# All Questions

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### I need to find the cardinality of this set : {x+y\sqrt2}+z\sqrt3} | x,y,z \in \mathbbQ}

I would really appreciate it if someone can help me start this problem. not asking for solution of course.
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### Sample the vectors in the plane

Let n be a non zero natural number and n vectors in plane $\vec{v_{1}},\vec{v_{2}},...,\vec{v_{n-1}},\vec{v_{n}}$.Prove that there exist $a_{1},a_{2},a_{3},...,a_{n}\in\left \{ 1,-1 \right \}$ for ...
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### Collecting the most presents possible with a limited distance that you can travel.

The following map shows towns (in purple) and the number of presents at each town (also in purple). The towns are connecting by roads (in orange) and the length of those roads is shown in orange (the ...
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### How to prove that R is a binary relation?

R={(x,y)|x,y are 5 bit strings with equal zeros number} Prove that R is Equivalence relation. How much classes does it have?
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### evaluation of ODE

I'm currently going through analysis. While learning ODE's I found some step I can't understand. $\dfrac{dp(x)}{p(x)}=\dfrac{d x}{1+x} \Rightarrow \ln p(x)=\ln(x+1)+\ln C$
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### First hitting time inequality

Let $X_t$ be a continuous time stochastic process. Define the first hitting time of $X$ as $\tau = \inf~\{ t \leq t': \lvert X_t \rvert \geq \lambda \}$ for some $\lambda$, $t'$ It is a known fact ...
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### Is the endomorphism of M (where M is an A-module) a non-commutative ring or not?

I was reading the "Undergraduate Commutative Algebra". It formalises the definition of Module. Consider M, an A-module where A is a ring. It defines $\mu_f : M \to M$ for the map $m \mapsto fm$, ...
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### A system of linear equations with solution $x=20$ and $y=20$

I am trying to come up with a system with solution $x=20$ and $y=20$. This is what I have now: $$\begin{array}{|l} 3(x-3y)-(2x-3y)=-100 \\ 4(x-3y)+2(2x-3y)=-200\end{array}$$ The expected solution is ...
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### Do $f=o(g)$ and $\lvert h\rvert\leq f$ imply that $h=o(g)$?

Suppose (1) $f(x)=\mathcal{o}(g(x))$ as $x\to\infty$ (2) $\lvert h(x)\rvert\leq f(x)$. Does this imply that $h(x)=\mathcal{o}(g(x))$ as $x\to\infty$? My would say: Yes! Let $\varepsilon >0$ ...
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### Which is the function that the zero-set of which is Möbius strip?

Many surfaces can be expressed by implicit functions. As described later, the spherical surface, the side surface of the cylinder, and the torus can be expressed by implicit functions. However, I have ...
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### Possible problems with definied log(z).

I am trying understand why I consider some of special contours from: $$(z)^{s-1}=e^{(s-1) \log (s)}$$ and in this contour. Why I could define $z^{s-1} ?$ reference
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### Expectation of stochastic integral

I'm not able to understand why $$E\bigg[\int_0^tB_s^3dBs\bigg]=0$$ it would be true if $B_s^3 \in \Lambda^2$ but it seems to me that $B_s^3\in \Lambda^2_{loc}$
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### What is the angle between the asymptotes of the hyperbola $5x^2-2\sqrt 7 xy-y^2-2x+1=0$?

What is the angle between the asymptotes of this hyperbola? $$5x^2-2\sqrt 7 xy-y^2-2x+1=0$$ I used $S+\lambda=0$ and used straight line condition to find combined equation to asymptotes. Then how ...
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### How to calculate this integral with residues $\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+a^2} dx$ [duplicate]

Anyone can help me with this integral? $\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+a^2} dx$ My professor tell me the answer is $\frac{\pi}{a} e^{-a}$
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### On the biconditional $I(n^2) = 2 - \frac{5}{3q} \iff (k = 1 \land q = 5)$, where $q^k n^2$ is an odd perfect number

MOTIVATION Let $N$ be an odd perfect number given in the so-called Eulerian form $$N = q^k n^2,$$ i.e., $q$ is the special / Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. ...
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### How to find the rotation matrix to rotate 3-d vectors such that the z-component of the rotated vectors is approximately constant?

I have an N-body simulation. Each body in the simulation has an array of positions as a function of time. For example, the body Earth has the following positional ...
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### Numerically stable algorithm to find most likely

I have $N$ Normal random variables $x_i$, each one with mean and variance $\mu_i$, $\sigma^2_i$. Given a value $L$, is there a numerically stable way to compute which of those has the higher ...
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### Having trouble proving that a series is divergent

I am trying to prove that $$\sum_{n=1}^\infty \frac{n}{\sqrt{n+1}}$$ diverges without checking the limit, bounds or doing any other lengthy steps, as it should be seen as divergent "immediately", ...
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### Compute $\int_Rx^2+y^2$, where $R=\{(x,y)\in\mathbb{R}^2:|x|\leq|y|\leq2\}$.

Compute the integral of $f(x,y)=x^2+y^2$ over $R$, where $R=\{(x,y)\in\mathbb{R}^2:|x|\leq|y|\leq2\}$. Writing the region as $R=\{(x,y)\in\mathbb{R}^2:-2\leq y\leq2, -y\leq x\leq y\}$, we have that: ...
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### Numerical Simulation Streamfunction

So I'm given the task of calculating the streamfunction with given angles of the flow on different carthesian coordinates (as shown in the picture)TaskAerodynamics. The translation of exercise one: ...
Given vector $\mathbf{v} = \begin{bmatrix} 5 \\ -1 \end{bmatrix}$, $\mathbf{b_1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\mathbf{b_2} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ all written in the ...