# All Questions

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### rewriting one expression as other expression

Can anyone explain as how we can rewrite the first expression as second ? I am not able to pick the step done to change from 1 to 2. ...
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### I need help to verify 5 order relations

Which the following relations are order relations on the set $\Bbb M$? $$\Bbb M: \{1, 2, 3\}$$ $$R:\{(1, 1),(3, 3),(1, 2),(2, 3),(1, 3)\}$$ It is not an order relation because $(2,2) \not \in R$ => ...
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### Developed a function optimization strategy - need opinions

I've developed a function optimization strategy which is close to evolutionary optimization strategies. It works fine for various functions, but cannot be used with thorough success for functions with ...
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### Considering the complex number $z = m+i$ for which values of $m$ do we have $\left|\overline{z}+\frac{2}{z}\right| \ge 1$

Good evening to everyone. I have the following problem that I tried to solve but my mathematical instinct tells me that I didn't solve it right: Considering the complex number $z = m+i$ for which ...
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### Why $\sup\limits_{n\ge 1} \sum_{k=1}^n a_k = \lim\limits_{n\to\infty} \sum_{k=1}^n a_k$?

If $\{a_n\}$ is a positive numbers sequence, then $\sup\limits_{n\ge 1} \sum_{k=1}^n a_k = \lim\limits_{n\to\infty} \sum_{k=1}^n a_k$. Is this wrong or right? And why?
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### Russel's paradox: what is the contradiction with $R \not\in R$?

Let the Russel's Set be: $$R = \{S | S \notin S\}$$ Where $S$ is a set Suppose $R \in R$, but by definition $R \not\in R$, contradiction. Suppose $R \not\in R$... (I am not sure what should be the ...
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### Proving that if $A\subseteq B$ and $B\subseteq C$ then $A\subseteq C$.

. If $A\subseteq B$ and $B\subseteq C$ Prove that $$A\subseteq C$$
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### Is there a method that determines an unknown permutation better than $(n+1)!/2^n$ steps on average?

Suppose I have a random permutation $s \in S_n$ that is unknown to me. However, suppose I can make a query where when I ask if $i$ is in the $j$th position in the permutation, I receive a yes or no ...
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### Poincaré–Bendixson theorem on the torus

I was reading the paper A Generalization of a Poincaré-Bendixson Theorem to Closed Two-Dimensional Manifolds by Arthur J. Schwartz which proves the following theorem: THEOREM. Let $M$ be a ...
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### distribution and density of maximum minus element

I am a bit rusty in probability, and for a project I am studying the random variable $Z = max(X_1, \ldots, X_n) - X_i, i = 1, \ldots, n$ where the $X_i$ are positive independent random variables. In ...
### Dual of the Banach space of $k$-times continuously differentiable functions.
Let $C^k([0,1])$ denote the Banach space of $k$-times continuously differentiable functions $f:[0,1]\to \mathbb R$ with norm $$\|f\|_{C^k}:=\max_{i=0,\dots,k}\sup_{x\in [0,1]}|f^{i}(x)|.$$ I'm trying ...