# All Questions

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### Permutation of Indistinguishable Objects

How many number of two digit numbers can be formed using $\{4,5,6,6\}$ without repetition? I know that $\{45,46,54,56,65,64,66\}$ are the possible answers, but I am wondering if there is any formula ...
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### How to find the eigenvalues of this matrix?

Regarding this quadric: $x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3-2=0$ I am asked to rotate, translate and then classify the geometric object. First, I want to remove the mixed terms of ...
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### Multinomial Coefficients Definition in expansion of $(1+x+x^2+\cdots+x^l)^n$

The literature defines multinomial coefficients (or extended bnomial coefficients) as $$\binom{n}{r_1,r_2,\cdots,r_l} = \frac{n!}{r_1!r_2!\cdots r_l!}$$ where $$r_1+r_2+\cdots+r_l = n$$ Which is ...
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### Proof Matrix L with respect to Basis B and C

Good one guys! I'm doing the conceptual exercises of my Linear Algebra book, and I ran up to the following exercise: I tried to use the following theorem: That came from: But it got messy ...
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### Can we view the connected component of the Picard scheme $\text{Pic}_0(X)$ as a “kernel” of the first Chern class?

So on a curve, $\text{Pic}_0(X)$ is just the Jacobian variety, and just correspond to degree $0$ divisors. One way to extend the notion of divisors corresponding to a vector bundle is taking the first ...
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### Underlying utility function behind a linear two-product demand curve

I am trying to find the underlying utility function behind a linear two-product demand model. For that, I use two methods considering the following utility function: U(q_1,q_2) = ...
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### verifying Green's Theorem between 2 Circles

I get the same answer as my textbook only with a negative sign , so I am wondering who is right ...: Verify Green's theorem in the plane for {line integral of} x^2ydx + (y^3- xy^2)dy , where C is the ...
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### Is there a name for this type of online optimization problem?

I have a sequence of items $1\leq i \leq n$ that arrive to me one at a time. Each item has a weight $w_j\geq 0$. If I pick up one item, I will not be allowed to pick up any of the next $k$ items ...
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### Learning if the possible roots of an equation are different without resolving it

Is there anyway to know if a given equation will have different roots (all of them different to each other). Say: $x^3 - 17x^2 + 5x - \pi = 0$ Is there any property or theorem to know this for ANY ...
I have a positive sequence which converges to zero, i.e. $a_k \geq 0 \;, \forall k \in \mathbb{N}$ and $\lim_{k\rightarrow \infty} a_k = 0$. Does there exist another sequence $b_k$ with the property ...