Articles of self learning

Venn diagram question

Here is my question. A math examination has three questions. Twenty-six students took the examination, and every student answered at least one question. Six students did not answer the first question; twelve did not answer the second question; and five did not answer the third question. If eight students answered all three questions, how many […]

Baker's transformation: continuity, orbits of irrational and rational points

I’ve reading the Pugh’s Analysis book and I have problems with one exercise. This says: The baker’s transformation: a rectangle of dough is stretched to twice its length and folded back on itself. Is this transformation continuous? A formula for the baker’s transformation in one variable is $f(x)=1-|1-2x|$ (…) The orbit of a point $x$ […]

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it’s the same or am I missing something else. A book I’m reading used a rule called Fatous lemma to give the […]

Open sets in the topology of weak convergence

I do have various questions regarding the topic of probability measures on polish spaces in general, thus I am trying to divide them in “small” subquestions. Hence, this is my first question on this issue. Notation: $\Omega$ is a Polish space; $\mathcal{B}(\Omega)$ is the Borel $\sigma$-algebra on $\Omega$; $\Delta (\Omega)$ is the set of probability […]

Probability returning to initial state

Let $P=\begin{bmatrix}0&\frac{1}{2}&\frac{1}{2}\\\frac{1}{2}&0&\frac{1}{2}\\\frac{1}{2}&\frac{1}{2}&0\end{bmatrix}$ and $P^{(n+1)}=P^{(n)}P.$ I know that if you start in any vertex the probability of return after $n$ hops is the same $P_{11}^{(n+1)}=P_{22}^{(n+1)}=P_{33}^{(n+1)}$ then $$p_{11}^{(n+1)}=\frac{1}{2}p_{12}^{(n)}+\frac{1}{2}p_{13}^{(n)}$$ since $p_{12}^{(n)}+p_{13}^{(n)}=1$ $$p_{11}^{(n+1)}=\frac{1}{2}\left(1-\frac{1}{2}p_{12}^{(n)}\right)$$ but I do not know how to solve this recurrence relation or if is right.

A Counter Example about Closed Graph Theorem

This is an example I read around closed graph theorem. Let $Y=C[0, 1]$ and $X$ be its subset $C^\infty[0, 1]$. Equip both with uniform norm. Define $D: X \to Y$ by $f \mapsto f’$. Suppose $(f_n, f_n’) \to (g, h)$ in $X \times Y$. Thus, $f_n \to g, f_n’ \to h$ both uniformly on $[0, […]

system of congruences proof

I’ve checked a lot of the congruency posts and haven’t seen this one yet, so I’m going to ask it. If there is a related one, I’d be happy to see it. Let $x \equiv r\pmod{m}, x \equiv s\pmod{(m+1)}$. Prove $$x \equiv r(m+1)-sm \pmod{m(m+1)}$$ So with the given conditions, we know $$x=mk_1+r$$ Then $mk_1+r \equiv […]

Determinant of an almost-diagonal matrix

I would like to compute the determinant of the $(k+1)\times (k+1)$ matrix below $$J=\begin{vmatrix} y_{k+1}& 0 & \ldots & 0 & y_1 \\ 0& y_{k+1}& \ldots& 0& y_2 \\ \vdots& \vdots& & \vdots &\vdots \\0 & 0&\ldots& y_{k+1} &y_k \\ -y_{k+1} & -y_{k+1} &\ldots &-y_{k+1}& \left(1-y_1-\ldots-y_k \right) \end{vmatrix} $$ The matrix is diagonal, if you […]

Primitive Recursion Functions (Programs)

The set $F_{n}$ of primitive recursive function symbols of arty $n$ can be defined inductively as \begin{array}[lr] & Z, \text{Succ} \in F_{1} & \\ \pi_{j}^{n} \in F_{n} \quad \text{for each} \quad j=1,\dots, n \\ &\text{if} \quad f \in F_{n} \quad \text{and} g_{1},\dots, g_{n} \in F_{m}, \text{then} \circ_{n}^{m}[f,g_{1},\dots,g_{n}]\in F_{m} & \\ &\text{if} \quad f \in F_{n+2} \quad […]

Chain rule (proof verification)

Hi everyone I’m asking two thinks is this proof correct (my other idea was using limit of sequences)? and are there a simpler alternative than this using Newton’s approximation? If someone could help me I’d be so thankful. Proposition: Let $X,Y$ be subsets of $\mathbb{R}$, $f:X\rightarrow Y$ and $g:Y\rightarrow \mathbb{R}$ be functions, $x_0\in X$, $y_0\in […]