Proof of Lebesgue Density Theorem
Lebesgue density theorem in the line
Is Angular Momentum truly fundamental? This may seem like a slightly trite question, but it is one that has long intrigued me.  Since I formally learned classical (Newtonian) mechanics, it has often struck me that angular momentum (and generally rotational dynamics) can be fully derived from normal (linear) momentum and dynamics. Simply by considering circular motion of a point mass and introducing new quantities, it seems one can describe and explain angular momentum fully without any new postulates. In this sense, I am lead to believe only ordinary momentum and dynamics are fundamental to mechanics, with rotational stuff effectively being a corollary.  Then at a later point I learned quantum mechanics. Alright, so orbital angular momentum does not really disturb my picture of the origin/fundamentality, but when we consider the concept of spin, this introduces a problem in this proposed (philosophical) understanding. Spin is apparently intrinsic angular momentum; that is, it applies to a point particle. Something can possess angular momentum that is not actually moving/rotating - a concept that does not exist in classical mechanics! Does this imply that angular momentum is in fact a fundamental quantity, intrinsic to the universe in some sense?  It somewhat bothers me that that fundamental particles such as electrons and quarks can possess their own angular momentum (spin), when otherwise angular momentum/rotational dynamics would fall out quite naturally from normal (linear) mechanics. There are of course some fringe theories that propose that even these so-called fundamental particles are composite, but at the moment physicists widely accept the concept of intrinsic angular momentum. In any case, can this dilemma be resolved, or do we simply have to extend our framework of fundamental quantities? I have these two CREATE TABLE statements:   CREATE TABLE GUEST (   id int(15) not null auto_increment PRIMARY KEY,   GuestName char(25) not null );  CREATE TABLE PAYMENT (   id int(15) not null auto_increment   Foreign Key(id) references GUEST(id),   BillNr int(15) not null );   What is the problem in the second statement? It did not create a new table. Is there a clear set of conditions under which lasso, ridge, or elastic net solution paths are monotone? The question  demonstrates solution paths for the lasso estimator that are not monotonic. That is, some of the cofficients grow in absolute value before they shrink.  I've applied these models to several different kinds of data sets and never seen this behavior "in the wild," and until today had assumed that they were always monotonic.  Is there a clear set of conditions under which the solution paths are guaranteed to be monotone? Does it affect the interpretation of the results if the paths change direction? After Doomsday was defeated, a spec-ops team sent to the Kryptionian ship in Metropolis found Luthor submerged in water, in front of a... thing, which "held" three metal cubes. The thing looked up, roars, and dissipates into thin air.    What the hell was that all about? I am really confused. What was that creature and what was Luthor doing with it? How align 2 adjacent divs horizontally WITHOUT float? av_acharya@AnoxBotox:/$ wireshark qt.qpa.plugin: Could not find the Qt platform plugin &quot;xcb&quot; in &quot;path/to/plugins&quot; This application failed to start because no Qt platform plugin could be initialized. Reinstalling the application may fix this problem.   sudo apt-get install --reinstall qt5-default Reading package lists... Done Building dependency tree        Reading state information... Done The following NEW packages will be installed:   qt5-default 0 upgraded, 1 newly installed, 0 to remove and 23 not upgraded. Need to get 0 B/24.4 kB of archives. After this operation, 170 kB of additional disk space will be used. Selecting previously unselected package qt5-default:amd64. (Reading database ... 241279 files and directories currently installed.) Preparing to unpack .../qt5-default_5.12.8+dfsg-0ubuntu1_amd64.deb ... Unpacking qt5-default:amd64 (5.12.8+dfsg-0ubuntu1) ... I've sectioned of this area in the sea near my base, and I need to get rid of all the water from it.  From what I've tried the only way I can see to clear it is fully cover one layer, delete the blocks, then go down to the next layer. Is that the only way I can clear the area or is there a better approach?  Ideally I'd like a survival method (if there is a better one), but any commands what could be used I'd like to hear just so I can learn more commands :) Transferring Lyx Keyboard Shortcuts I recently updated my version of Lyx from 1.6.7 to 2.0.6. I would like to transfer my old keyboard shortcuts to this new version, but I am having difficulty because I was inconsistent with the way that I set them up:   Some were defined in the specified .bind file,  but others were modified in the lyx preferences menu.   I found that when importing my old .bind file into this updated version of lyx, it imported the keyboard shortcuts that were in the .bind file, but not those that I had added in the preferences menu.  Does anyone know if these keyboard shortcuts that were added via the preferences menu are stored in another .bind file somewhere in the lyx directory? Has anyone else successfully transferred keyboard shortcuts that had been implimented in the preferences menu? I read the following dev email  and when i read this part:  Note that this does not completely remove the ability for extensions (e.g. Dropbox) to add emblems programmatically by using the libnautilus-extension library, just the user-added emblems from the properties dialog/emblems sidebar.   There is ability to bring emblems programmatically. So, How can i use libnautilus-extension library to add some emblems like Dropbox does? Or is there any tweak tool for this? I have read that a holomorphic function in an open disc has a primitive in that disc. Does this mean that for any holomorphic function $f$ in any open set $U,$ that $f$ has a primitive? I say this because I can take a large enough open disc to cover the open set $U$ and find its primitive there. Is this a correct way of thinking? Explicit formula for Bernoulli numbers by using only the recurrence relation It is not hard to show, by induction on $m\in\mathbb N$, that there exist a sequence $(B_n)_{n\geq0}$ of rational numbers such that  $$\sum_{k=1}^nk^m=\frac1{m+1}\sum_{k=0}^m\binom{m+1}kB_k\,n^{m+1-k},\ \style{font-family:inherit;}{\text{for all}}\ n\geq1\,.$$  These are the Bernoulli numbers of the second kind, that is, $B_1=+1/2$. A corollary of the proof (by induction) of the fact above is a recurrence formula for such numbers $B_n$, which are known as Bernoulli numbers:  $$\sum_{k=0}^{n-1}\binom nkB_k=0\,.\tag{$\ast$}$$  On the other hand, there are a number of explicit formulae for $B_n$, which are obtained using the equality $\frac x{e^x-1}=\sum_{n=0}^\infty B_n\,\frac{x^n}{n!}$; see  and . For example:  $$B_n=\sum_{k=0}^n\frac1{k+1}\sum_{r=0}^k(-1)^r\binom krr^n\,.\tag{$\ast\ast$}$$  Is there some clever way to manipulate the recurrence $(\ast)$ in order to obtain $(\ast\ast)$? Get exit status of process that's piped to another I have two processes foo and bar, connected with a pipe:  $ foo | bar   bar always exits 0; I'm interested in the exit code of foo. Is there any way to get at it? A Terms of Service update restricting companies that scrape your profile information without your permission Determine $4$ specific digits in $34!$ Find the values of $a,b,c,d\in\mathbb{N}$ such that      $$ 34!=295232799cd9604140847618609643ab0000000 $$   My Attempt:  The factorial of $34$ contains a $3$, so the RHS must be divisible by  $3$. Similarly, it must be divisible by $7$, $11$, $13$, $19$ etc.  But I do not understand how can I calculate $a,b,c,d$ in this equation. If $|f(z)|\le |g(z)|$, can we say $f$ is constant? Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?  Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$? Inequality Regarding e Over at , there is overlapping text in the Database Schema under Close-As-Off-Topic-Reason-Types:  that makes it really hard to read.  This is also happening underneath &quot;Pending Flags&quot;, and seemingly with every long line in the schema.  This would be fine if you could increase the width of the list as you can with the height, which would in theory separate the lines, but you can't.  Can we fix this, please?  UPDATE: I have found that  post is almost exactly the same as mine, but steady on with duplicate votes.   comment says that it will be fixed, but that question was asked more than five years ago, and in that time the problem has not been resolved.  UPDATE 2: It seems that  would help.  I changed a little CSS/HTML, and came up with this:  Although mine is a little cruder than it could be.  This is just an example. How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $? In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $ P \rightarrow Q $ is equivalent with $\neg P \lor Q$.   that gives as answer: 'just compare the truth tables and you can see that they are equivalent'. However, I would like to know how to prove this using formal propositional logic. This should be possible, right? I'm very confused about tenses.  I have examples in two different situations here. Situation 1: I went to New York two months ago and was talking to someone about it.  A. This is the first time I've been to New York. B. That was the first time I'd been to New York.  Which is correct in a daily conversation?  Situation 2: I'm going to New York next month and talking to someone.  A. This will be the first time I've been to New York. B. This is the first time I've been to New York. C. This will be the first time I go to New York.  Which sentence is appropriate?  According to a grammar article I've read before, I think it says that I can use this sentence in both cases:  This is the first time I've been to New York.  But I'm not sure if this is really so. So I'd like to know which tense I should use in each case. Could you explain it? I am trying get an unused server up and running, it booted fine but i was greeted with the login window and none of the known passwords seem to work. The server I am trying to get to work is not a Domain Controller and has not been used in a long time.  OS: Windows 2003 Server R2 Enterprise Edition  I tried Ophcrack XP (there is no WinServer version of Ophcrack) - it couldn't locate the partitions with hashes. I tried mounting the partitions myself (#mount /dev/hda*x* /mnt/ntfs), but it gave me an error saying /dev/hda*x* didn't exist.  Next I tried booting into a Ubuntu 10.04.3 LiveCD so that I could install and use chntpw - no go, Ubuntu Live wouldn't boot.  Based on what I've read up the following tools may be of help:   Windows Password recovery Petter Nordahl-Hagen's Offline NT Password &amp; Registry Editor Openwall's John the Ripper EBCD – Emergency Boot CD - Bootable CD (R)ecovery (I)s (P)ossible Linux rescue system   I am hoping that someone here has prior experience with Win2k3 Administrator password resets so that I don't end up having to try all of the above to find out which one works.