# All Questions

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### quastion in character theory of group

let X be an irreducible character of G and C be a conjugacy class of G. for any g of C, if (o(g) , |C|) = 1 then X(g)=0 or |X(g)|= X(1).
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### Integral of floor function.

How can I solve this integral? $$\int\frac{\mathrm dx}{\sqrt{\lfloor 1+ \sqrt{1+x}\rfloor}}$$
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### How to show that SVM is convex problem

It's well-known fact that SVM is convex problem $min \frac{1}{2} \left \| w \right \|^2$ s.t. $(wx_i+b)y_i \geq 1$ I don't understand how given the LP formulation of SVM I can coclude that it's ...
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### Probability of a sequence of urn draws having some pair of draws with a minium number of “matches”?

I have $U$ urns. Each urn contains some sequentially numbered balls (not necessarily the same count between urns) $1, 2, 3,... N_u$. I draw one ball from each urn $1, 2, 3,...U$ in turn, and note ...
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### Limit of a sequence of a supremum.

Problem: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a\leq c < d \leq b$ such that $f(c)=f(a)$ and $f(d)=f(b)$ and ...
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### Little o(h) limit about h=0

I understand that generally if a function $f(h)$ is described as $o(h)$ that $f(h)$ has a smaller rate of growth than $h$ (like it would have to be $\sqrt{h}$). i.e. $\sqrt{h} = o(h)$, just like (for ...
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### How to convert this equation to close form (matrix form)

I have read that the matrix form for the following summation $$Error(w) = \sum_{i=0}^{m} W^{T}x_i - y_i$$ $W^T$ is the transpose of weights vector in linear regression $x_i$ is the ith input in ...
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### Law of Iterated Expectation Proof

I have a proof that needs to be done: $$\mathbb{E}(XY) = \mathbb{E}[\mathbb{E}(Y|X)\,X]$$ So I start with the following \begin{align} \mathbb{E}(XY) &= \mathbb{E}(X)\cdot\mathbb{E}(Y)\\ ...
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### If $N=a^2+b^2=c^2+d^2$ then $N$ cannot be a prime number.

The problem says that if $N$ can be expressed in two ways as the sum of two squares then $N$ is not prime. Clearly the first idea is to try and express $N$ as a product of two expressions containing ...
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### Absolute square in deriving Fourier transform variance

I'm having some trouble understanding how to derive the variance of the Fourier transform. This is for an image, i.e., it's a 2D transform. The variance is $|\hat{I}(\xi,\eta)|^2$, the absolute ...
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### dividing a chebyshev polynomial by another polynomial

If I took a Chebyshev polynomial, is it possible to divide it completely by something that isn't a chebyshev polynomial?
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### convex function divided by convave function is quasiconvex

$p(x) \geq 0$ is convex, and $q(x) > 0$ is concave. How to prove $f(x) = \frac{p(x)}{q(x)}$ is quasiconvex? My proof is using t-sublevel set: $\{x | \frac{p(x)}{q(x)} \leq t\}$ is ...
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### Is $[a,\infty )$ closed

In standard topology, of course $[a,\infty )$ is closed since its complement is open. But I don't know how to prove closeness of $[a,\infty )$ in Real Analysis using just the definition of closeness, ...
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### given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once.

given a circle $(x-1)^{2}+ y^{2}=1$, find $b$ such that the line $y=x+b$ intersects with the circle just once. This question is for a precalculus class so setting the derivative of the positive ...
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### Trouble understanding some basic concepts of measure theory [on hold]

I am currently undergoing a course in Measure Theory. The book is "Principles of Real Analysis" by Charalambos D. Aliprantis and Owen Burkinshaw. The approach is little difficult for me to grasp and I ...
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### Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3$ permutation representations. For example, we have  \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
A sunflower or $\Delta$-system is a collection of sets $\mathscr{F}$ whose pairwise intersections are all the same set $S$, possibly empty. Elements of the collection of sets $\mathscr{F}$ are called ...
Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...