Let $W$ be a standard, one-dimensional Brownian motion and $0 &lt; T &lt; \infty$. Show that  $$\lim_{\beta \to \infty} \sup_{0\leq t \leq T} |e^{-\beta t }\int_0^t e^{\beta s } dW_s| = 0$$ a.s.
How to show that  $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$  where $\left (B_s \right)_{s\geq 0}$ is a real standard brownian motion starting from zero ?
Can't watch youtube videos with chromium on ubuntu 13.04 64-bit I've digged all over google but I can't find any solution to my problem.   Firefox works pretty well. I have this issue only with chromium and only with youtube. I have no problem watching videos at any other site with chromium.   When I try to watch a video a black screen occurs and after a while it gives me the warning "an error occurred". Weirdly, sometimes when I try a couple of refreshing it starts to play but then the issue occurs again.  I tried removing-reinstalling chromium or the flash plugin but it did not do any good. I've started putting into practice the answer to . Yet I would like to change this so that there are brackets around the citations:   in the superscript in text,     in the superscript in footnote,  in the bibliography.   I've managed to get the first one and the last one working. The second one however, I don't know how to do this.   Note that I mainly write in english and in dutch, so I don't think there is much babel interference.  Also, would there be a way to order the footnotes so that first all ''normal'' \footnote{} are printed and below that all \sfcite{} citational footnotes (per page that is)?  Here's what I've managed to concoct:  \documentclass{article} \usepackage[english]{babel} \usepackage[pdfborder={0 0 0}]{hyperref} \usepackage[citestyle=numeric-comp,sorting=none,hyperref]{biblatex} \usepackage{filecontents}  % No brackets around the number of each bibliography entry \DeclareFieldFormat{labelnumberwidth}{\textbf{[#1]}}  %--------------------------------------------------------------- % Essentially verbatim from Joseph Wright % http://www.texdev.net/2010/03/08/biblatex-numbered-citations-as-footnotes/ \makeatletter \ExecuteBibliographyOptions{citetracker,sorting=none}  \DeclareCiteCommand{\notefullcite}[\mkbibbrackets]     {\usebibmacro{cite:init}%         \usebibmacro{prenote}}     {\usebibmacro{citeindex}%         \usebibmacro{notefullcite}%         \usebibmacro{cite:comp}}     {}     {\usebibmacro{cite:dump}%         \usebibmacro{postnote}}  \newbibmacro*{notefullcite}{%     \ifciteseen         {}         {\footnotetext[\thefield{labelnumber}]{%             \usedriver{}{\thefield{entrytype}}\addperiod}}}  \DeclareCiteCommand{\sfcite}[\cbx@superscript]%     {\usebibmacro{cite:init}%         \let\multicitedelim=\supercitedelim         \iffieldundef{prenote}         {}         {\BibliographyWarning{Ignoring prenote argument}}%     \iffieldundef{postnote}         {}         {\BibliographyWarning{Ignoring postnote argument}}}     {\usebibmacro{citeindex}%         \usebibmacro{sfcite}%         \usebibmacro{cite:comp}}     {}     {\usebibmacro{cite:dump}}  \newbibmacro*{sfcite}{%     \ifciteseen         {}         {\xappto\cbx@citehook{%             \noexpand\footnotetext[\thefield{labelnumber}]{%             \fullcite{\thefield{entrykey}}\addperiod}}}}  \newrobustcmd{\cbx@superscript}[1]{%     \mkbibsuperscript{[#1]}%     \cbx@citehook     \global\let\cbx@citehook=\empty} \let\cbx@citehook=\empty \makeatother %---------------------------------------------------------------  \begin{filecontents}{\jobname.bib} @article{ref1,     hyphenation = {american},     title = {Natural Products as Sources of New Drugs over the Last 25 Years},     author = {Newman, D.J. and Cragg, G.M.},     journal = {J. of Nat. Prod.},     volume = {70},     number = {3},     pages = {461--477},     year = {2007}} @article{ref2,     hyphenation = {american},     title = {Natural products as leads to potential drugs: an old process or the new hope for drug discovery?},     author = {Newman, D.J.},     journal = {J. of Med. Chem.},     volume = {51},     number = {9},     pages = {2589--2599},     year = {2008}} @online{ref3,     author = {Quintin, C.},     title = {Les eponges},     url = {http://christophe.quintin.pagesperso-orange.fr/Liste_Doc.html},     urldate = {2009-05-03} } \end{filecontents}  \addbibresource{\jobname.bib}  \begin{document} First citation.\sfcite{ref1,ref2} Some text\footnote{A footnote}.   \pagebreak Second citation\sfcite{ref1}, second citation\sfcite{ref2}, first citation.\sfcite{ref3} \printbibliography \end{document} When should I create a new tag? How do I request a new tag if I don't have enough rep? Can someone please create the tag  on our site? We need to have it because        Foo is a recent/upcoming release / edition / model / episode of a popular, established OS / board game / bike brand / TV show   Foo is a hot new language / ingredient / game   we're currently using some other tag for questions about Foo, but it's causing ambiguity with a separate thing that's also named Foo      This is not a joke post. We get a lot of feature requests that ask for this or that tag to be created, and every time, a discussion ensues wherein people explain that there is no "tag creation department" and that community members create tags just by using them. I'm posting this "question" so that future requests of that type can be closed as a duplicate of this, saving time and eliminating redundant conversations for everyone involved.     See also: How to get my 4yo daughter's fixation on my belly button under control? I don't know where to begin with this question. It's awkward and to be honest, it makes me feel rather uncomfortable. Simply put, I'm led to believe that my four year old daughter is obsessed with my belly button. It all started one day during the summer when she was three. I decided to take my family to the pool, and I guess I didn't see my daughter follow me into the changing room. Mind you, she's still at the age where it's appropriate to see her parents naked, so I wasn't surprised when, after I finished changing into my bikini, I turned around to see her staring at me.  I followed her gaze and saw her looking directly at my belly button. I asked her what she was looking at and she screamed "mommy, I can see your belly button!" which made me giggle. I picked her up and kissed her forehead, telling her that everyone has a belly button.  I think I might have accidentally given her a green light, because over the next year she constantly asks me if she can stick her finger in my belly button, and if I don't let her, she begins to scream.  For her fourth birthday, she requested that I wear nothing but a sports bra and underwear (in toddler speak. It sounded like "mommy, show your belly button for my birthday!" I could have simply worn a crop top but i didn't want to risk her throwing a temper tantrum)  My husband obviously doesn't mind, because he gets to look at my belly button all the time but it really upsets me that my daughter's finger is constantly in my belly button.  Tomorrow is her fifth birthday and I want to discuss with her that it's not appropriate to always be poking my belly button. Any ideas on how i can talk to her without her throwing a screaming fit? I flagged a post today, typed out a fairly thorough explanation, clicked the button... and got told that I'd already flagged this post (for something else, which I'd forgotten about) so this flag wasn't allowed.  (This happens if the first flag is still active.)  The dialogue with my explanation was gone at this point, so if I wanted to communicate with the moderators about that, I was going to have to start over.  (In case you're wondering how this could happen, one flag was related to the post itself and one was related to the user -- there being no way to flag a user directly, you have to hang it off of a post.  But I've also seen this with cases where the first flag hangs around unhandled for a week or two and I've forgotten about it.)  Since the system is going to detect that I've already flagged a post and reject a second flag, can it do that after I click the "flag" link instead of waiting until the end of the process?  (Or gray out the link so I can't click it in the first place?)  2015-12-18: someone suggested to me in chat today that this had been implemented, so I tested again.  The behavior has not changed: I flagged a post using "other", then flagged it again (also "other") before the first flag was handled, and the flag was rejected only after I'd typed the message.  (I did this on a site where I'm a moderator so I could then clean up after myself.) How are release codenames chosen? Is there a way to combine custom settings with custom labels ? So that the output from a setting value would be a custom label ?   To explain:  In example, we have an international organisation, and say countries are stored as an ISO code. As this is a limited and stable set of data, using custom settings would be the appropriate way to deal with say the full country name.   The issue arises when I do not just want to store the English country name, but want to have it translatable, presenting the right translation depending on the user's locale. Now, this is the point where I'd like to integrate custom settings with custom labels.  So that say the custom setting with name 'some iso code' doesn't return the full country name, but Label.Country'some iso code', which is than automatically translated in the right version.   Sounds good right. But as custom settings are fixed to certain types, as far as I've gotten is put the label name in the settings as a string value. And then .. i'm stuck :).  No way to cast a string to a Custom label, or have it be interpreted (and translated) as such, or is there ?   I'm aware this can be done in a custom object &amp; apex, I was hoping for a more automated, "cleaner", less resource consuming way, better integrated, configurable way.  Additional clarrification: *The goal is to extend the use of custom labels (translation workbench), by adding a mapping from any data-based value to a custom label.*   While we could do a long matching process in apex, it would be faster &amp; neater if the custom settings mechanism could be used. There are disadvantaged to custom objects, they require soql &amp; apex, while custom settings &amp; custom labels are cached. They too can be used in many other places. Countries were an example, I'd use this mechanism for every variable value that needs to be translatable. How do I mount a qcow2 disk image? Will I die if I lose enough life to kill me from an effect that also heals me? What happens if I dip below 1 life from an effect that will also heal me after the damage is dealt?  For example:   If I bolt myself while I'm at 3 or less with a  on the board? Will I die while at 3 or less if I -2 , targeting myself? Will I die if a creature with my  on it damages me? Will I die if I target myself with enough damage to 'kill' me with a ?   and so on and so forth. You get the idea. I'm pretty sure there's a relatively simple answer that I just don't know about. Re-Structuring a JSON Below, the  of second-order logic is used.    My question is about a second-order analogue of $ZFC$ other than the usual "second-order $ZFC$." Rather than define the latter, I'll just state that its (set-sized) models are exactly the $V_\kappa$s with $\kappa$ strongly inaccessible (or if you prefer, the uncountable Grothendieck universes).  Specifically, I'm interested in the second-order theory consisting of the usual (first-order) axioms of Pairing, Extensionality, Union, Choice, Infinity, and Powerset, and the Separation and Replacement schemes ranging over second-order formulas. I'll call this "$ZFC_2^{scheme}$."    Specifically, for each formula $\varphi(x, y_1,...,y_k)$ of second-order logic with only the displayed variables, each of which is first-order, we have a corresponding Separation and Replacement instance. We do not allow second-order parameters here, so formulas with second-order free variables don't yield Separation/Replacement instances.   The passage from $ZFC$ to $ZFC_2^{scheme}$ has a nice uniformity: it's an example of a more general operation $\mathcal{ZFC}$ which spits out a $ZFC$-analogue given a logic at least as strong as first-order logic.    This is the reason for the parameter restriction above: I want $ZFC_2^{scheme}$ to look just like $\mathcal{ZFC}(\mathcal{L})$ for any other logic $\mathcal{L}$, and not take into account the particular feature that we now have a new kind of variable at our disposal - and in particular, take as "syntax-free" an approach to logics as we can get away with. (This is a choice of arguable value, but it's the one I'm making for now.)   It's easy to show that every model of $ZFC_2^{scheme}$ is well-founded, so to understand its models we only need to look at transitive sets. A natural hope at this point is that $ZFC_2^{scheme}$ is just second-order $ZFC$ in disguise, that is, that for a transitive set $M$ we have $M\models ZFC_2^{scheme}$ iff $M=V_\kappa$ for some strongly inaccessible $\kappa$. However, this isn't at all obvious to me (although it is easy to show the right-to-left direction).     Question. What are the (set-sized) models of $ZFC_2^{scheme}$?   (Note by contrast that the "arithmetic analogues" $PA_2^{scheme}$ and second-order $PA$ are equivalent since each pins down $\mathbb{N}$ up to isomorphism - although that leads to .)  (I'm happy to drop Choice if that would help.)  EDIT: As Hanul Jeon pointed out below,  shows that consistently $ZFC_2^{scheme}$ (there called "$ZFC_2^{def}$") has countable models, so we have a partial answer. However, I don't see at the moment a way to get an outright $ZFC$ proof that $ZFC_2^{scheme}$ is strictly weaker than second-order $ZFC$. What is the benefit of launching a satellite "a few days before the monthly report" I've seen this hint in the game as well as on this site and I've been wondering if there is any specific advantage to launching satellites closer to the monthly report rather than any time during that month.  The way I see it, monthly maintenance costs are taken out during the reports (correct me if i'm wrong), so regardless of when you launch it during the month it would cost the same.   Is the benefit of holding on to them longer simply a safety net in case some countries go into the red during that time? Smoothing polygons in contour map? Here is a contour map for which all the polygons of levels are available.  Let ask how to smooth the polygons keeping all vertices preserved in their exact locations?    Indeed the contour is made on top of a grid data, you may suggest then to smooth the grid data and hence the resulting contour will be smoother. Note that this is not working as my desire since the smoothing function such as Gaussian filter will remove small packs of data and will change the range of the third variable e.g., the height which are not allowed in my application.    Actually I'm looking for a piece of code (preferably in Python) which can do the smoothing of 2D polygons (any type: convex, concave, self-intersecting etc) reasonably painless (forget pages of codes) and accurate.   FYI, there is  that does this perfectly, but using third party commercial applications is not my choice for this question.        1)    Scipy.interpolate:        As you see the resulting splines (red) are not satisfactory!  2)    Here is the result using the code given in . It is not working well!      3)    To me the best solution should be something like the following figure in which a square is being smoothed gradually by changing only one value. I hope for similar concept for smoothing any form of polygons.       Satisfying the condition that spline passes the points:     4)    Here is my implementation of "whuber's idea" line by line in Python on his data. There are possibly some bugs since the results are not good.         K=2 is a disaster and so for k>=4.   5)    I removed one point in the problematic location and the resulting spline is now identical to whuber's. But it is still a question that why the method does not work for all cases?        6)    A good smoothing for whuber's data can be as follows (drawn by vector graphics software) in which an extra point has been smoothly added (compare with update   4):      7)    See the result from Python version of whuber's code for some iconic shapes:     Note that the method seems doesn't work for polylines. For the corner polyline (contour) green is what I want but got red. This needs to be addressed since contour maps are always polylines although closed polylines can be treated as polygons as in my examples. Also not that the problem emerged in update 4 has not been addressed yet.  8) [my last]    Here is the final solution (not perfect!):        Remember that you will have to do something about the area pointed by stars.  There is perhaps a bug in my code or the proposed method needs further development to consider all situations and to provide desired outputs. I have a bunch of raw data values that are dollar amounts and I want to find a confidence interval for a percentile of that data.  Is there a formula for such a confidence interval? In H. P. Lovecraft's The Case of Charles Dexter Ward, Dr. Willett      Enters Joseph Curwen's dungeon and resurrects a dead person identified as No. 118 by reciting the incantation of the Ascending Node over his salts.   Dr. Willett is told by      The resurrected Joseph Curwen, impersonating Charles Dexter Ward, who he secretly murdered,   that   " 'twas Number 118, and I conceive you would have shook had you looked it up in my list in t'other room."  No. 118 himself gives Dr. Willett a note asking the doctor to      kill Joseph Curwen.   It soon becomes clear that No. 118 is opposed to the nefarious dealings that have been transpiring and desirous of being an ally of Dr. Willett.  Who is No. 118?  Earlier in the story,      The grave of Ezra Weeden, a primary instigator in Joseph Curwen's initial downfall and the romantic rival of Curwen, is robbed. There doesn't seem to be any other mention of Weeden in the 20th century occurrences, so without a further connection the passage seems superfluous.   The shared goals of these two people seem to indicate that No. 118 may, in fact, have been that person, but the      Note given by No. 118 to Dr. Willett in eighth century Latin seems to indicate someone much older than Ezra Weeden.   It also occurred to me that No. 118 could be      The recently murdered Charles Dexter Ward himself. Ward's reference to killing Dr. Allen (who is actually an alias of Curwen) and dissolving his body with acid and No. 118's mention of doing the same to Curwen seem to support the idea, but that still doesn't explain the archaic note. Ward is described as writing in a somewhat modern style, at least as opposed to Curwen. It also doesn't explain why Curwen would want to murder Ward if he was going to resurrect him later anyway. $P$ is a prime ideal of $R$ iff $R/P$ is an integral domain. : $P≠R$ Here is the theorem and proof in my textbook.     Thm.  Let $P$ be an ideal of a commutative ring $R$ with identity $1$.  Then $P$ is a prime ideal of $R$ if and only if $R/P$ is an integral domain.   Proof.   Suppose that $P$ is a prime ideal of a commutative ring $R$ with $1$.  Then $P$≠$R$ implies $1+P≠0+P$. Hence $R/P$ is a commutative ring $R$ with identity.  Assume that $(a+P)(b+P)=0+P$.  Then $ab+P=0+P$ and $ab∈P$.  By the definition of a prime ideal P we get $a∈P$ or $b∈P$.  That is, $a+P=0+P$ or $b+P=0+P$.  Thus $R/P$ is an integral domain.   Conversely, if $R/P$ is an integral domain, then $1+P≠0+P$ and $R/P$ is a commutative ring $R$ which has no zero divisors.  Hence $P≠R$.  Assume $ab∈P$.  Then $ab+P=0+P$ and $(a+P)(b+P)=0+P$.  Since $R/P$ is an integral domain, we get $a+P=0+P$ or $b+P=0+P$.  So $a∈P$ or $b∈P$.  Thus $P$ is a prime ideal.    I am having problems understanding proof with "Conversely, if $R/P$ is an integral domain, then $1+P≠0+P$.".  I know that $R/P$ has an identity. (cause it is ID)  But, why doesn't "$P$" have an identity?  Isn't there a case $P=R$?  Please help me understanding. I think $P≠R$ should be with condition of Thm..  Thank you in advance. Is there an official rule that the minimum damage can't be negative? If a function returns no value, with a valid return type, is it okay to for the compiler to return garbage? Why don't we treat rubbish the same as spam? I have three partitions on my drive, arranged something like the following:   ___________ | 1 |  2 | 3| -------------   I wish to merge partitions 1 and 3.  Is there a way to do this, or do I have to manually "slide" partition 2 over to partition 3, and then merge the newly open space where partition 2 used to be?  Note: I do not have the space to copy partition 2 to partition 3 (2 is 200GB and 3 is 65GB).