1:23 AM
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To give an example of a topology on a set of four points, different from discrete and anti-discrete, in which every open set is closed, and every closed set is open.I have 0 ideas at all, I thought that just the same, that only a discrete topology has such characteristics for sets.

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what is the number of lattice paths of length 16 from $(0,0)$ to $(8,8)$ that go through $(4,4)$, don't go through $(1,1), (2,2), (3,3)$, and don't go over $y=x$? Here's what I tried: since we can't go through those 3 points above, we can first go to $(3,2)$ from $(1,0)$ (there is one forced move...

3 hours later…
3:57 AM
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In one of his interviews, Clip Link, Neil DeGrasse Tyson discusses a coin toss experiment. It goes something like this: Line up 1000 people, each given a coin, to be flipped simultaneously Ask each one to flip if heads the person can continue If the person gets tails they are out The game contin...

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In a recent complex analysis exam, we were asked which step(s) in the proof of the Riemann Mapping Theorem fail, when you replace every instance of $\mathbb{E}$ with the square $$\mathbb{S} = \{z\in\mathbb{C}:|Re(z)|, |Im(z)| < 1 \}$$ One error that arose is that the square root function is not a...

4:40 AM
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I know that set of nilpotent elements in commutative ring forms an ideal, but set of zero divisors does not form a well known structure. I am wondering if there exists a ring $R$ such that all elements of $R$ are zero divisors. For example we can take $S=\lbrace 0,2,4,6 \rbrace$ as a subring of $... 1 hour later… 6:07 AM 1 Let be$X=[x_j(t_i)]_{i,j} \ M_{n,p}(\mathbb{R})$a state matrix of$nx(t_i) \in M_{1,p}(\mathbb{R})$observations of$p$sensors$X_j$representing the normal conditions of a system for some timesteps$t_1, ..., t_n \in T_n \subset T. The linear estimator of the Kernel Regression is : \fo... 3 hours later… 8:58 AM 3 Suppose X is a compact Hausdorff topological space with a basis of clopen sets. Let A be a closed subspace of X and let A^{(0)}=A, A^{(1)}=A^\prime, etc. My question is: it is true that A^{(n)}=A\cap X^{(n)} for all n\in\mathbb{N}? Since A\subset X, then A^{(n)}\subset X^{(n)}. ... 2 hours later… 11:26 AM 0 Consider a radial, nonlinear, 2D-heat equation \begin{align*} u_t &= u_{rr} + \frac{1}{r}u_r + F(t,r,u,u_r), \quad (t,r) \in [0,+\infty) \times [0,1] \\ u(0,r) &= f(r) \in C^{\infty}([0,1]), \quad f(0) = 0 \\ u_{|[0,+\infty) \times \{0,1\}} &= 0 \end{align*} where F(t,r,p,q) is a smooth functio... 3 hours later… 1:57 PM 4 I'm sorry if this question (or possibly multiple depending on how long this I intend for this to be) is a little too elementary, but I've been seriously struggling with this for the past week. I never really questioned this until recently and it has me stumped and I can't move on with math witho... 3 hours later… 4:27 PM 2 I was this problem:\int\frac{dx}{\sqrt{x^4+2x^3+3x^2+2x+1}}$$I solved this question because I just knew that (1+x+x^2)^2=x^4+2x^3+3x^2+2x+1 but this made me wonder is there is a way to know if the n-th root of a certain polynomial is a polynomial? Given a polynomial how to determine n w... 5 hours later… 9:49 PM 3 My question is the following. Given that we have n i.i.d. random variables X_1,...,X_n with distribution f(x)=\frac{2}{\lambda^2}x\mathbf{1}_{[0,\lambda]}(x), where \lambda> 0 is some parameter, how do I calculate the the conditional expectation$$E[X_i|X_{\max}],$with$X_{\max}=\max\{X...