Given two elements $a,b$ in the Euclidean ring $R$ their least common miltiple $c\in R$ is an element in $R$ such that $a\mid c$ and $b\mid c$ and such that whenever $a\mid x$ and $b\mid x$ then $c\mid x$.   a) Prove that any two elements in the Euclidean ring $R$ have a least common multiple in $R$.  b) If the least common multiple of $a$ and $b$ is denoted by $[a,b]$, prove that $[a,b]=ab/(a,b).$   Proof:   (a) Consider the ideals $(a)$ and $(b)$, where $(x)=xR$. Their intersection $(a)\cap (b)$ is also ideal of $R$ since Euclidean ring is the ring of principle ideal then $(a)\cap (b)=(c)$ for some $c\in R$. I proved that this $c$ has all needed properties of least common multiple. Thus the LCM of $a$ and $b$ exists.  (b) We know that $\text{gcd}(a,b)$ exists in Euclidean ring for any $a,b\neq 0 \in R$. But I have no thoughts how to prove that $[a,b]=ab/(a,b)$.   So I would be very thankful answer.
Let $R$ be a UFD. Then for any two elements $a,b \in R \setminus \{0\}$ $ab \sim \gcd (a,b){\operatorname{lcm} (a,b)}$.   My attempt :  Let $d = \gcd (a,b)$.Then $d|a$ and $d|b$.Then $\exists x,y \in R$ such that $a = dx$ and $b = dy$.Now it is to be shown that $c = dxy$ is the lcm of $a,b$.Which I find difficulty to show.It is clear that $a|c$ and $b|c$. Now if $u$ is any common multiple of $a$ and $b$.Then $dx|u$ and also since $y|b$ we have $y|u$ i.e. $dxy|u^2$ $\implies c|u^2$.But I have to show $c|u$ to complete the proof which I fail to prove.Please help me to complete the proof.  Thank you in advance.
What do *args and **kwargs mean? What exactly do *args and **kwargs mean?  According to the Python documentation, from what it seems, it passes in a tuple of arguments.  def foo(hello, *args):     print hello      for each in args:         print each  if __name__ == '__main__':     foo("LOVE", ["lol", "lololol"])   This prints out:  LOVE ['lol', 'lololol']   How do you effectively use them? I am confused by the orientation of a topological manifold. My understanding is: An orientation of a topological manifold is a choice of generator of the $H^n(M,\mathbb Z)$. So given a manifold, we could have 2 orientation defined on the manifold. For $\mathbb{CP}^2$ on orientation is determined by the complex structure, the other orientation is denoted by $\overline{\mathbb{CP}}^2$. And it is well known that there is no orientation reversing map from $\mathbb{CP}^2$ to itself.  My question is: are they homeomorphic? I guess they are not homeomorphic. But I am confuesed, doesn't the orientations defined on the same manifold, how come after reversing the orientation they become not homeomorphic? For bash script, I can use "$@" to access arguments. What's the equivalent when I use an alias? I've lost my original power adapter, and I can't find the proper one in any of the stores currently available.  I've HP Compaq 6730b, and as per  it outputs 18.5V 3.5A. However, I've found an older power adapter I have which outputs 19.5V 4.62 A.  So, my question is: how much of a difference between those two it really is, and more importantly, how much of a risk would it be if I tried the power adapter I already have.  P.S. I know pretty much nothing about electricity/electronics. Why do computers only use 0 and 1? Why do computers only use 0 and 1? Won't the addition of other numbers such as 2 or 3 speed up computers? Also, 2 and 3 can be used to shorten the bit-length of integers (2 and 3 can be used to end an integer, so that the number 1 only needs one two bits.)..  Why is binary computer more preferred? WinForms: How to programmatically fire an event handler? I want to programmatically invoke an event handler for a control. For example:  DateTimePicker dtpLastConsummated;   I want to trigger the TextChanged event handler for the dtpLastConsummated, how can i do it?  In other languages I would call something akin to:  dtpLastConsummated.TextChanged(this, new EventArgs());   but in .NET you can have multiple event handlers:  dtpLastConsummated.Click +=new EventHandler(dtpLastConsummated_TextChanged); dtpLastConsummated.Click +=new EventHandler(dtpLastConsummated_AnotherHandler); dtpLastConsummated.Click +=new EventHandler(dtpLastConsummated_MoreHandlers); ... dtpLastConsummated.Click +=new EventHandler(dtpLastConsummated_Nminus1);   so you need a way to trigger all the attached event handlers.    Answer  The following code will fire the event:  Toolkit.FireEvent(dtpLastConsummated, "TextChanged", new EventArgs());   And here's the code of the static toolkit function:  /// &lt;summary&gt; /// Programatically fire an event handler of an object /// &lt;/summary&gt; /// &lt;param name="targetObject"&gt;&lt;/param&gt; /// &lt;param name="eventName"&gt;&lt;/param&gt; /// &lt;param name="e"&gt;&lt;/param&gt; public static void FireEvent(Object targetObject, string eventName, EventArgs e) {    /*     * By convention event handlers are internally called by a protected     * method called OnEventName     * e.g.     *     public event TextChanged     * is triggered by     *     protected void OnTextChanged     *      * If the object didn't create an OnXxxx protected method,     * then you're screwed. But your alternative was over override     * the method and call it - so you'd be screwed the other way too.     */     //Event thrower method name //e.g. OnTextChanged    String methodName = "On" + eventName;     MethodInfo mi = targetObject.GetType().GetMethod(          methodName,           BindingFlags.Instance | BindingFlags.NonPublic);     if (mi == null)       throw new ArgumentException("Cannot find event thrower named "+methodName);     mi.Invoke(targetObject, new object[] { e }); }     Note: I'm not creating a subclass of every control in the .NET framework, and every 3rd party control, and convincing an enterprise worth of developers to retrofit every form to use my custom controls. Why is the unit for capacitance so large? Is there any way to increase the number of candy made? In Candy Box 2, the candies are being produced at a rate of 1 candy per second. Is there any way to increase the potential amount of candy made at a faster rate? (notice that I am only up to the part where the person with the cellar in their house gives me a map. Factorial of $i$ With $i^2=-1$, what is $i!$ ?  I'm not sure if it is $i$ or if it is an infinite product $i(i-1)(i-2)(i-3)(i-4)(i-5)\dots$  It would make sense either way.  Is it not defined? Fully transparent windows in Pygame? Is it possible to get a fully transparent window in Pygame (see the desktop through it)?  I've found how to create a window without a frame, but there doesn't seem to be any obvious way to make it transparent.  I'd be willing to tie into system-specific technology/frameworks as long as there are solutions for both Windows and Mac OS X, but I'm not sure which direction to be looking.  The only topic I was able to find , which isn't something I can do for this particular project (needs to be Pygame). How do I install Waterfox? Dealing with splits in my ground plane Whenever I'm making a cheap, 2-sided PCB, I run into this problem:    Where I have signal lines (on the top layer here) running over top of a big fat power trace (on the bottom layer).  Now, this is obviously terrible because there is no good return path on the bottom side.  For issues like this, Hott consultants has a good recommendation:    However, I wonder if a solution like the one below is also acceptable.    My intuition tells me yes, this solution is ok.  Over the break in the ground plane, the current will hop over the vias and then back down to the lower plane.  Some back of the envelope math tells me that this adds a trivial amount to the loop area, and therefore a trivial amount to the return impedance.  Can someone tell me...   How right / wrong my intuition is here about using vias? If my intuition is wrong, why? Also, if my intuition is wrong, what is the correct solution in this sort of situation (2 layer, with power traces under signal)?   For the record, these particular signal lines aren't terribly fast (500 kHz), but I would like to know how to deal with this situation when I have signals in the 5 - 10 MHz range.  Edit  Below is a screenshot of about 60% of the board.  Maybe it will help with the critique My table is like this:  Sitecode    Month  Amount --------    -----  ------ XX          Jan     1000 XX          Jan     3000 XX          Apr     3000 XX          Apr     1000   What I want is, to show the result something like this:  Sitecode    MonthJAN   MonthAPR --------    --------   -------- XX          4000       4000 An example of a regular function over an open set I'm trying to setup OpenVPN using NetworkManager. The GUI seems buggy and unresponsive. Are there any other tools for setting up a VPN client? An excited electron loses energy in the form of radiation. The radiation constitutes photons which move at a speed $c$. But is the process of conversion of the energy of the electron into the kinetic energy of the photon instantaneous? Is there a a simple way to visualize this process rather than math? I plugged in my external harddrive (which was formatted on my Mac into HFS+ journaled) to my Ubuntu desktop 9.04 64bit.  I am not able to get the drive to mount with write capability, how do I do that?  Right now all I'm getting is read access, I tried  sudo mount -t hfsplus /dev/sdf2 /media/"Portable HD"   but that still gave me only read access...  ideas?? Boats from Manaus, Brazil to Leticia, Colombia? I've heard of and met a few people who have done this route, and I hope to do it at the end of this year or early next year.  However, the descriptions were mostly sketchy, and I'm wondering if it's a regularly scheduled service, or more ad-hoc.  So I guess the question is, how can I find out which boats go from Manaus to Leticia, and when they go?  Ideally this wouldn't be a tour, but a 'locals' boat. I look today on a few fresh question, and the view count was 0. Even after I refreshed the page a few time. This is definitely a bug, since the question isn't mine.  Also I don't think the all page was cached, since that the DateTime mark was change every reload.  But maybe the SQL result is cached. Does C1 code coverage analysis exist for Ruby? I'm currently using  to get C0 code coverage analysis for a rails project that I'm working on.  However, those results are practically meaningless- I have 100% coverage according to rcov (as it only covers C0 analysis) and I've barely written half the test cases for the functionality that exists thus far.  I'm used to the useful results from the code coverage in Visual Studio 2008 Team, which has C1 coverage.  Are there any tools that provide similar coverage for ruby?