Notify me when a bounty is placed on one of my questions
Can there be a notification for bounties started on questions you've asked or answered?
Suppose $a_n>0$ and $\sum_{n=1}^{\infty}{a_n}$ diverges. Determine convergence of $\sum_{n=1}^{\infty}{\frac{a_n}{s_n^2}}$, where $s_n=\sum^n a_n$. I am a computer scientist, and one of my professors today used the symbol $\propto$. I tried to search that using google, but it returns no results, and I do not even know its name.  So, I would like to ask, what does the operator $\propto$ mean? What is its name, and how is it used? Double decimal formatting in Java Why does the loss/accuracy fluctuate during the training? (Keras, LSTM) how I can draw an skew cylinder (rotate around x axis) and fill it (in 3D tikz)? Does copying a spell from a scroll into your spellbook consume the scroll? The text on page 32 of the Basic Rules that talks about copying spells does not explicitly state that the process consumes the scroll. Does it? I'm using NetBeans, trying to change the familiar Java coffee cup icon to a png file that I have saved in a resources directory in the jar file.  I've found many different web pages that claim they have a solution, but so far none of them work.  Here's what I have at the moment (leaving out the try-catch block):  URL url = new URL("com/xyz/resources/camera.png"); Toolkit kit = Toolkit.getDefaultToolkit(); Image img = kit.createImage(url); getFrame().setIconImage(img);   The class that contains this code is in the com.xyz package, if that makes any difference.  That class also extends JFrame.  This code is throwing a MalformedUrlException on the first line.  Anyone have a solution that works? Is it possible to recover an email that has been clicked "delete forever" in the trash folder? Show that the boundary of a set equals the boundary of its complement $\newcommand{\bdy}{\operatorname{bdy}}$  I'm trying to show that $\bdy(A) = \bdy(A^c)$.  I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \setminus \operatorname{int}(A)$, but I don't know where to go from there.  Any help or hints would be very much appreciated. I need to match a space character in a PHP regular expression. Anyone got any ideas?  I mean like "gavin schulz", the space in between the two words. I am using a regular expression to make sure that I only allow letters, number and a space. But I'm not sure how to find the space. This is what I have right now:  $newtag = preg_replace("/[^a-zA-Z0-9s|]/", "", $tag); Minimal polynomial of $\sqrt[3]{2} + \sqrt{3}$ Suppose I want to find the minimal polynomial of the number $\sqrt[3]{2} + \sqrt{3}$.   Now that means I want to find a  unique polynomial that is irreducible over $\Bbb Q$ such that $f(x)=0$. Now I know that because   $\sqrt[3]{2} + \sqrt{3}$ belongs to $\Bbb Q( \sqrt[3]{2} , \sqrt{3})$  it might be degree $2$, $3$ or $6$ and does not belong to $\Bbb Q( \sqrt[3]{2})$ so it cannot be of degree $3$ or  $\Bbb Q( \sqrt{3})$ so it cannot be of degree $2$.  So it is of degree $6$. I think my sayings are a bit intuitive and not formal and lack rigorous. Couldn't it belong to another extension of degree $2$? or $3$? I can't answer that. Why checking only those $2$ is enough? Or is it wrong at all to say that? The code for reference is this  @article{matcontcl,  title= {MATCONT and CL MATCONT},  author = {A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and  B.Sautois},  year = {2006},  journal = {Continuation toolboxes in matlab}, }   This is the error that follows  This is BibTeX, Version 0.99d (TeX Live 2015/W32TeX) The top-level auxiliary file: PropuestaInvestigacion.aux The style file: abbrv.bst Database file #1: biblio.bib Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1049 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1049 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1090 of file abbrv.bst Too many commas in name 1 of "A. Dhooge, W. Govaerts, Yu.A. Kuznetsov, W. Mestrom, A.M. Riet and B.Sautois" for entry matcontcl while executing---line 1090 of file abbrv.bst (There were 4 error messages)   How could I fix it? Can anyone recommend a cheap/free utility that can do this without very much effort?  My preference is that it would be really easy.    Boot into Windows Pick drive to move Pick target drive It copies everything over, and reboots to the correct partition. I am reading through Folland's Real Analysis: Modern Techniques and Their Applications, and they have the following proposition and proof:     Proposition: If $\{f_{j}\}$ is a sequence of $\bar{\mathbb{R}}$- valued measurable functions on $(X, \mathcal{M})$, then the functions:      $$g_{1}(x)=\sup_{j} f_{j}(x)$$ $$g_{2}(x)=\inf_{j} f_{j}(x)$$ are   measurable.       Proof:      We have       $$g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}  f_{j}^{-1}((a,\infty])$$ $$g_{2}^{-1}([-\infty,a))=  \bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$$      Where the result follows from the fact that $g_{1}^{-1}((a,\infty])\in  \mathcal{M}$ for all $a \in \mathbb{R} \iff g$ is measurable, and   $g_{2}^{-1}([-\infty,a))\in \mathcal{M}$ for all $a \in \mathbb{R}  \iff g$ is measurable.   I understand the last part of the proof, but how do we know $g_{1}^{-1}((a,\infty])= \bigcup_{j=1}^{\infty}f_{j}^{-1}((a,\infty])$ and $g_{2}^{-1}([-\infty,a))=\bigcup_{j=1}^{\infty} f_{j}^{-1}([-\infty,a))$? I would appreciate any help with this. Why do professional print houses use gamut-limiting CMYK? I recently had some business cards printed, and was warned to use the CMYK colour space if providing TIF files.  When I tried saving from Paint Shop Pro to a CMYK, of course the colours came out very much duller, so I provided a different file type that preserved my colours using RGB. To my horror, the cards came back nevertheless converted to the CMYK dull shades.    A few months previously I had been in a hurry and had printed my own one-sided quasi business cards using glossy photo paper and a cheap home printer.  This reproduced my desired colours perfectly.  Since the whole point of using professional printers is quality (though I admit bulk as well), why do they not use printers that can cope with the RGB colour range?   FYI: the main colour in question, which I chose very carefully, is a slightly yellow-tinged lawn green, which I understand doesn't feature in the CMYK gamut.  So why use that gamut, and is there anything I can do to overcome this issue and get cards printed in the colour I have chosen? I'm looking at buying a second-hand Mac, probably one that does not include an OS installation disk.  How can I scan it to make sure that it has not had security threats installed on it?  Is there a CD or DVD image I can boot it up with to perform this scan offline?  Edit: prospective machine would be a 2011- or 2012-vintage MacBook Pro.  Also note that I don't have reliable access to high-speed internet, so solutions that involve downloading 5GB over the Internet are not ideal.  For others, though,  may be helpful. In C, how do I restrict the scope of a global variable to the file in which it's declared? How to prove $a_1^m + a_2^m + \cdots + a_n^m \geq \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}$ Permanent background ssh connection to create reverse tunnel: what is correct way? Supremum and infimum of sets I've recently started to study supremum and infimum, and I'm having trouble determining the supremum and infimum of the following sets:  $ A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\} $ $ B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\} $ $ C = \left\{ {m\over \vert m\vert +n} \mid m \in \mathbb Z, n \in \mathbb N \right\} $ $ D = \left\{ {n\over 3} - \left[{n\over 3}\right] \mid m, n \in \mathbb N \right\} $   In $D$, the $[\; ]$ denotes the integral part of ${n\over 3}$.   $E = \{x \mid x \text{ is decimal fraction between } 0 \text{ and } 1 \text{ that has the digits } 0 \text{ and } 1\}$.  In my progress so far I was able to verify that sup $\{x \in A\} = +∞$  Setting $m = (n-1)$ to get  $$ \frac{mn}{1+m+n}= \frac{n(n-1)}{1 + n + n-1} = \frac{n(n-1)}{2n} = \frac{n-1}{2} $$  That way $\frac{n-1}{2}\in A$ for every $n\geq 2$. Thus, the supremum of $A$ is bigger than $\frac{n-1}{2}$ for every integer $n$.  I don't know how to go past this. x_x  Any help would be greatly appreciated!