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 Jan 6 comment How to show that $(H,\cdot)$ is normal subgroup of $GL_2(\Bbb R)$? I think that it is not true. Jan 5 comment ${\left| {\sum\limits_{i =1}^n {{{\left( {\frac{{{x_i}}}{{{x_1}}}} \right)}^m}} } \right|^{\frac{1}{m}}}\mathop \to \limits^? 0$ I think you can use the fact that $\lim_{p\rightarrow\infty}||x||_p=||x||_{\infty}$. See en.wikipedia.org/wiki/Lp_space. Jan 5 comment How to draw complex function graphics Looking at the first graph, it appears that they're imagining it as a function $f:\mathbb{R}\rightarrow\mathbb{C}$, and treating the y-axis as the imaginary axis when referencing the orange line. Notice that the blue line remains at $0$ between $(-2,2)$, indicating that the real part is $0$ for those values. Honestly this is pretty cool and I have not seen it before. Jan 5 comment “Non-trivial element” in factor group terminology I think they just mean besides the identity element. Jan 5 comment Determine if the improper integral converges you can bound this integral by $\int_3^{\infty}e^{-x}dx$. Jan 2 revised Switching limit and integral in a min function added 7 characters in body Jan 2 comment Defining the pre-image topology for $f:A\rightarrow B$ your continued harping on one silly little sentence fragment I made off-hand is ridiculous, like I said I bet you know the answer to question 1 without having to do any work. Jan 2 comment Defining the pre-image topology for $f:A\rightarrow B$ except clearly I have done work and put considerable thought into formulating what I wrote (e.g. I wrote subbase and not base for a reason). My understanding was that Math.SE was a place to learn about mathematics above and beyond just a place where you get help when you're stuck on a proof, maybe I'm wrong about that. Not every one has the time to do a proof for everything they want to know. Jan 2 comment Defining the pre-image topology for $f:A\rightarrow B$ Because I doubt the details of the proof are in this case particularly enlightening. I put it in the format I did to make it easy on the reader, but it appears that approach has backfired and made it look like a homework question, either way if you don't want to answer it then you've down voted it and voted to close and done everything in your power to make it so others won't answer so now why don't you kindly go away. Jan 2 comment Defining the pre-image topology for $f:A\rightarrow B$ @Rob Arthan well I imagine most people with expertise in topology (probably you) could just tell me off hand that it's true, it's not like they'd have to do the proof themselves. They're certainly welcome to give me the details if they wish, but I'm not requesting them. Sometimes you just want to verify your intuition without laboring through the details of every single potential fact you come across. If I had access to people in real life with this knowledge I'd just ask them, but since I don't Math.SE is all I have so if you want to be an asshole and close the question then that's on you. Jan 2 comment Defining the pre-image topology for $f:A\rightarrow B$ @LeeMosher $A=f^{-1}(B)$, where $B$ must be open for it to have a topology so I'm not sure what you mean. Jan 2 asked Defining the pre-image topology for $f:A\rightarrow B$ Dec 28 comment What is tangent to a curve or function? I would suggest just taking a look at the wikipedia article on the concept of a tangent Dec 28 comment What is tangent to a curve or function? after all if tangency required that it not intersect at any point then all you'd be left with are the convex/concave functions. In fact you can't even restrict non-intersection to a local neighborhood, e.g. $y=x^3$ at $(0,0)$. Dec 28 comment What is tangent to a curve or function? I don't know what texts you're reading but the definition you give is non-standard as far as I'm concerned. Dec 28 comment A bijective linear map with no inverse, how is it possible? that's not what onto means, onto means that it hits every point in the co-domain, which it clearly doesn't. Dec 28 comment A bijective linear map with no inverse, how is it possible? bijective requires that a map be one-to-one and onto, your matrix $A$ is not onto since it doesn't send anything to points off the x-y plane. You can of course restrict the co-domain of an injective map to its range and turn it into a bijective mapping, which in the case of $A$ would be the identity function. Dec 28 comment What is tangent to a curve or function? it just means it doesn't cross at the point at which it is tangent. It may or may not cross at other points. Dec 27 comment Find the condition that fourth degree equation $x^4+rx+s=0$ will have no real roots. do you know calculus? Dec 20 comment Understanding the random variable definition of Markov chains If I understand you correctly you're talking about just looking at a particular path of the chain so the probability of the path would just be the product of each transition probability for that path. If there are $n$ states and you want to look at paths of length $N$ then I assume the (inherited) state space would be $\{1,...,n\}^N$ and the random variables would be $N$-variate, but I'll say no more because this is getting beyond my depth now.