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I encountered an 'errorUnknown graphics extension: .eps.' I tried to load an .eps picture, but an error occured,      errorUnknown graphics extension: .eps. I don't really understand regular expressions.  Can you explain them to me in an easy-to-follow manner?  If there are any online tools or books, could you also link to them? Existence of minimal $\sigma$-algebra and transfinite induction Quote environment with reference at the end right This is an exercise from textbook Analysis I by Amann/Escher.  Let $G$ be a group and $X$ a nonempty set. Then $G$ acts (from the left) on $X$ if there is a function $$G \times X \rightarrow X, \quad(g, x) \mapsto g \cdot x$$ such that: GA1: $e \cdot x=x$ for all $x \in X$. GA2: $g \cdot(h \cdot x)=(g h) \cdot x$ for all $g, h \in G$ and $x \in X$. Let $\mathrm{S}_{m}$ be the set of all permutations of $\{1,2,\cdots,m\}$. Then $(\mathrm{S}_{m},\circ)$ is a group where $\circ$ is function composition operation. Show that $$\mathrm{S}_{m} \times \mathbb{N}^{m} \rightarrow \mathbb{N}^{m}, \quad(\sigma, \alpha) \mapsto \sigma \cdot \alpha := \left(\alpha_{\sigma(1)}, \ldots, \alpha_{\sigma(m)}\right)$$ defines an action of $\mathrm{S}_{m}$ on $\mathbb{N}^{m}$.  I am stuck at proving the GA2, i.e. $\pi \cdot (\sigma \cdot \alpha) = (\pi \circ \sigma) \cdot \alpha$.  My attempt: Clearly, $\operatorname{id}$ is the identity element of $\mathrm{S}_{m}$ and $\sigma \cdot \alpha = \alpha \circ \sigma$. Because $\operatorname{id} \cdot \alpha = \left(\alpha_{\operatorname{id}(1)}, \ldots, \alpha_{\operatorname{id}(m)}\right) = \left(\alpha_1,\cdots,\alpha_m\right) = \alpha$, GA1 follows. For $\pi,\sigma \in \mathrm{S}_{m}$ and $\alpha \in \mathbb{N}^{m}$, we have $$\pi \cdot (\sigma \cdot \alpha) = (\alpha \circ \sigma) \circ \pi = \alpha \circ (\sigma \circ \pi) = (\sigma \circ \pi) \cdot \alpha$$  It is clear that $\sigma \circ \pi = \pi \circ \sigma$ if and only if $(\mathrm{S}_{m},\circ)$ is Abelian, which is not true in general. Please help me figure out where I am wrong! Good-morning Math Exchange (and good evening to some!)  I have a very basic question that is confusing me.   At school I was told that $\sqrt {a^2} = \pm a$  However, does this mean that $\sqrt {a^2} = +2$ *and*$-2$ or does it mean: $\sqrt {a^2} = +2$ *OR*$-2$  Is it wrong to say 'and'? What are the implications of choosing 'and'/'or'  Any help would be greatly appreciated. Thanks in advance and enjoy the rest of your day :) I noticed that Synonymizer is only a bronze badge. Yet in order to obtain this badge you need to have 2500 reputation or more to even create tag synonyms. Then, on top of that you also need to be able to find a tag synonym.  Only 514 of these have been awarded... This makes the BRONZE badge Synonymizer rarer to have than most the gold medal badges. So my question is, why is this a bronze badge? $$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} &amp; (x,y)\neq(0,0) \\ f(x,y) = 0 &amp; (x,y)=(0,0) \end{array}\right.$$  In order to verify if this function is differentiable, I tried to prove it by the theorem that says that if $\frac{∂f}{∂x}$ and $\frac{∂f}{∂y}$ exist and are continuous at the point $(x_0,y_0)$, then the function is differentiable at this point. So I did:  $$\frac{\partial f}{\partial x}(0,0) = \lim_{h\to 0}\frac{f(0+h,0)-f(0,0)}{h} = 0$$ $$\frac{\partial f}{\partial y}(0,0) = \lim_{h\to 0}\frac{f(0,0+h)-f(0,0)}{h} = 0$$  so we have that the partial derivatives at point $(0,0)$ is $0$. Now, if we take the derivative at $(x,y)\neq (0,0)$ and then take the limit of it as $(x,y)\to(0,0)$, we can see if the derivatives are continuous or not. So here it is:  $$\frac{\partial f}{\partial x}(x,y) = \frac{y(y^2-x^2)}{(x^2+y^2)}$$  but   $$\lim_{(x,y)\to(0,0)} \frac{y(y^2-x^2)}{(x^2+y^2)} $$ does not exist (by wolfram alpha... but can anybody tell me an easy way to prove this limit does not exist? easier than taking the limit in different directions?), therefore the derivative is not continuous at $(0,0)$, so we can't say $f$ is differentiable at $(0,0)$, but for $(x,y)\neq (0,0)$ the function is continuous, as it is a quotient of continuous functions. So $f$ is at least differentiable at $(x,y)\neq (0,0)$.   Now, to verify differentiability at $(0,0)$ I think we must use the limit definition of differentiablity: A function is differentiable at $(0,0)$ iff: $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = 0$$  Let's calculate this limit:  $$\lim_{(h,k)\to (0,0)} \frac{f(0+h,0+k)-f(0,0)-\frac{\partial f}{\partial x}(0,0)-\frac{\partial f}{\partial y}(0,0)}{\|(h,k)\|} = \\ \lim_{(h,k)\to (0,0)} \frac{\frac{hk}{h^2+k^2}}{\sqrt{h^2+k^2}} = \\ \lim_{(h,k)\to (0,0)} \frac{hk}{(h^2+k^2)\sqrt{h^2+k^2}}$$  which I think, it's a limit that does not exist, therefore the function isn't differentiable at $(0,0)$ Infinitely differentiable function with given zero set? How can I display site logo, name and slogan? how can i show site logo, name and slogan Separately in custom area in my site theme, I now use this code to display theme, buts this code don't allow me to edit each of them Separately.  {{ page.header }} How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards? I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by profession. Knowing full well that my proof may be fallacious, erroneous, or simply lacking mathematical formality, what advice would you give me? Why the Galaxy has 25GB instead of 32GB I just purchased a Samsung Galaxy S3 32GB, when I check the storage I have only 25.5GB. Where is the other 6.5GB? Are there dextral/sinistral higher animals? For example, almost every dog I've met in my life (more than 20 subjects) uses their right paw touching one while asking for attention. Can it be used as a criterion for calling them dextral? If not, what?  However, I realize that even those dogs who live in the street might get influenced by dextral people:     There is no domestic animal which has so radically altered its whole   way of living, indeed its whole sphere of interests, that has become   domestic in so true a sense as the dog (Konrad Lorenz)   Is there any research about it? Stringification of a macro value I have been developing in the windows space with Visual Studio for a while now with work, but I have also been using Ubuntu for a while and am keen to get into some software development for linux.  I should also note. I am not looking for .NET and I am aware of mono. I am also familiar with c++ development and some python, so the language isn't so much relevant as the "all in one" aspect.  I was interested to know if there is a useful all in one code/debug/design(gui) IDE similar to something like Visual Studio but for linux? I would like to know how potential difference is created between two points?Is potential difference created by adding more electrons at one side?If that's the case how to add more electrons at one side?  I'm really looking to know what really constitutes potential difference? Everytime, I read about blackbody, I always get confused at the point where it is written     Under thermal equilibrium conditions , the blackbody radiation depends only on temperature. ..... . At equilibrium , electromagnetic waves bounce around with the walls of the cavity.   What is this equilibrium all about? Why is it important here?? If there is thermal equilibrium, why does the body radiate?? Please help. Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that How to debug a Custom Timer Job in SharePoint 2010? How can I convert string to datetime with format specification in JavaScript?