# All Questions

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### Behavior of eigenvalues of certain matrices

I am trying to analyze the behavior of the 2 highest eigenvalues of matrices of this form : Symmetric $n*n$ matrices that contains only : $1/k$ (for fixed k), -1,1 and 0. My hope is to find some ...
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### Solving differential equation $x''(t)=x^6$.

Solve the following differential equation $$x''(t)=x^6(t)$$ If I had $x'(t)$ instead of $x''(t)$ the exercise would have been easier for me. I would appreciate some help with this problem. Thank ...
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I'm Xavier Vigan, a physical oceanographer. I've been using your $f(x)=\dfrac 12 \times \left(X+C-\sqrt{S+(X-C)^2}\right)$ function to calibrate quantile vs quantile plots. Because of the shape of ...
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### Instalments ( simple interest)

The price of a T.V set worth Rs. 20,000 is to be paid in 20 instalments of Rs. 1000 each. If the rate of interest be 6% per annum, and the first instalments be paid at the time of purchase, then the ...
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### Maximum of given expression?

Suppose $a,b,c>0$ and further that $a^{2} + b^{2} + c^{2}=2abc + 1$. The problem is to find $\max \big(a-2bc\big) \big(b-2ca\big) \big(c-2ab\big)$. Give me some help. I've tried $X=a-2bc$, ...
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### If I have a matrix M=[A,B;0,C], how do I prove that rank(A)+rank(C)<=rank(M)?

. . . . . . . A . . B . . . . . . . 0 0 0 . . . 0 . 0 . C . 0 0 0 . . . If I have a matrix $M$ as displayed in the text above ($A$ ...
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### $\Bbb{Z}_{2}(\alpha)$ as splitting field

i have problems with an exercise: let $\alpha$ be a root of the polynomial $X^{3}+X^{2}+1$ in $\Bbb{Z}_{2}$. Prove that $\Bbb{Z}_{2}(\alpha)$ is the splitting field of this polynomial over ...
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### How to prove a point in a set is an extreme point of the set ?

Def: an extreme point of a set $K$ is the point that cannot be expresssed as a convex combination of other points in $K$. Apart from the definition, what else arguments can we use to prove that a ...
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### Gradient w.r.t. boundary conditions in PDE

I am trying to solve the following problem. Suppose I have a field $\Phi(r)$, which is the solution to a partial differential equation: $\mathcal{L}\Phi(r) = s(r)$, as long as $r \neq r_0$ Here ...
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### if the integrals of two function are equal then the functions are equal almost everywhere. true or false?

Actually I know that if the integral of a non negative function is equal to zero then that function is equal to zero almost everywhere. Can I use that to prove or is there a counter example for my ...
Is there any way to get the probability of a draw outcome using ELO formula as it only gives the Win probability ELO formula is given by $E = \frac{1}{1+10^\frac{d}{a}}$ where d is the difference in ...