# All Questions

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### Proving that there exists a saturated set with given highest weight

This is an question about an exercise in Humphreys book on Lie algebras. First of all a bunch of definitions and notation, see §13 in Humphreys for details. Let $\Phi$ be a root system, $\Delta$ a ...
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### Which harmonic numbers have prime numerators?

Let $$1+\frac12+\frac13+\cdots+\frac1n=\frac{q}{p}$$ with $(p,q)=1$ and $q$ is a prime number. (I) Prove or disprove that the quantity of $n$ is limited (II) Find all of the $n$ I use the matlab ...
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### A problem with sign of coefficients of a polynomial expression

Let $f$ be a real coefficient homogeneous polynomial in $n$ undeterminates, such that $f(x_1,\cdots,x_n)>0$ whenever $x_1,...,x_n$ are non-negative real numbers, not all $0$. Then how to show ...
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### For every $a>0$, show that $\langle f_a, \psi\rangle= \int_{|x|>a} \frac{\psi(x)}{|x|}dx+\int_{|x|>a} \frac{\psi(x)- \psi(0)}{|x|}dx$

To prove that the inner product is a distribution it must satisfy the following property" $$|T(\phi)|=|\langle T,\psi\rangle| \leq C_N \sum_{|\alpha| \leq N} \|\partial^\alpha \psi\|_\infty$$ Part ...
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### what is the limit of $\theta$ if $r\to 0$ when we have these conditions?

the problem is based on this picture. at beginning or we say $t=0$, $P$ is a circle of which the center is at the point $(0,r)$, $r_0=1$ is the initial radius of this circle. $AB$ is a vector which ...
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### Hyperplanes and Support Vector Machines

I have the following question regarding support vector machines: So we are given a set of training points $\{x_i\}$ and a set of binary labels $\{y_i\}$. Now usually the hyperplane classifying the ...
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### Formula for adjusting font height

INTRODUCTION AND RELEVANT INFORMATION: I am a software developer that needs to implement printing in my application. In my application user can choose different paper sizes ( A3, A4, A5 ...) which ...
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### Interchanging pointwise limit and derivative of a sequence of C1 functions

Assume the following: $f_n$ is a sequence of C1 functions on $[0,1]$ $f_n(x) \rightarrow 0$ for pointwise. $f'_n(x) \rightarrow g(x)$ pointwise. Is it true that $g(x) = 0$ almost everywhere? I think ...
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### Lenstra's Elliptic Curve Algorithm

I am currently trying to understand Lenstra's Elliptic Curve Algorithm. As a source I use "Rational Points on Elliptic Curves" by Joseph H. Silverman and John Tate. They describe the algorithm as ...
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### Solving system of two linear odes

I am trying to solve \begin{align} y_1'+B_{12}y_1=\beta_{12}y_2\\ Ay_2'+B_{21}y_2=\beta_{21}y_1, \end{align}with $y_1(0)=y_2(0)=y_0$. I find the eigenvalues to be ...
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### Proof of external product theorem using K-theory

I am using Hatcher's K-Theory book to work through the proof of the external product theorem: $\mu:K(X) \otimes \mathbb{Z}[H]/(H-1)^2 \to K(X) \otimes K(S^2) \to K(X \times S^2)$ is an isomorphism ...
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### Verifying Stokes' Theorem

Could someone help me set up this problem? I'm not sure where to start... I know I have to prove that the line integral equals the surface integral of the curl, but how do you figure out what line to ...
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### Seeking proof for the formula relating Pi with its convergents

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via A002485(n)/A002486(n) ? (-1)^n\cdot(\pi ...