# All Questions

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### What is the meaning of $1_{a>b}$?

What would this mean: $1_{a>b}$ .. Based on the context, it could mean "$1$ if $a>b$ else $0$", but it's the first time I see it so help would be appreciated.
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### $Z$ score probability

I was given a question where I was supposed to find the probability of obtaining $y$ between two scores, however when I input my answer it tells me that I'm wrong, the question is given below along ...
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### Proof a function is continous.

Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, where $f(x_1,x_2) = x_1^2 + x_2^2$, a continuous function? My attempt: Suppose that $\forall \varepsilon > 0$ $\exists \delta >0$ such ...
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### Integral curve for vector field tangent to sphere

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field ...
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### Asymptotic behaviour of e * !n - n! , n tends to infinity

What is the asymptotic behaviour of the function $e !n-n!$ , where $!n = n! \sum_{k=0}^n \frac{(-1)^k}{k!}$ is the subfactorial of $n$. I tried Wolfram Alpha but the series for n=$\infty$ is ...
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### Fourier transform of a function over finite group

Let $G$ be finite abelian group and $\hat G$ be its character group. The Fourier transform of a function $f:G \to \mathbb C$, is the function $\hat{f}:\hat{G}\to \mathbb C$ defined by ...
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### Books about Turing machines and undecidability

I need help with finding literature about Turing machine and undecidability. First book I was suggested is Introduction to Automata Theory, Languages, and Computation by Hopcroft, Motwani and Ullman. ...
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### Validity of a Limit Proof

I am trying to show that if $\displaystyle{\lim_{s\to\infty}} s_n = s$, $\displaystyle{\lim_{s\to\infty}} \sqrt{s_n} = \sqrt{s}$ for some sequence $s_n$. We must note that $s_n > 0$ for all $n$ ...
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### Solve NC + BN =F

I asked this in the computer section; someone suggested asking in the maths section: Is there a simple way to solve the following matrix equation for N: NC + BN = F The matrices B, C, and F are ...
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In the last days I've been studying the tensor algebra $T(V)$ of a vector space $V$ over the field $K$ and I've realised that what I'm not understanding hasn't to do with tensor products, but rather ...
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### Homology and cohomology are basically the same

Is my following understanding correct: A chain complex $(C,\partial)$ is a family $\{C_i\}_{i\geq 0}$ of $R$-modules ($R$ is a given ring) together with a family of $R$-module homomorphisms ...
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### Proof of the properties of limits of CDFs

The cumulative distribution function is defined as $F(a) = \mu((-\infty,a])$ where $\mu$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Given this definition, it is easy to prove ...
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### Finding $\lim_{x \to 0} x^x$ without l'Hôpital [closed]

I have to find the limit of $x^x$ as $x$ approaches $0$ without derivatives.
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### Prove that the eigenvectors are independent.

Given two vectors $\boldsymbol\alpha=\left(\alpha_1,...,\alpha_N\right)^{T}$ and $\boldsymbol\beta=\left(\beta_1,...,\beta_N\right)^{T}$, let $M$ be the $N\times N$ matrix whose entries are expressed ...
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### Test for convergence/divergence of $\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$

Given the series $$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$ I need to test for convergence/divergence. I think the divergent test might work here. I could see that the ...
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### What is the meaning of “integral point”?

While reading this paper (http://cowles.econ.yale.edu/P/cd/d04b/d0473.pdf) I encountered the concept of "integral point", used first in definition 5.1, on page 34. Does anybody know more details about ...
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### Solving the differential equation $y'' + 2y' + 2y = 0$ given constraints

How can I solve this initial value problem? $$y'' + 2y' + 2y = 0,$$ given $y\,(\pi/4)=2$ and $y'(\pi/4)=0$. I've found $y(t)=e^{-t} \left(C_1\cos t + C_2\sin t \right)$ but I wasn't able to find ...
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### Properties of Lie derivative

Let's have Lie derivative: $$L_{V}\varphi = V^{\mu}\partial_{\nu}\varphi , \quad L_{V}A_{\mu} = V^{\nu}\partial_{\nu}A_{\mu} + (\partial_{\mu}V^{\nu})A_{\nu}.$$ How to show that for scalar and ...
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### Continuity of functions from complex numbers

i have a question about continuity. Suppose i have a function from $\Bbb{C}$ into a Banachalgebra $A$ for example $r\mapsto exp(ra)$ for a fixed $a\in A$. Do we have to prove continuity by ...
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### Finding limit of a sequence [duplicate]

Hi I'm having a lot of trouble with this: I need to find the limit of the following sequence: There's a hint that I should use the following formula: Any help is much appreciated!
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### Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$

Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1}$ Thanks in advance, my professor asked us to this a couple weeks ago, but I was enable to get to the right answer. Good luck! Here is ...
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### Equivalent Definition of the Closure of a Set

I want to prove the following Lemma: Let $(X,\tau)$ be a topological space and $B\subseteq X$ be a subset. Then holds $$\overline{B}=\bigcap_{X\setminus A\in\tau \atop B\subseteq A}A.$$ I have ...
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### Conformal map between annulii

Is there any conformal map between $D_1= \lbrace z \in \mathcal{C} \; ; \; 1 \leq |z| \leq 2 \rbrace$ and $D_2 = \lbrace z \in \mathcal{C} \; ; \; 1 \leq |z| \leq 3 \rbrace$. By the Schottky theorem ...
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### Find the general formula of a the series.

If we are given that the first 4 terms of a series are 1,2,4, and 8. And the rest of the terms are summation of the previous 4 terms i.e. 5th term is 8+4+2+1 and so on. So how can we find the general ...
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### estimation of a parameter

The question is: $x_i = \alpha + \omega_i,$ for $i = 1, \ldots, n.$ where $\alpha$ is a non-zero constant, but unknown, parameter to be estimated, and $\omega_i$ are uncorrelated, zero_mean, ...
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### Let $\Omega$ be a countable set $\mathcal A=2^{\Omega}$ be the collection of all subsets of $\Omega$.Then $\dots$

Let $\Omega$ be a countable set $\mathcal A=2^{\Omega}$ be the collection of all subsets of $\Omega$.Then If $\mu:A\rightarrow [0,\infty]$ is defined by $\mu(E)=|E|$ that $|E|$ is number of ...
I am writing an algorithm that evaluates the square root of a positive real number $y$. To do this I am using the Newton-Raphton method to approximate the roots to $f(x)=x^2-y$. The $n^{th}$ iteration ...
How many ways there are to organize $7$ men in a row, if two insist on not standing next to each other? How do I approach this?