "noun"? I'm wondering whether or not the verb form that follows a catenative verb has the grammatical function of a noun or of a verb, and whether or not it depends on the first catenative verb.  "I like to run" "I like running"  the -ing and the to infinitive can easily be switched in this sentence, and it seems to be implying "I like [the act of[running]]", so both the infinitive and the gerund are nouns.  "I like to run marathons"/"I like running marathons"   These seem like they might still be nouns but need "verb powers" to take the object, and then these sentences really turn into "I like [the act of [running marathons]]"/ "I like [to be in the state of [running marathons]]".  Is this true for both the -ing (and so it would be a gerund) and the to-infinitive? Are they, sort of, initially verbs so that they can take the object, and then the verb + object becomes a noun phrase?  If this is true, I'm also wondering if the same rules apply for a catenative verb such as "start".  "I started to run in the mornings" "I started running marathons in 2008"  "start" is a catenative verb that isn't a cognitive/emotion verb, and specifically suggests action. So what is the grammatical function of the to-infinitive and -ing verb forms? are they verbs? are they nouns? It just seems incorrect to say that their grammatical function is a noun, but everything I've seen on the internet is pretty vague. I've also seen that to-infinitives are never verbs, but only from one source and it just doesn't seem right to me.  So any ideas on the true grammatical function of the -ing and the to-infinitive after a catenative verb? And is it universal across catenative verbs, or does it depend on the verb category? Failed to load map. Error contacting Google servers. This is probably an authentication issue Proof of Frullani's theorem Question  Let $X_n\geq 0$ be independent for $n\geq 1$. The following are equivalent.  $\sum_{n=1}^\infty X_n&lt;\infty$ a.s. $\sum_{n=1}^\infty[P(X_n&gt;1)+E(X_nI(X_n\leq 1)]&lt;\infty$ $\sum_{n=1}^\infty E(X_n/(1+X_n))&lt;\infty$   My attempt I was able to prove $(1)$ implies $(2)$ and $(2)$ implies $(3)$ but not $(3)$ implies $(1)$. For $(1)$ implies $(2)$, observe that by Borel-Cantelli $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$, else $X_n&gt;1$ infinitely often w.p.1 so that $(1)$ is violated. To see that $\sum_{n=1}^\infty E(X_nI(X_n\leq 1)]&lt;\infty$, apply Kolmogorov three-series theorem (I think). For $(2)$ implies $(3)$ we can write that $$ \begin{align} \frac{X_n}{1+X_n}&amp; =\frac{X_n}{1+X_n}I(X_n\leq 1)+\frac{X_n}{1+X_n}I(X_n&gt;1)\\ &amp;\leq X_nI(X_n\leq 1)+ I(X_n&gt;1). \end{align} $$ Now take the expectation of both sides and sum on $n$ using $(2)$ to conclude $(3)$. For the last implication $(3)$ implies $(1)$, I tried to use Kolmogorov three series theorem. To this end put $Y_n=X_nI(X_n\leq 1)$. First note that $\sum_{n=1}^\infty P(X_n&gt;1)&lt;\infty$ since $$ P(X_n&gt;1)\leq P\left(\frac{X_n}{1+X_n}&gt;\frac{1}{2}\right)\leq 2E\frac{X_n}{1+X_n} $$ by Markov's inequality. But I am unable to show that $\sum EY_n&lt;\infty$ and $\sum\text{Var}(Y_n)&lt;\infty$. Any help is appreciated. Algorithm to find Largest prime factor of a number What is the best approach to calculating the largest prime factor of a number?  I'm thinking the most efficient would be the following:   Find lowest prime number that divides cleanly Check if result of division is prime If not, find next lowest Go to 2.   I'm basing this assumption on it being easier to calculate the small prime factors. Is this about right? What other approaches should I look into?  Edit: I've now realised that my approach is futile if there are more than 2 prime factors in play, since step 2 fails when the result is a product of two other primes, therefore a recursive algorithm is needed.  Edit again: And now I've realised that this does still work, because the last found prime number has to be the highest one, therefore any further testing of the non-prime result from step 2 would result in a smaller prime. Closed unit disk homeomorphic to $\mathbb{R}^2$? I've already shown the existence of a homeomorphism between the open unit disc and $\mathbb{R}^2$ and now I'm trying to work out whether the closed unit disc is homeomorphic to $\mathbb{R}^2$ or not. Clearly the only real difference is the fact it includes the points actually on the circle, but because the continuous, invertible, bijective map I defined for the open disc was $\dfrac{x}{1-|x|}$ this clearly is not defined for the points on the line.   Intuitively I'm thinking it's not possible to form a homeomorphism with the closed disk precisely because of these points on the circle but can't really form a decent argument. I was thinking along the lines of showing that the non-open, closed subset $\bar{D}(0,1) \backslash D(0,1)$ (which is just the set of points on the actual circle i.e the boundary/frontier of the closed disk) can't possibly bijectively map to a closed subset of $\mathbb{R}^2$ - which clearly would have to be the case in a homeomorphism.   Any advice would be greatly appreciated. Why does my front derailleur not work when it is cold? When it is warm outside my front derailleur works just fine. But when it gets colder (40F) the lever on the handle bar does not catch and I can move it without any resistance the whole way without the derailleur shifting.  Update: The wire is not moving and the shifter is a Shimano Deore.  Any idea what can cause that and how to fix it?