# All Questions

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### Equation of tangent line for $y' = \frac{x}{(1-x)^2}$ at point $(0,0)$

I tried to solve this by plugging zero into x the $x$ values and I end up getting $\frac{0}{1}$, which obviously is $0$. From there I multiply out and get all zeros. What am I doing wrong? More ...
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### Continuity and other properties of complex exponential

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces ...
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### An open cover $\{U_\alpha\}$ of $X$ is locally finite iff each $U\alpha$ intersects $U_\beta$ for only finitely many $\beta$

I am trying to prove: Lee, Smooth Manifolds, Exercise 2.9. Show that an open cover $\{U_\alpha\}$ of $X$ is locally finite if and only if each $U\alpha$ intersects $U_\beta$ for only finitely many ...
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### Proving the zeros of the chebyshev points

I am trying to prove that the zeros of $T_n(x)$, also called the chebyshev points are, $x_i = \cos ((2i + 1)\frac{π}{2n}) \in (−1, 1), i = 0, 1, . . . , n − 1.$ I believe I have to use the fact that ...
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### how to prove this limit, convolution?

I am wondering how to prove that $\vert (f*g)(x) \vert \rightarrow 0$ when $\vert x \vert \rightarrow \infty$ if we assume $f \in L^{p}(\mathbb R)$ and $g \in L^{q}(\mathbb R)$ where $1/p+1/q=1$ ...
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### Integration involving $\log_2(x)$

Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to: ...
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### Let G be a group and let H,K be subgroups of G where |H|=12 and |K|=5. Then the intersection of H and K = {e}.

The question is as stated in the title. I used LaGrange's theorem to show that |H|||G| and |K|||G| so 12||G| and 5||G|, and that 12 and 5 are relatively prime. I'm not sure if this has gotten me ...
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### Pairs of natural numbers $(a,b)$ such that $\frac{a^{2}+b}{b^{2}-a}$, $\frac{b^{2}+a}{a^{2}-b}$ are natural numbers [closed]

$a$,$b$ are natural numbers such that both of $~\dfrac{a^{2}+b}{b^{2}-a}$, $~\dfrac{b^{2}+a}{a^{2}-b}$ are natural numbers. Find all pairs $(a,b)$.
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### Exponentiation in terms of Summation

For positive integers, $a \times b=\sum\limits^{b}{a}$, correct? So therefore exponentiation where n is also a positive integer should be something like $a^n=\sum\limits^n{\sum\limits^a{a}}$ This is ...
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### Center of Mass with two functions

I am having trouble trying to figure out how to go about this problem. I can do problems with single variables but I can not solve this one. I think I would need to subtract the functions from one ...
We can easily find the correct derivative of $x^x$ with logarithmic differentiation as follows. $$\begin{eqnarray*} y & = & x^{x}\\ \ln(y) & = & x\ln(x)\\ \dfrac{dy}{dx}\dfrac{1}{y} ... 1answer 24 views ### Searching Natural Numbers a,b,c,d are natural numbers satisfying the following ~1\leq a,b,c,d\leq 9. a+b+c\equiv 1 (mod d) a+c+d\equiv 1 (mod b) a+b+d\equiv 1 (mod c) b+c+d\equiv 1 (mod a) ... 1answer 32 views ### Volume using shells I'm working on a problem of finding volume between two functions using the shell method. The functions given are f(x) = 2x - x² and g(x) = x. It is reflected across the x-axis. I solved this ... 1answer 212 views ### Invariance of Brownian motion under orthogonal transformations Let \left(B_t\right)_{t \in [0,\infty)} be an n-dimensional Brownian motion with start at x \in \mathbb{R}^n, and let A be an orthogonal n \times n real matrix. I'm trying to show that AB ... 2answers 66 views ### How many way can we obtain 0? You are walking in road and you have only two directions,forward and back.Your nth step has length n. How many way can you return your starting point after n steps ? It is equivalent to say ... 1answer 102 views ### Topological contraction on compact spaces This is a follow up question. You can see the original here. I have the following problem. Let X be a compact Hausdorff space and let f:X\to X be continuous. Show that there exists a ... 1answer 149 views ### Cardinality of the set of all straight lines in \mathbb R^2 Find the cardinality of the set of all straight lines in \mathbb R^2. Here's what I did: Let M be the given set.$$M \sim\{y=ax+b, \ a,b\in \mathbb R \}\cup\{x=c, \ c\in\mathbb R \} ...
It is established that Equivalence of atlases is an equivalent relation. Now consider the real line $\mathbb{R}$ and the following one chart atlases $\mathcal{A} = \lbrace (\mathbb{R},Id)\rbrace$, ...