# All Questions

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### Supermodularity and n-increasingness

Let $\geq$ be the usual partial order over $\mathbb{R}^n$ (i.e. if $x,y\in\mathbb{R}^n$, $x\geq y$ iff $x_i\geq y_i \forall i=1,\dots,n$). Definition 1: Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$. ...
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### Conditions on $f$,$g$ such that $\int f(x) - g(x)\,\mathrm{d}x$ converges given $f \sim g$

Let $f$, and $g$ be two functions. I am trying to study under what conditions the integral $$\int_C f(x) - g(x)\,\mathrm{d}x$$ converges. Where $C$ is some open, half open or closed interval. ...
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### Divisors of Fermat Curves.

I've been studying Algebraic Geometry (in coding theory), and my book has been very ambiguous about what some of these ideas actually look like. So I was curious as to what some of the Divisors of ...
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### what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
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### MLE of discrete uniform distribution

Assume that $X$ is a discrete random variable with uniform distribution on the set $\{1,2,3,\ldots, N\}$, where $N$ is an unknown positive integer. Find the MLE $\hat{N}_k$ of $N$, assuming that ...
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### Contraction Mapping Conditions

Consider $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ continuous and uniformly bounded, and consider the iteration $$x^{(k+1)} = \frac{1}{k} \sum_{i=1}^{k} f\left( x^{(i)} \right) + x^{(i)}$$ for ...
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### Assistance solving $x'(t)=t-x(t)^2$

I'm taking a second level ODE class and for part of some problem I need to solve a nonlinear first-order differential equation, but I've never worked with nonlinear problems before (there was no ...
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### Conditional convergence of $\sum _{n=1}^{\infty} a_n$ and $\sum _{n=1}^{\infty} \log(1+a_n)$

In the Ahlfors Complex Analysis book section on infinite products, there is a result that the the series $\sum _{n=1}^{\infty} |a_n|$ converges exactly when $\sum _{n=1}^{\infty} | \log(1+a_n)|$ does. ...
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### Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
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### Question about integrals of an unkown function and then differentiating to it

In my thesis I encountered the following problem: I have an unknown function y(x) and I need to calculate the following combination of integrating and differentiating: ...
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### General 2D taylor surfaces from axial behaviour and discrete points

I have a problem as follows: I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete ...
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### Why is tree traversal the fastest ray-box method?

I'm learning ray tracing (the problem of intersecting a ray, aka a vector, against a 3D box defined by a max and a min point) and I'm wondering: why is a tree traversal (e.g. bounding volume ...
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### Pointwise convergence to $\ln(x)$

I came up with a stepfunction on $(0,1]$: $s_n = \sum_{i=1}^{n} \ln (\frac{i}{n}) \chi_{(\frac{i-1}{n},\frac{i}{n}]}$, where $\chi$ denotes the characteristic function. I need to show that this ...
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### Multiple measurements per person per treatment

Suppose I wish to assess reaction time of individuals before and after treatment. Now to analyse the results I could use a paired t-test or if I had additional treatments, I could use a repeated ...
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### How to find the axis of rotation needed to rotate a $3d$ vector to another $3d$ vector?
I have two vectors $(a,b,c)$ and $(d,e,f)$. How can I find the axis of rotation needed to rotate the first vector to be parallel to the other vector? Thanks