# All Questions

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### Splitting Probability Within a Finite Time

Let $\mathcal{X}=\{x_i\}_{i=1}^{N}$ be a subset of $\mathbb{Z}$. For $j\in(1,N)$, what is the probability that the first element of $\mathcal{X}$ encountered by a simple 1D random walk is $x_j$ and ...
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### Problem in finding examples of linear operators.

Find the example of two linear operators $T$ and $U$ such that $TU = O$ but $UT$ is not a zero operator. But I fail to find out proper example.Please help me in finding the example.Thank you in ...
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### Factorize a third degree polynomial

It's my first time posting here so I'm not used to describing my problem in mathematics. I'm currently trying to solve a problem which asks if a 3x3 matrix is diagonalizable, I know the method but ...
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### What is the formal definition of repeated limit?

The basic question is what has been asked in the title. I looked for the definition here, here and here but no definition uses quantifiers. I tried to do so but couldn't.
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### When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this ...
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### Bounded linear operator counterexample?

Does there exist a continuous linear operator $T: (C[0,1],||.||_2) \to (C[0,1],||. ||_2)$ such that $T$ is discontinuous if $C[0,1]$ is considered with $||.||_\infty$ instead of $||.||_2$ ? I think ...
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### Real analysis and Topology book recommendetion

I hold a bachelor's degree in mathematics and I have taken undergraduate real analysis course. But I'm interested to know more about it, especially the stuff that will help me to understand more of ...
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If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\... 0answers 10 views ### Relations, Ordered Pairs, Naive set theory by Halmos I quote: "Explicitly: a set R is a relation if each element of R is an ordered pair;" The question is: "what about the converse? is a set of ordered pairs could be considered a relation?" 1answer 15 views ### Question of P &C There are 3 pots and 3 coins. All thesecoins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed such that no pot ... 0answers 6 views ###$\mathbf{E}\left[\frac{(U_1+c)^2}{\max((U_1+c)^2, U_2^2)} \right] \ge \mathbf{E}\left[\frac{U_2^2}{\max((U_1+c)^2, U_2^2)} \right]$We consider two i.i.d. random variables$U_1$and$U_2$such that$\mathbf{E}[U_1] = \mathbf{E}[U_2] = 0$and$\textrm{Var}[U_1] = \textrm{Var}[U_2] < \infty$. Prove that for any$c > 0$the ... 1answer 17 views ### Proving homotopy equivalence of a torus with points removed Suppose I'd like to show that a Torus with$n$points removed is homotopy equivalent to a wedge sum of$n+1$circles. I depict it in a usual way - as a rectangle with$n$points removed inside. Now it ... 2answers 23 views ### Notation for a sum over a set of variables I have a vector of variables$y=(y_1, \ldots, y_n)$whose elements are either zero or one. I would like to express the sum over all variables belonging to a subset$S$. For example, if$n=4$and$S=\{...
The well-known Pólya-Vinogradov Inequality states: $\forall m, n \in \mathbb{N}: \displaystyle \left|{\sum_{k \mathop = m}^{m+n} \left({\frac k p}\right)}\right| < \sqrt p \ \ln p$. I would like ...