All Questions

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To find a $y_n$

I need to prove this theorem for positive series: $\forall {x_n}$ such that $\displaystyle\sum_{i=1}^{\infty} x_n$ diverges, $\exists y_n$ such that: 1) $\displaystyle\sum_{i=1}^{\infty} y_n$ ...
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Poisson actions defined in terms of coactions.

If $(M,\{ \cdot,\cdot \}_{M})$ and $(M',\{ \cdot,\cdot \}_{M'})$ are two Poisson manifolds, then a smooth mapping $\varphi: M \to M'$ is called a Poisson map if it respects the Poisson structures, ...
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Forty poor thieves

Forty thieves have 4000 gold coins to split between them. A group of five thieves is $poor$ if together they have less than or equal to 500 gold coins. Let N be the minimum number of poor groups of ...
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Solve the differential equation $y'-xy^2 = 2xy$

I get it to the form $\left | \dfrac{y}{y+2} \right |=e^{x^2}e^{2C}$ but I'm not sure how to get rid of the absolute value and then solve for y. I've heard the absolute value can be ignored in ...
42 views

Determine whether $\sum_{t=2}^n\frac{\log t}{\Omega(t)}\sim n\log n$

Determine whether $$\sum_{t=2}^n\dfrac{\log t}{\Omega(t)}\sim n\log n$$ Let $r=n^{1/\Omega(n)}$, where $n$ is a positive integer and $\Omega(n)$ is the total number of prime factors of $n$. If $r$ is ...
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Why is $\int$ $1\over(a^2+b^2)^{3/2}$ $da$ $=$ $a\over b^2\sqrt {(a^2+b^2)}$

$$\int\frac{da}{(a^{2}+b^{2})^{3/2}} =\frac{a}{b^{2}\sqrt{(a^{2}+b^{2})}}.$$ Found this in the solution to a problem in my physics textbook, and left clueless.
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For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B.

I am an undergraduate student. Please tell me if my proof is correct. Thanks! For a, b ∈ N, let A and B be the sets of all integer multiples of a and b. Prove that for all a,b ∈ N, a = b iff A = B. ...
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Finding the Legendre transform of an “entropy type” functional

I want to find the Legendre Transform of $$T(f) = \int_{\mathbb{R}^2} f \log \left(\frac{f}g{}\right) \, dx$$ on a set $H_M = \{ f: f \ge 0 \text{ and } \int_{\mathbb{R}^2} f =M\}$, where g is some ...
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Why are people interested in solving the Navier-Stokes equations if people can find an good approximate solution?

Why are people interested in solving the Navier-Stokes equations if people can find a good approximate solution? Also especially when people have supercomputers?
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Prove that $X_n/\lambda_n \to$ 1 in probability for $X_n \sim \text{Pois}(\lambda_n)$

Let $X_n \sim \text{Pois}(\lambda_n)$, where $\lambda_n \to \infty$ as $n \to \infty$. Prove that $X_n/\lambda_n \to 1$ in probability Should I solve this problem using chebychev's inequality? I'm ...