For the function $F(x) = \int_0^x |\cos(1/u)| du$, I want to determine if the derivative exists at $x=0$ and find $F'(0)$.  I figured this out when the absolute value is removed using integration by parts:  $$\frac{F(h) - F(0)}{h} = \frac{1}{h}\int_0^h\cos(1/u)du = \frac{1}{h}\int_{1/h}^\infty\frac{\cos(x)}{x^2}dx \\= - h \sin(1/h) + \frac{1}{h}\int_{1/h}^\infty\frac{2\sin(x)}{x^3}dx = -h \sin(h) + \frac{1}{h}O(h^2)$$  So $F'(0) = 0$ here.  This breaks down for $F(x) = \int_0^x \cos(1/u)du$.  Thank you for any help.
Does the limit $\lim _{x\to 0} \frac 1x \int_0^x \left|\cos \frac 1t \right| dt$ exists ? If it does then what is the value ?   I don't think even L'Hospital's rule can be applied . Please help . Thanks in advance
Unable to install any updates through update manager/apt-get upgrade I have been running Ubuntu 12.04 for the past few weeks. I recently tried to download and install some files through the Update Manager. It located a few files, downloaded them, but then failed to install them. It gave the error:  installArchives() failed: (Reading database ...  (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error   I tried using apt-get upgrade but a similar error was produced:  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be upgraded:   icedtea-6-jre-cacao icedtea-6-jre-jamvm openjdk-6-jre openjdk-6-jre-headless   openjdk-6-jre-lib 5 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/34.6 MB of archives. After this operation, 1,024 B of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2)   Any help would be greatly appreciated. Thanks in advance.  Edit: Here's some extra information that may be useful. My original problem was that the Ubuntu Software Centre would not open. I found that the Update Manager and Synaptic Package Manager also failed to open. They all gave an error along the lines of:   "installArchives() failed: dpkg: error: reading package info file '/var/lib/dpkg/available': Input/output errorError in function:"   After much googling I seemed to fix that problem as the package managers would open again. When I then tried to download and install any updates from the Update Manager I got the error at the beginning of this post. Firmware vs. Software I'm trying to typeset SuperCollider code using the listings package.  Any "freestanding" identifiers that begin with a capital letter are class names, and should be highlighted; by "freestanding", I mean outside of strings or comments. I would be tickled pink if listings could let you define identifiers in terms of regexp, e.g. [A-Z][A-Za-z0-9_]* but the manual doesn't suggests that as a possibility.  SuperCollider has some 2300 classes... I really don't relish listing all of them literally in my preamble. I guess I could extract them by hand for each individual listing, but I'd rather not have to.  Is there a way to do this without a really massive morekeywords expression? (Perhaps using another package?)  Sample SuperCollider code:  p.clear;  ~grains.addSpec(\tfreq, [1, 40, \exp]); ~grains.addSpec(\overlap, [0.1, 10, \exp]); ~grains.addSpec(\pos, [0, b.duration]);  // 3.43 is nice! ~grains.addSpec(\rate, [0.5, 2, \exp]); ~grains = { |tfreq = 25, overlap = 6, pan = 0, amp = 0.2, pos = 3.43,    rate = 1|    var trig = Impulse.ar(tfreq);    TGrains.ar(2, trig, b, rate, pos, overlap / tfreq, pan, amp) }; ~grains.play;   Impulse and TGrains should be highlighted in blue. I have dozens of other listings using different capitalized keywords. Why can't I use the Delete key to delete in Photoshop CS6? The previous version of Photoshop I used was CS4 and whenever I selected a portion of an image with the magic wand and hit Delete, it simply deleted that portion of the image (rendering it transparent). Easy!  In CS6, this is not the case. Now when I select a portion of an image and hit Delete, it brings up the Fill dialogue.  For example I have this picture (it's a Vauxhall Corsa). I want to use the Magic Wand to render anything that isn't part of the car transparent, so I start by highlighting some clouds with the Magic Wand like so:     Click on images to see larger versions  Now when I hit Delete, instead of that area of the picture being deleted as I would expect based on my experience with CS4, I get this dialogue:    My questions are:   Can this dialogue be used to achieve the functionality I want? If so, how? If not, how do I delete the background in CS6? comparing the first column of two files and printing the entire row of the second file if the first columns match I have two files in these formats:   file1 :  air smell hand dude road cat   file 2:   air,4,21,01,13,3,2 smell,21,4,2,5,6 dude,1,31,42,1 road,1,4,2,1,4 cat,1,5,6,3,1 hand,1,4,2,1,6 mouse,1,3,5,6,2   what I want to do is print the entire row of file 2, if the first string in column 1 of file 2 is found in file 1, and I want to keep the order of file 1.   expected output:   air,4,21,01,13,3,2 smell,21,4,2,5,6 hand,1,4,2,1,6 dude,1,31,42,1 road,1,4,2,1,4 cat,1,5,6,3,1 How do I say what is the current UTC in eastern zone I'm curious to know how important the bias node is for the effectiveness of modern neural networks. I can easily understand that it can be important in a shallow network with only a few input variables. However, modern neural nets such as in deep learning often have a large number of input variables to decide whether a certain neuron is triggered. Would simply removing them from, e.g., LeNet5 or ImageNet have any real impact at all? In a  a doubt was voiced about my answer, which claims GCM to requires $2^{128}$ for a successful forgery. The doubt was that the square root needs to be taken meaning the security would be $2^{64}$.  So of course I immediately checked the relevant security theorem ():  $$\mathbf{Adv}^{\text{auth}}_{\operatorname{GCM}[\operatorname{Perm}(n),\tau]}(\mathcal A)\leq \frac{0.5(\sigma+q+q'+1)^2}{2^n}+\frac{q'(\ell_A+1)}{2^\tau}$$  with $\sigma$ being the total plaintext size in blocks, $q$ being the total number of encryption queries, $q'$ being the total number of decryption queries, $n$ being the size in bits of the underlying permutation, $\ell_A$ being maximum authenticated input length in blocks and $\tau$ being the tag size in bits.  Now clearly from this we can see that performing $2^{n/2}=2^{64}$ queries yields a strong enough advantage for a break.  Now my question is: When talking about "n-bits of security" or "needs $2^n$ operations to break", do we usually talk about "online" security, ie queries to an oracle can be done and cost $1$ or "offline" security where no oracles are available or do we need to make this decision depending on the situation?  TL;DR: Does GCM offer 64 or 128 bit security? Should 'fuck' be present in review audits (intentionally fake suggested edit reviews)? Why would Voldemort spare Lily? In one of Harry's dementor-induced flashbacks, he presumably hears his mother protesting against Voldemort:  Harry was falling again through thick white fog, and his mother’s voice was louder than ever, echoing inside his head – ‘Not Harry! Not Harry! Please – I’ll do anything –’ ‘Stand aside – stand aside, girl –’ ‘Harry!’  Voldemort tells her to stand aside - why did he not just kill her? Approximation of real number to the power of irrational Say I have two numbers, $c\in\Bbb R^+\,,q\in\Bbb{R\setminus Q}$.  Is there a way to approximate the value of $c^q$?  This question bugging me because of how simple it sound in compare to how complicated it is. First I want to understand what a power of irrational number is, when talking about $\Bbb R^+$(I use this domain for sake of simplicity) we have very well defined actions for power a rational number, we can even use some theorems to get $c^{a/b}=\sqrt[b]{c^a}$, but when I am dealing with irrational number I can not do this.  I know that we have some definition for this because the solution of the equation $2^x=3$ is irrational.  So what exactly does it mean to take something to a irrational power and is there a way to approximate a numeric value for this? Is there a way to recover deleted data from a SSD / USB Flash drive?  All I could find were recovery for damaged drives, but I want to recover deleted data.  So how do I do this? Does someone know if the following is true:  Let $\mathbb{X}$ be some arbitrary Banach space. $\{x_k \}_{k=1}^{\infty} \in \mathbb{X}$ is a sequence chosen from $\mathbb{X}$.  Now, if the series  $$\sum_{k=1}^\infty \|x_k\|_X$$ converges, would the "more generic" series $$\sum_{k=1}^\infty x_k$$  also converge?  If yes, could you please give the proof (or just mark the proof steps) ?  Thank you. Suppose we have a set $S$ of real numbers. Show that   $$\sum_{s\in S}|s-x| $$   is minimal if $x$ is equal to the median.    This is a sample exam question of one of the exams that I need to take and I don't know how to proceed. I have a list of cards for a deck I want to build or buy. What should I do to check whether that deck is legal in a particular format (Standard, for example)? How to ignore everything in the document environment? Why my differential equations become True? It's become clear that last night's (Tues Jan 8th 2019) patches to Windows 7 broke SMB2 networking in Lanman Server.  One or both of KB 4480970 and KB 4480960 results in any OS that uses SMB2 (such as Windows XP, some Windows 7, and same generation server OS' and Mac OS X machines (certainly 10.10, 10.11 not sure about others yet)) being unable to connect to Windows 7 Pro shares.  Client machines repeatedly prompt for login despite credentials being correct. I have 16GB RAM. Do I need 32GB swap? I read many places that the rule of thumb for swap space is to double the amount of physical RAM. However, 32 GB does seem a LOT. Do I need that much? Do I need it at all with this high amount of physical RAM? Where are some good places to find royalty free stock images?