Show that $\mathfrak{so}(4)\cong \mathfrak{so}(3)\oplus \mathfrak{so}(3)$
Isomorphism between $\mathfrak o(4,\mathbb R)$ and $\mathfrak o (3,\mathbb R) \oplus\mathfrak o (3,\mathbb R) $
If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$ If $I$ and $J$ are ideals in a ring $R$ with 1 which are  co-maximal, i.e $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n$ in $\mathbb{N}$  Work done:     Should I proceed using    By taking $$K=I^m+J^m \neq R$$  Let $$S=\{I,J \in R \mid I^m+J^n \neq R \}$$ be the required partially order set ordered by inclusion then  $$I+J \subseteq I^n+J \subseteq I^n+J^m \subseteq \ldots $$  Then, I think this chain doesn't have an upper bound...as $I^m+J^n \neq R$ for all $ m,n \in \mathbb N$. Am I right??  Thanks for any help!! The title pretty much says it all: how can I type Unicode characters - specifically the em-dash —? Mean and variance of inverse of a normal RV My random variable $X$ is normally distributed with mean $b$ and variance $p$. I defined a new RV, $Y$, such that $Y=1/X$. Does anyone know how to find the mean of $Y$ and $Y^2$? Because I'm still living like it's 2008, I run a website with a .in TLD. The website has nothing to do we India and does not target Indian users.  Unfortunately, when I go to Google Webmaster Tools I see that Google have locked my geotargeting status to India:    On the , Google state that:     Sites with country-coded top-level domains (such as .ie) are already associated with a geographic region, in this case Ireland. In this case, you won't be able to specify a geographic location.   This seems implausibly inflexible, so I want to know if there are any sneaky ways of removing auto-geotargeting, or changing it to a more appropriate region? I have a dual boot setup with Ubuntu and Windows 7. I need to shrink my root Ubuntu volume in order to get some free space to create a new partition. Are there any Windows based tools that will do this for me? I tried booting up in gparted to do this but I cannot tell which partition is which since I cannot find the size (it displays everything in blocks). Cardinalities of the intervals $(-1,1)$ and $[-1,1]$ Proper way to handle multiple forms on one page in Django Python: Finding differences between elements of a list Good practices for organizing notes I am doing some arcpy -functions and afterwards I'm trying to add the calculated results from one table to another table. I'm using the arcpy.AddJoin_management() function.   As input table I'm using a table which is stored in a personal geodatabase. The personal geodatabase is not the workspace environment! As "JoinTable" a Table of a file geodatabase is used. This file geodatabase is the workspace geodatabase environment.  The initialized function looks like this:  (%self.getWorkspaceEnvironment => "W:/a18fb/data/mypersonalgeodatabase.mdb" and tFactsheetHochwasser => table in the personal geodatabase)  arcpy.AddJoin_management("%s/tFactsheetHochwasser"%self.getWorkspaceEnvironment(),"KPG", "GebaeudeKtn_BWVred1Stat","KPG","KEEP_ALL")  If the code is executed the following error-message appears:  Failed to execute.Parameters are not valid. The value cannot be a table     ERROR 000840: The value is not a Raster Layer.      ERROR 000840: The value is not a Raster Catalog Layer.      ERROR 000840: The value is not a Mosaic Layer.      Failed to execute (AddJoin)`. Can the links in the "possible vandalism: edits" flag point to the revision rather than the post When we get the following flag:     possible vandalism: edits (auto)   We see a set of links to the posts that have been edited.  This means that to check that edit is valid we have to click the link to the post then click the link to the revisions on that post and then go back twice to get back to the flag to check the other posts.  It would be far more efficient if the links were to the revisions page rather than the post itself. It would make checking these flag much quicker. Jackknife vs. LOOCV Numbering a set of horizontally distributed equations Why are the dineutron and diproton unbound? It is known that there are no diproton or dineutron nuclei.  Does this mean that two protons or neutrons are not actually attracted to each other? Even if the attraction was weak, wouldn't it cause bound states anyway?  Related: What is the standard way to add N seconds to datetime.time in Python? Given a datetime.time value in Python, is there a standard way to add an integer number of seconds to it, so that 11:34:59 + 3 = 11:35:02, for example?  These obvious ideas don't work:  &gt;&gt;&gt; datetime.time(11, 34, 59) + 3 TypeError: unsupported operand type(s) for +: 'datetime.time' and 'int' &gt;&gt;&gt; datetime.time(11, 34, 59) + datetime.timedelta(0, 3) TypeError: unsupported operand type(s) for +: 'datetime.time' and 'datetime.timedelta' &gt;&gt;&gt; datetime.time(11, 34, 59) + datetime.time(0, 0, 3) TypeError: unsupported operand type(s) for +: 'datetime.time' and 'datetime.time'   In the end I have written functions like this:  def add_secs_to_time(timeval, secs_to_add):     secs = timeval.hour * 3600 + timeval.minute * 60 + timeval.second     secs += secs_to_add     return datetime.time(secs // 3600, (secs % 3600) // 60, secs % 60)   I can't help thinking that I'm missing an easier way to do this though.  Related Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable Problem : Given $f(x) = x + |x|$ for what values of $x$ is $f$ differentiable?   For the sake of generality, let's assume that it is unknown to us that $|x|$ is not differentiable at $x = 0$  Attempted Solution :  Using the definition of differentiability, a function is differentiable over an interval $I$ $\text{iff}$  $$f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}, \ \ \forall x\in I$$  Now implicitly this definition of differentiability requires $$\lim_{h\to 0^+} \frac{f(x+h)-f(x)}{h} = \lim_{h\to 0^-} \frac{f(x+h)-f(x)}{h}$$  Therefore $f$ will be differentiable only when  $$\lim_{h\to 0^+} \frac{(x + h + |x+h|)-(x + |x|)}{h} = \lim_{h\to 0^-} \frac{(x - h + |x-h|)-(x + |x|)}{h}$$  But it is unclear what to do next as $x$ is just arbitrary $$ \begin{equation} \begin{aligned} |x+h| &amp;= \begin{cases}x+h &amp; \text{if} &amp; x \geq -h\\ -x-h &amp; \text{if} &amp; x &lt; -h \end{cases} \\ &amp;= \begin{cases}x+h &amp; \text{if} &amp; x \geq 0\\ -x-h &amp; \text{if} &amp; x &lt; 0 \end{cases} \end{aligned} \end{equation}$$    A Wrong Solution :  We could take the derivative of $f$, (and evaluate the domain of the derivative $f'(x)$ to values for which it is defined), using the rules of differentiation, but we would get a wrong answer.  $$f'(x) = \frac{|x| + x}{|x|}$$ $$\implies f'(x) = \begin{cases} 2 &amp; \text{if} &amp; x \geq 0 \\ 0 &amp; \text{if} &amp; x &lt; 0 \end{cases}$$  This implies the derivative $f'(0)$ exists, when it does not as per the definition of differentiability. Why is that so?    Questions:   How can a solution be found using the definition of differentiablity? Why does taking the derivative of $f$, $f'(x)$ and evaluating it's domain, not give the correct values of $x$ for which $f$ is differentiable? A term for adoption of a made up term Is there a term for the acceptance of a made up term that then became the name of the object/idea/action once it was invented?   Example: Cyberspace. It was first used by William Gibson in his cyberpunk novels, but when actually invented, it now is widely used across the world (or at least across the United States).  Is there a term for this phenomenon? How could I define function with function such as Coefficient? Does an electrolytic capacitor degrade each time it receives reverse voltage? In the ,  can monitor some things like the total traffic on the site. However, it seems to be broken at the moment, all figures are almost zero. This is how it looks for Meta Stack Exchange (other sites show similar pictures):