# All Questions

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### Compute limit of a function

Compute: $$\lim_{x \rightarrow 0^+} \frac{\arctan(e^x+\arctan x)-\arctan(e^{\sin x}+\arctan(\sin x))}{x^3}$$ WolframAlpha tells me it's 1/6. Any nice idea how to rewrite that expression? Thanks!
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### Find the acute angle made by vector $OC$ and the x-axis.

Given that vector $OA$ = $3i+5j$, $OB$ = $-2i+6j$ and that $OC$ = $OA + OB$, calculate i) |OC|, ii) the acute angle made by vector $OC$ and the x-axis. I found i) $\sqrt122$ Please help me in ...
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### How to get the following limit into indeterminate form?

I am struggling to get the following limit into its indeterminate form so that i can apply the l'Hopitals rule: $$\lim_{x\to 0^+}(\sin x)^x$$ A solution would be greatly appreciated, been struggling ...
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### Crossing edges at space

Let's say I have Graph $G(v, e)$ I want to draw the graph without crossing edges on space. By giving $(x, y, z)$ for any Vertex. How can I check if one edge crosses another?
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### Bifurcation in PDE

How do we characterize bifurcation in nonlinear PDE instead of ODE i.e. ht=f(x,h,hx,hxx,hxxx,...)? For example, study the temporal evolution of a regular pattern into a chaotic one. Can someone please ...
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### Characterization of Sobolev Space

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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### Irreducible action of a group on a set

Let $p$ be a prime. I am solving a problem and I am told that I should use that the action of $\text{SL}_2(p)$ on $\mathbb{F}_p^2$ is irreducible. But I don't know what this means?
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### Find the probability $P[ x(t) \le 1]$ where $x(t)$ is a filtered Poisson process (rect pulses)

I can't understand the following question: "The random process x(t) is defined as $$x(t) = \sum_{n=- \infty}^{+\infty} rect(\frac{t-\tau_{n}}{T}) \quad ,\quad t \ \epsilon \ (R)$$ where {$\tau_{n}$} ...
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### Proving a set is of measure zero.

Let $C\subset A\times B$ be a set of content zero. Let $A'\subset A$ be the set of all $x\in A$ such that $\{y\in B: (x,y)\in C\}$ is not of content zero. Show that $A'$ is a set of measure zero. ...
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### Is $\text{Id} = \chi_{\{ |u| \leq k\}} + \chi_{\{|u| > k\}}$ well defined for $u \in L^p(0,T;L^q)$?

Is the decomposition $$\text{Id}(z) = \chi_{\{ |u| \leq k\}}(z) + \chi_{\{|u| > k\}}(z)\tag{1}$$ well defined for $u \in L^p(0,T;L^q(\Omega))$? I guess (1) holds a.e. So the problem is, is the set ...
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### Parametric formula for figure 8 mobius strip

I'm making 3D prints with Mathematica, and am interested in a parametric formula for a mobius strip that is in the form of a figure 8, rather than simply a circle with a twist in it. Can someone help ...
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### How to solve Absolute Value Inequality: |x-1| ≥ 3-x

I am learning the topic of solving absolute value inequality question. I had mostly understood the steps in order to solve for an inequality. However, I'm still clueless of a step to solve the ...
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### Integral closure of 1-dimensional noetherian local domains

Let $(R,m)$ be a $1$-dimensional noetherian local domain and $S$ its integral closure. Clearly $S$ is $1$-dimensional noetherian semi-local domain. Is $mS=J(S)$, where $J(S)$ is the Jacobson radical ...
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### Are distance-related paradoxes limited by the size of an atom?

See these 2 paradoxes: Coastline paradox The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. ...
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### Bound for this integral

Give $h,k:[a,b] \rightarrow \mathbb{R}$ functions of class $C^1$ coinciding at the end points of $[a,b]$ and $h\leq k$ on $[a,b]$. Assume that the graph of $h$ is an arc of circle of radius $R$. Show ...
I'm currently reading the following paper by Richard Skora, entitled Cantor sets in $S^3$ with simply connected complements found here, and on page 2, just before Theorem 1, it says "the homeomorphism ...