# All Questions

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### Solving differential equation using Laplace transform

Can this DE be solved using Laplace transform? $\frac{\mathrm{d} y}{\mathrm{d} x}\cos x=y\sin x+\cos ^{2}x$
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### Can we obtain the pair $(1,50)$ with these following operations?

It's a problem from some russian competition: We're given a card with two positive integers $(a,b)$ and we have tree machines which generate another card from the one we insert on it(I assume we ...
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### Showing that $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$

I have problems to demonstrate: $\nabla\times(\nabla\times\vec{A}) = \nabla(\nabla\cdot\vec{A})-\Delta\vec{A}$. I don't have any clue how can I start to work with it. Any hint will be helpful.
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### Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
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### Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$

My question is, Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$ for some $a_k$ where $a_k>0$ and $a_k \rightarrow 0$ as $k \rightarrow 0$. ...
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### What is the limit of this sequence as n->infinity? [on hold]

Find the limit of the following sequence $n^{\ln(n)/n}$ as $n\to\infty$? Please answer without using L'Hopital
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### What do we know about the first occurrences of prime gaps?

Are there any conjectures from which we can infer something about the first occurrences of prime gaps length $n$ and their distribution? I've made an interesting graph of these values to make this ...
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### Number of ways to get from top to left through a blocked Grid

There is a Grid and a guy , guy can move either down or right in a Grid as shown from starting point (1,1). Let the Grid Dimensions be N x M . (The length of Grid x The Breadth of Grid) There are ...
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### Find limit of the following sequence?

Find the limit of $\frac{\log(n+1)}{\log(n)}$ where $n\rightarrow\infty$. Here $n$ is a natural number so I guess we can't use L'Hopital
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### Direct Limit of finitely generated groups

Is every group the direct limit of its finitely generated subgroups? This is true for abelian groups, I have not seen this statement for nonabelian groups, so i am wondering if this is true. Seems ...
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I am working on solving complex system of 2D PDEs governing the behaviour of plasma in a gas lamp during discharge. Recent tests have shown that because of steep gradients in temperature field and ...
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### Direct Sum Proof

I am reading Axler's Linear Algebra Done Right Book and I am on the part about the direct sum. He gives the following proposition: When he assumes that $a$ and $b$ hold to prove that the proof gives ...
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### A question regarding harmonic function.

Can any one provide some hint on the following question? I have being thinking about this for a while but cannot figure out where to start. I have been thinking about Taylor expansion but it seems not ...

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