# All Questions

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### Finding the range of $f(x) = 5-x$

Let $a < b$ and $f: (a, b] \to R$, $f(x) = 5 - x$. What is the range of the function? How do you find the range of an equation with unknowns?
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### Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
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### Sturm Liouville form

How do you put $u'' +c u' +d =0$ into regular SL-form? Can not see how it's an eigenvalue problem without a first order term. But the theorem states EVERY second order operator can be put into SL ...
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In these notes, an extension of the subgradient method is presented in Section 8 (page 30). The method is described so quickly and neither convergence analysis (compared to classical subgradient for ...
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### 2-dimensional Cohen-Macaulay domain

I am searching for a $2$-dimensional Cohen-Macaulay (normal or not) domain. Thanks in advance for any suggestion.
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### How do I differentiate ${(e^e)}^x$?

I know how to differentiate $e^x$ (it's just $e^x$), but how do I differentiate ${(e^e)}^x$? any hints would be welcome.
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### Is this module noetherian?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Assume that $A$ is a noetherian ring, and let $I\subseteq A$ be a proper ideal. Consider the ideal $I\otimes_k A \subseteq A\otimes_k A$....
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### Are random walk variations Markov-Chains?

Let $S_{n}:= S_0 + \sum_{i=1}^{n}X_i$ be a simple random walk, $X_i$ are independent random variables with $P[X_i=1] = p, P[X_i = -1] = 1-p$. Let $M_n:=\max\{S_0, \dots, S_n\}$. The task at hand is ...
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### Sin along a vector?

I have a 2D vector that I currently use to plot a line in its direction. I calculate the next point using: p_new = p_old + v * i Were i going from 0 to some ...
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### Deduct percentage from x to reach desired value

Let's suppose I need \$200,000 to pay a debt, no more, no less. So I placed a property to sell with an agent, but I know the agent will deduct 6% for himself, which will amount to \$12,000. Since ...
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### In Dijkstra algorithm, it takes the source, what about the sink?

I'm studying the Dijkstra algorithm, but in my book, the algorithm takes as input only the graph and the source. Why it doesn't ask for the destination vertex? How can it work? Thanks a lot.
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### Proving closure of unit space of a Hausdorff groupoid

For Hausdorff topological groups, the set $\{e\}$ containing only the identity is closed. This is because Hausdorff implies T1 which implies singletons are closed. For topological groupoids, defined ...
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### Bayes Probability Problem

I need a small confirmation regarding a probability problem: We estimate that 5% of Americans spent their holidays in Texas, this proportion reaching 40% among Texans. Texans represent 2% of the ...
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### How do I form these quadratic equations

A car travels a distance of $1200 \ km$ at a speed of $a \frac{km}{h}$, while a train travels the same distance at $(a – 20) \frac{km}{h}$. If the time taken by the train is $5$ hours more than the ...
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### Show that, $Z$ is $\mathcal N(0,1)$

If $Y\sim\mathcal N(0,1)$ and let $a>0$. Let $$Z=\ \begin{cases} Y&\text{if } |Y|\le a\\ -Y &\text{if }|Y|> a\\ \end{cases}\$$ Show that $Z\sim\mathcal N(0,1)$ I ...
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### Collection vs set in this textbook about category theory, and some related questions.

What is the meaning of collection in this context ? Is it here a synonym of set ? Can someone please explain what the author means by "A moment's though shows that, as sets of functions, these two ...
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### Conditional probability and distribution

Let Y ∼ Exp(1/5). Find P(Y ≤ 18|Y > 13). Could anyone give me any hints?
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### How to resolve multiply differentiation function algorithms?

My simple function is $f(x)=\frac{1}{2}e^{-x}\sin(2x)$; Can I resolve for multiply differentiation $f^{(n)}=?$ algorithm? Thx for answer.
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### Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. My ...
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### Combination problem technique

How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the units place must be greater than that in the tenth place? It can be easily solved that, the total ...
I have just correctly solved this programming problem. The problem is the following: $N$ teams play a round-robin tournament, i.e. each pair of teams plays exactly one game and the winner gets 3 ...
Let $R$ be a reduced root system. ($R$ is a finite set spanning $V$, $\alpha \in R \rightarrow -k\alpha \in R$ iff $k=1$, $s_{\alpha}(R)=R$, $s_{\alpha}(\beta)-\beta=k\alpha$ whit $k$ integer). ...