If $f$ is Riemann integrable on $[a,b]$ then so is $|f|$   Here is my attempt:  Since $f \in \mathscr{R}[a,b]$, then $f$ is continuous almost everywhere on $[a,b]$ and is bounded. Let $V$ be a set of points where it is discontinuous so $V$ is of measure zero.  Let $S$ be the set of intervals where $f$ is negative. That is, $S=\{I_1, I_2,...,I_n\}$.  Let $T=\{x_0,x_1,...,x_p\}$ be the set of endpoints of these intervals.  Then $|f|$ is discontinous on $V \cup T$. Since $V \cup T$ is of measure zero, then $|f|$ is continuous almost everywhere on $[a,b]$. It is also bounded since $f$ is bounded.  Hence, $|f| \in \mathscr{R}[a,b]$  $\Box$  Is my proof correct? How can I improve it? What's another way to prove this?
Show that if $f$ is integrable on $[a, b]$ then $\lvert f \rvert$ is also integrable. Hint: Show that $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ I have: $$U (P , \lvert f \rvert) \ge U (P , f )$$ and $$ L(P , \lvert f \rvert) \ge L(P,f)$$ Thus, $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$
How to get the full executable name of a running process in Linux Adding feature class to Map in CURRENT project using ArcPy in ArcGIS Pro? About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$ What's wrong with var x = new Array(); I remember watching this one show when I was a kid in the 80's/early 90's in Canada. It was a live-action show about these two kids and had some puppet/muppet-like characters, but it kind of had a dark/mysterious element to it. I can't remember too much about it except portions of some episodes, but the main characters were kids, and they had some mysterious artifact (kind of like a compass?) that took them to this other world where there were giants and other mysterious beings that we rarely saw. The two kids weren't the only one's going to this place either, they find out kids from all around the world that also had this same artifact (and they all fit together like a round puzzle?). I seem to recall that the giants in this other world were the ones who made the artifacts, and were dying/or in desperate need of the help of the kids.  A couple episodes I remember:   There was one where the leaves were falling, and the world was doomed to end as soon as the last leaf touched the ground, so they're trying to figure out how to prevent all the leaves from falling, but after a long and suspenseful episode (for a kid) the last leaf dramatically falls, but out of no where the main girl catches the leaf and saves the world (it was very exciting for my young self, and I remember being very impressed with the pretty girl for thinking to catch the leaf).     There was another episode where the kids meet these two old guys with long white hair (muppets) sullenly working in a mine/tunnel, they're in tattered old clothes working away with their pick axes or hammers and they are connected to each other at the beard. This long old white beard grows out of one of their faces right into the other's. The kids save the day by cutting the old guy's beard in the middle, freeing them from each other. They celebrate being free from their bonds and excitedly thank the kids, but after the kids leave they continue to work away, this time in much brighter spirits. Seemingly unbeknownst to them though, as they work, their beards slowly and subtly grow back together.   Anyone else remember this show? I'm training a neural network but the training loss doesn't decrease. How can I fix this?  I'm not asking about overfitting or regularization. I'm asking about how to solve the problem where my network's performance doesn't improve on the training set.    This question is intentionally general so that other questions about how to train a neural network can be closed as a duplicate of this one, with the attitude that "if you give a man a fish you feed him for a day, but if you teach a man to fish, you can feed him for the rest of his life." See this Meta thread for a discussion:   If your neural network does not generalize well, see: How to choose trace width for very high current PCB? I am currently designing a PCB layout for my team's high current, high voltage project and am having trouble with choosing my trace width.  The circuit is composed of 12 inductors placed in series with short traces connecting them together (about 5mm max length). We are feeding 10μs pulses at ~300A/1000V at a maximum rate of 10 pulses per second by discharching capacitor banks into the coils. The capacitors are recharged by an external source.  I have looked for resources on the subject, but nothing I could find applies to the level of current and voltage we are applying.  Is there a rule for determining trace width for very high current pulses? How to align right within equation array Checking host availability by using ping in bash scripts In a complete metric space with no isolated points, any countable intersection of open dense sets is uncountable? Choosing between meaningful and meaningless hostnames ASP.NET -> WCF concurrency problem SSH Permission denied (publickey) I am trying to connect to a Linode (running Ubuntu 12.04 LTS) from my local machine (also running Ubuntu 12.04 LTS)  I have created a private and public key on my local machine and copied my public key to my Linode's authorized_keys file.  However, whenever I try to ssh to my Linode I get the error message Permission denied (publickey).    It's not a problem with how ssh is set up on my Linode because I can ssh to it from my Windows machine using key authentication.    In my .ssh directory on my local Ubuntu machine, I have my id_rsa and id_rsa.pub files.  Do I need to create an authorized_keys file on my local machine?    EDIT:  This is what I get when I run ssh -vvv -i id_rsa [youruser]@[yourLinode]:  debug3: authmethod_lookup publickey debug3: remaining preferred: keyboard-interactive,password debug3: authmethod_is_enabled publickey debug1: Next authentication method: publickey debug1: Offering RSA public key: id_rsa debug3: send_pubkey_test debug2: we sent a publickey packet, wait for reply debug1: Authentications that can continue: publickey debug2: we did not send a packet, disable method debug1: No more authentication methods to try. Permission denied (publickey). Jointly Complete Sufficient Statistics: Uniform(a, b) Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a &lt; b$.   Let $Y_1$ and $Y_n$ be the largest and smallest order statistics.  Show that the statistic $(Y_1, Y_n)$ is a jointly complete sufficient statistic for the parameter $\theta  = (a, b)$.     It is no problem for me to show sufficiency using factorization.   Question: How do I show completeness? Preferably I would like a hint.    Attempt: I can show $\mathbb E[g(T(x))] = 0$ implies $g(T(x)) = 0$ for the one parameter uniform distribution, but I am getting stuck on the two parameter uniform distribution.   I tried playing around with $\mathbb E[g(Y_1, Y_n)]$ and using the joint distribution of $Y_1$ and $Y_n$, but I am not sure if I am going in the correct direction, as the calculus is tripping me up. Did you know that :  Map&lt;Object,Object&gt; m1 = new HashMap&lt;Object, Object&gt;(); Map&lt;Object,Object&gt; m2 = new HashMap&lt;Object, Object&gt;(); System.out.println("m1.equals(m2) = "+m1.equals(m2)); System.out.println("m1.keySet().equals(m2.keySet()) = "             +m1.keySet().equals(m2.keySet())); System.out.println("m1.entrySet().equals(m2.entrySet()) = "             +m1.entrySet().equals(m2.entrySet())); System.out.println("m1.values().equals(m2.values()) = "             +m1.values().equals(m2.values()));   would output :  m1.equals(m2) = true m1.keySet().equals(m2.keySet()) = true m1.entrySet().equals(m2.entrySet()) = true m1.values().equals(m2.values()) = false   This is caused by the fact that AbstractCollection (which HashMap$Values inherits from) does not overrides #equals().  Do you have an idea why this is so ? Is it possible to take an image from the clipboard and output it to a file (using X)?  I can do this with text easily:  $ xclip -selection c -o &gt; file.text   But when I try the above with an image nothing is written.    The reason I want to do this is I don't have an image editor installed, and it got me thinking whether I could do this without installing one. How to maintain a file's 'last modified' date? I found native people prefer to use passive voice in a sentence. For example:  Tonight's moon can be seen from anywhere worldwide. We have reason to believe that progress can be made.  Be honest, it's difficult for an English learner to understand. Why not phrase it this way:  We can see the moon from anywhere worldwide.  In which cases do people prefer to use passive voice rather than active voice? From , page 209:     Dividends   Dividends are income paid by investments. Both bond funds   and stocks can  pay dividends. Bond fund dividends (the interest paid   by the individual  bonds in a fund) tend to be higher (as a percentage   of the amount you have  invested in a fund). When a dividend   distribution is made, you can receive  it as cash (which is good if   you need money to live on) or reinvest it into  more shares in the   fund. In either case, the share price of the fund drops to  offset the   payout. So if you’re hoping to strike it rich by buying into a bunch    of funds just before their dividends are paid, don’t bother. You’ll   just end up  paying more in income taxes.    It is talking about mutual funds.   I've never played with stocks.  However, I would like to know more about it.  Does the bold text imply the stock price will drop because of a company giving out a dividend? Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$ I recently came across this problem     Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times  }} = 3$   After trying it I looked at the solution from that book which was very ingenious but it was incomplete because it assumed that the limit already exists.  So my question is     Q2 Prove that the sequence$$\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots,\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}$$ converges.   Though I only need solution for Q2, if you happen to know any complete solution for Q1 it would be a great help .  If the solution from that book is required I can post it but it is not complete as I mentioned.   Edit: I see that a similar question was asked before on this site but it was not proved that limit should exist.