Spivak's Calculus on Manifolds - Problem 1-30.
Let $f$ be increasing on $[a,b]$ and $a < x_1 < \dotsb < x_n < b$. Show that $\sum_{k = 1}^n [f(x_k^+) - f(x_k^-)]\leq f(b^-) - f(a^+)$.
What does the Recycle Bin in Windows actually do?  Is it just a glorified folder and a holding place for soon to be deleted files, or does it do something specific?  Specifically, are files that are "moved" to the Recycle Bin actually moved on the hard disc, or are just the pointers to the files moved?  I am a fairly experienced user, I just wanted a more depth explanation of the Recycle Bin. DNS settings on managed switches in LAN We have a Windows 2008 server that is connected to a managed switch with about 15 Cat5e connections and three fiber branches that go to different parts of campus. At each of these parts there is another managed switch that branches out to the LAN. Our cable Internet comes into one of these locations, goes through the supplied modem and into a Sonicwall.  Question: Under DNS settings on all the switches, should I list the LAN Server IP and/or just the Gateway IP (Sonicwall)? How to download packages from a command-line given repository? It is not about the common download &amp; install thing, what apt (aptitude, etc) does, it is a scripted download a package file.  I found the dget tool for this task, which works quite well. But, unfortunately, it doesn't have any option to specify a different repository URL as it is given in the /etc/apt/sources.list.  Is it somehow possible? Actually, a dget-like functionality would be the best, but where I can also specify the repo URL.    P.s. thank you very much the scripts, but I would like to prefer a debian tool for this very simple task. It will be part of a buildscript, for external usage, and any unneeded complexity has intolerable collateral costs. It must be done by a debian tool.  P.s.2. Finally I solved this with updating the system-wide repos, and downloading the packages with dget. Thank you very much the scripts! How can I rewire half switched receptacles? In my bedroom the bottom half of all receptacles are controlled by one switch. Is it possible to have just one receptacle with top and bottom hot all the time?  The top half is hot all the time the bottom half is controlled by one switch.  Keep in mind there are six other receptacles in this room that are the same way controlled by the same switch. When I took the receptacle out the right side tab had been removed.  The top right screw had a black wire, the bottom right had a red wire, the top and bottom left screws had separate white wires.  I changed the receptacle out as suggested, I capped off the red wire and I capped off the white wire that was attached to the bottom left screw. I attached the other white wire to the top left screw and attached the black wire to the top right screw. I now have none of the receptacles working at all. What did I do wrong? Attacks of the MAC construction $\mathcal{H}(m\mathbin\|k)$ for common hashes $\mathcal{H}$? Consider a common practically-collision-resistant hash function $\mathcal{H}$ (e.g. , , , ), perhaps based on the  as are the first three. We define a  $\mathcal{C}$ $$(k,m) \mapsto \mathcal{C}(k,m)=\mathcal{H}(m\mathbin\|k)$$ where $\mathbin\|$ denotes concatenation, $k$ is a secret key (constant, or at least of fixed size), and $m$ is a message (possibly of variable length). Assume that an adversary can (iteratively) submit queries with $m_j$ and obtain $C(k,m_j)$, and wants to obtain $k$ or otherwise compute $C(k,m)$ for some $m\ne m_j$.  That MAC $\mathcal{C}$ is not trivially bad. In particular, if $\mathcal{H}$ was indistinguishable from a random function in the , $\mathcal{C}$ would be secure. And even though $\mathcal{H}$ may have the , it does not turn into a devastating attack on $\mathcal{C}$.  The less impractical generic attack that I see is that if a collision was known for $\mathcal{H}$ with the colliding messages of moderate identical length, it could be deduced countless collisions for $\mathcal{C}$. Hence security is demonstrably not better than collision-resistance of $\mathcal{H}$ (for identical-length messages). We could assume that $k$ is half the size of the result of $\mathcal{H}$, and hope that the security is about  or  or , 280, 2128, 2256 hash rounds for SHA-1, RIPEMD-160, SHA-256, SHA-512.   What are the known attacks against $\mathcal{C}$ (better than the above), and their cost, for each of these common hashes?  Is there hope for an argument that an attack against $\mathcal{C}$ would turn into an attack of similar cost against $\mathcal{H}$, or hint of the contrary?  Update: this  is of interest, but I fail to find that it really answers the present question.  Update 2: I am aware that the construction considered is weaker than , and in particular is vulnerable to collision on $\mathcal H$; I stated that, and that it is thus pointless to have the key wider hopeless to target security against some attacks better than half the hash's size. I'm asking exactly what cryptanalytic attack better than finding a collision on $\mathcal H$ there are. There is room for such an attack only by exploiting a weakness in the structure or/and the round function of a concrete $\mathcal H$. Already awarded bounty refunded on question migration? In March 2011 I offered a 500 bounty on one of my StackOverflow questions () and awarded it to Filip de Vos.   This question was later migrated to Database Administrators and has the following messages on it.     migrated to dba.stackexchange.com by Will♦ Dec 28 '11 at 12:25      This   question belongs on our site for database professionals who wish to   improve their database skills and learn from others in the community.   .     deleted Dec 31 '11 at 1:01   I appear to have been   on whereas Filip .  Is this a bug? Definitely doesn't seem fair or correct to me. The question still shows up in both our bounty tabs despite the migration. I would have hoped that the previously awarded bounty reputation points could still be honoured in this situation. There is a formal definition for the Big O notation in . Up to now I have come across Big O in Numerical Analysis, Calculus and Algorithms which all are pretty distinct fields. What I am wondering is if that definition is global and is the only one that is used in every field where the Big O is involved or there are other ways that the Big O is defined in other areas  of Mathematics. Automatically fit the graphics in the remaining space of a Beamer slide I seem to have screwed up something that allows you to inset evenly, I think I have tried every combination of settings in inset and pivot center and even is far from even.  I tried uninstalling and re installing blender 2.71 and get the same result.  The image attached is a simple default cube that I tried to inset evenly.  I have also attached the blend file link  help?? I showed up here to write a perfectly normal rant about misaligned pixels, and there's a big fat checkbox cluttering up the "Ask" page:    WHAT THE EEEEK IS THIS NEW ABOMINATION? I noticed with the holiday patch that there are new trees and shrubs that spawn on my land.  One popped up today, being specifically an X-Mas tree - which costs 25,000 gold to remove.   Is there any benefit to removing this X-Mas tree?  Is there any benefit to leaving it there?    It is noticeably the most expensive obstacle to remove yet. How can I install a Canon printer or scanner driver? My laptop has a function key which turns F keys into keys which change volume or brightness. I would like to swap a number of them so that they perform that function under normal operation and are an Fx key while the function key is being held.  I assumed that the function keys were just normal keys codes with the set of keys being large enough to include all the possible speical keys. I tried to so as this tutorial suggests () and use xev to obtain the key codes and then would use xmodmap to swap their value.  This does not seem to work though. The F12 key gives the xev output:  KeyPress event, serial 41, synthetic NO, window 0x4a00001,     root 0xb1, subw 0x0, time 468005, (23,-15), root:(89,37),     state 0x10, keycode 96 (keysym 0xffc9, F12), same_screen YES,     XLookupString gives 0 bytes:      XmbLookupString gives 0 bytes:      XFilterEvent returns: False   Which I think is to be expected. However, the Increase Volume button give this response:  KeymapNotify event, serial 41, synthetic NO, window 0x0,     keys:  65  0   0   0   0   0   0   0   0   0   0   0   0   0   0   0               0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0    And so does the Decrease Volume key. How can I perform the swapping of the keys that I would like?  Thank you Why isn't $\lim \limits_{x\to\infty}\left(1+\frac{1}{x}\right)^{x}$ equal to $1$? Using special auto start servlet to initialize on startup and share application data I read about Java's type erasure .  When does type erasure occur? At compile time or runtime? When the class is loaded? When the class is instantiated?  A lot of sites (including the official tutorial mentioned above) say type erasure occurs at compile time. If the type information is completely removed at compile time, how does the JDK check type compatibility when a method using generics is invoked with no type information or wrong type information?  Consider the following example: Say class A has a method, empty(Box&lt;?&nbsp;extends&nbsp;Number&gt;&nbsp;b). We compile A.java and get the class file A.class.  public class A {     public static void empty(Box&lt;? extends Number&gt; b) {} }     public class Box&lt;T&gt; {}   Now we create another class B which invokes the method empty with a non-parameterized argument (raw type): empty(new Box()). If we compile B.java with A.class in the classpath, javac is smart enough to raise a warning. So A.class has some type information stored in it.  public class B {     public static void invoke() {         // java: unchecked method invocation:         //  method empty in class A is applied to given types         //  required: Box&lt;? extends java.lang.Number&gt;         //  found:    Box         // java: unchecked conversion         //  required: Box&lt;? extends java.lang.Number&gt;         //  found:    Box         A.empty(new Box());     } }   My guess would be that type erasure occurs when the class is loaded, but it is just a guess. So when does it happen? Most of the time, it should be really easy to determine who is my ally and who is not. The people who try to kill me are most likely my foes, whereas the people who entered the dungeon with me are probably my allies.   But sometimes it’s not that simple. What if one of my party members got replaced by an evil doppelganger, who plans to kill me in the near future? What if one of my allies infiltrated the court of the evil king and I accidentally have to fight her, while she is (unbeknownst to me) under the effect of a  spell? What if a character fell in love with the BBEG and just wants to save him from himself, while the BBEG hates the character and actively tries to murder him?  Most spells and abilities that affect others specify the targets as “creatures” or “willing creatures”. Some explicitly affect allies only. That’s why I want to know, how to determine who’s allied with whom. Do I have to think that someone is my ally for someone to count as my ally? Does someone else have to believe to be my ally in order to count? Do both of us have to agree on being allies? Is there an objective observer who determines who’s allied? Can Person A be the ally of Person B, while Person B is not the ally of Person A?  There are many examples for spells/abilities that care about allies, among others:   the Exalted Champion feature from the  (SCAG, p. 132-133), that grants allies within 30 feet advantage on Wisdom saving throws. the Inspiring Surge feature from the  (SCAG, p. 128), that allows one or two allies within 60 feet to make a melee weapon attack as a reaction.   With features like this, you could accidentally affect someone who’s not actually your ally. You could fail to affect someone who is actually your ally, but you don’t know it (yet). Or you could affect someone who doesn’t want to be affected (because they don’t want to be allied with you). This all depends on the interpretation of the word “ally”. I’m not sure if "ally" is a defined game term (but I don’t think so), or if I have to rely on the dictionary definition. That’s why I want to ask:  How do I determine who’s allied with whom? Which derivation of drift velocity is correct? In the derivation of drift velocity I have seen two variations and want to know which one's correct.   $s=ut+\frac{at^2}{2}$ Assume that the drift velocity of any electron in any conductor is : $$\vec{v_d}=\frac{\vec{l}}{t}$$ Due to the electric field the acceleration of electrons in any conductor is: $$a=\frac{-e\vec{E}}{m}$$ Now the distance travelled by an electron after a long time (initial thermal velocity = 0) $$l=\frac{at^2}{2}\implies \frac{-e\vec{E}t^2}{2m}$$ the time between the collisions is $\tau$ $\therefore$ $$l=\frac{-e\vec{E}\tau^2}{2m}$$ thus the velocity is $$\vec{v_d}=\frac{-e\vec{E}\tau}{2m}$$ In another proof I saw the author using $v=u+at \implies v_d=\frac{-e\vec{E}\tau}{m}$   My question is which of the the two equations of motion can be used in the proof? Can they be used at all. Java generics super keyword How can I block app and game invitations once and for all in Facebook? I want to block app and game invitations once and for all in Facebook. How can I do it so they cannot disturb me anymore?