I Can chatgpt accurately calculate expected lengths in Pascal's triangle?

[Office_Shredder]

#8) Prove or give a counterexample:
Every ideal in a commutative ring is the intersection of the prime ideals containing it.

 

[fresh_42]

$$12\mathbb{Z}\subsetneq 6\mathbb{Z}=\bigcap_{p|12} p\mathbb{Z}$$

 

[Infrared]

#7,). Prove or give a counterexample: Every irreducible complex representation of a compact group is unitary

A lot of words there. Is irreducible necessary? Does the representation need to be continuous? Maybe someone will find out!

I think representations of topological groups are by definition continuous. If you don't require continuity, this looks very false. For example, consider the group $GL(n,\mathbb{C})$ equipped with the indiscrete topology, which trivially makes it compact. Then the identity map (which is not continuous) $GL(n,\mathbb{C})\to GL(n,\mathbb{C})$ can be considered as a representation, which is certainly not unitary. The righthand side of course has the standard topology.

Assuming continuity, if the group is also Hausdorff, then it possesses a Haar measure $\mu$ and any representation is unitary, using the inner product $\langle u,v\rangle=\frac{1}{\mu(G)}\int_G \langle gu,gv\rangle_0 d\mu(g)$ where $\langle \rangle_0$ is any fixed inner product.

Even if the group $G$ is not Hausdorff, the image of the representation $\rho:G\to GL(V)$ is a compact subgroup $\rho(G)\subset GL(V)$ which is Hausdorff (since $GL(V)$) is. So we just apply the above argument to the group $\rho(G)$ instead, thinking of $\rho(G)\hookrightarrow GL(V)$ as a representation.

I don't see where irreducibility should factor in.

 

[Office_Shredder]

I think representations of topological groups are by definition continuous. If you don't require continuity, this looks very false. For example, consider the group $GL(n,\mathbb{C})$ equipped with the indiscrete topology, which trivially makes it compact. Then the identity map (which is not continuous) $GL(n,\mathbb{C})\to GL(n,\mathbb{C})$ can be considered as a representation, which is certainly not unitary. The righthand side of course has the standard topology.

Nice example.

In the case where the representation map is continuous, I was thinking of a bit of a different argument. Let's suppose we have some representation, and the image is a Hilbert space (otherwise it doesn't make sense to ask if it's unitary). There is a natural norm defined on operators, $||A||=sup_{||x||=1} ||Ax||$. . This is only defined for linear operators that are continuous (in infinite dimensions this could be infinity, but then $A$ it's not continuous)

This norm defines a topology on continuous operators. If this is the topology with respect to which the representation is continuous (which requires the image to lie in the space of continuous operators), then claim, for $A=\rho(g)$ in the image of the representation, $||Ax||=1$ for all unit vectors. For finite dimensions this follows by compactness (there is a vector that maximizes the norm, you can show it's an eigenvalue, hence $A^n$ or $A^{-n}$ has arbitrarily large norm) therefore the representation is unitary. . I'm not actuality sure if it holds for infinite dimensional representations though.

Also, new question.

#9) Prove every lebesgue measurable set of positive measures contains measurable subsets of arbitrarily small positive measure, or provide a counterexample.

 

[Infrared]

there is a vector that maximizes the norm, you can show it's an eigenvalue

This doesn't look true to me: The matrix $A=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$ has only a one dimensional eigenspace $\text{Span}\left(\begin{pmatrix} 1 \\ 0\end{pmatrix}\right)$ but $\text{max}_{||x||=1} ||Ax||$ occurs when $x=\pm \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. This specific matrix can't come from a compact representation because it generates a non-compact subgroup though.

It looks like you're arguing that if $H$ is a Hilbert space, then every continuous representation $\rho:G\to GL(H)$ lands in $U(H)$ when $G$ is compact. Even in the finite-dimensional case, I think this is false. Since $U(n)$ is not a normal subgroup of $GL(n,\mathbb{C})$ pick an invertible matrix $A$ for which $A U(n) A^{-1}\neq U(n)$ and use the representation $U(n)\to GL(n,\mathbb{C})$ given by $U\mapsto AUA^{-1}.$

My argument was to show the weaker claim, which I interpreted as the problem statement, that if $\rho:G\to GL(V)$ is a continuous representation of compact $G$, then there exists an inner product on $V$ for which every $\rho(g)$ is unitary with respect to that inner product (but the inner product should not be set in advance).

 

[Infrared]

9. This seems suspiciously easy but I'll write something down: Let $A$ be a measurable subset of $\mathbb{R}$. For $x\geq 0$ define $f(x)=m(A\cap [-x,x]).$ Note that $f$ is (uniformly) continuous because $|f(x)-f(y)|\leq |x-y|.$ Since $f(0)=0$ and $\lim_{x\to\infty}f(x)=m(A),$ by the intermediate value theorem, $f$ should attain every value in $[0,m(A)),$ so $A\cap [-x,x]\subset A$ can be made to have arbitrarily small measure. The same argument works in $\mathbb{R}^n$ replacing $[-x,x]$ with the closed ball of radius $x.$

 

[Office_Shredder]

This doesn't look true to me: The matrix $A=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}$ has only a one dimensional eigenspace $\text{Span}\left(\begin{pmatrix} 1 \\ 0\end{pmatrix}\right)$ but $\text{max}_{||x||=1} ||Ax||$ occurs when $x=\pm \begin{pmatrix} 0 \\ 1 \end{pmatrix}$. This specific matrix can't come from a compact representation because it generates a non-compact subgroup though.

Shoot. I dug into this a little bit, and the maximum value occurs at an eigenvector of $A^*A$. The rest of my post quickly falls apart, whoops.

I like your answer for #9, and it gives a much stronger result than my approach which was to cover the whole space in $\epsilon$ width intervals and observe one intersection must have positive measure by additivity.

#10 feels like it might be a standard result every first year undergrad knows, but it doesn't quite match the ones I thought of when I saw these words z so might even be false.

#10) Prove or disprove: Every uniformly continuous function that is differentiable on a closed interval is necessarily Lipschitz continuous.

I think differentiable on a closed interval requires the one sided derivatives to exist on the endpoints, but if dropping that changes the answer I'm interested in that fact.

 

[Infrared]

#10) Prove or disprove: Every uniformly continuous function that is differentiable on a closed interval is necessarily Lipschitz continuous.

Let $f(x)=x^2\sin(1/x^2)$ for $x\neq 0$ and $f(0)=0,$ let's say defined on $[-1,1].$ We see that $f'(x)=2x\sin(1/x^2)-\frac{2}{x}cos(1/x)$ for $x\neq 0$ and $f'(0)=\lim_{h\to 0} \frac{h^2\sin(1/h^2)}{h}=0.$ As $f$ is continuous on a compact domain, it is uniformly continuous. Since $f'$ is not bounded, but any differentiable Lipschitz function has derivative bounded by its Lipschitz constant, this $f$ cannot be Lipschitz.

 

[Office_Shredder]

#11) given three points on a plane, not collinear, prove there exists a unique circle passing through them.

 

[Office_Shredder]

#12) Prove or disprove: Every graph of 30 vertices has two vertices with the same degree (same number of edges)

Is 30 important here? It actually asked me a question about friends that might have been a mix between a graph theory question and the birthday problem but this is what I reduced it to.

 

[Office_Shredder]

#13) Prove or disprove: Every infinite dimensional Banach space has a linearly independent sequence that is not a basis.

 

[Muu9]

TL;DR Summary: Prove you are smarter than the bot!

Chatgpt is actually pretty good at generating math problems. It's awful at solving them. I guarantee every question posted here cannot be solved by chatgpt, but maybe can be solved by a human? My plan is to spend a couple minutes getting a question I think it's cool and then posting it here - I don't know if I'll actually do it each day.

Since chatgpt is bad at knowing whether things are true or not, and I'm not going to try to solve all of these before posting, most will be of the form prove or find a counterexample
#5.) .I take a piece of string of length 1, and do the following k times: cut the string into a ratio of 2 to 1 (so a string of length 1 is cut into a string of length 2/3 and a string of length 1/3) I then throw out one of the pieces, leaving myself with a single piece that I can repeat this process on, until I have made k cuts.

What is the expected length of the final piece of the string in terms of k, if
a.) Each time I pick one of the two pieces to throw out randomly with equal probability
b.) Each time I "grab" a random spot on the string and throw that piece out - i.e. I have a 2/3 probability of throwing out the longer piece

For a, imagine a pascal's triangle of height k+1, but instead of the standard digits we have 1 at the top, then (2/3) and (1/3) on the second row, then (2/3)^2, (2/3)(1/3), and (1/3)^2 on the third row, and so on.
The k+1th row will have be (2/3)^k*(1/3)^0, then (2/3)^(k-1)*(1/3)^1, then (2/3)^(k-2)*(1/3)^2, and so on until finishing with (2/3)^0*(1/3)^k. To adjust for the fact that the middle numbers are more likely, each of these values is multiplied by the corresponding value of the normal Pascal's triangle. The mean of these adjusted values is the expected value of the final length. Anyone know how to express this in a more traditional, concise form?

For b, do the above but square each value before multiplying by the corresponding value of the normal Pascal's triangle.

Which prompt are you using to generate these?

 

[nuuskur]

A sandwich semigroup is a semigroup $S$ such that $S=SES = \{ses' \mid s,s'\in S, e\in E(S)\}$, where $E(S)$ is the subset of idempotents of $S$. An idempotent $e\in S$ is an element that satisfies $e^2=e$.

A semigroup $S$ is said to be closed if the map $S\otimes _SS\to S,\ s\otimes s'\mapsto ss'$ is bijective.

Claim. All sandwich semigroups are closed.
(False)

 

[Office_Shredder]

Which prompt are you using to generate these?

It's a bit of messing around. A common one for the more advanced questions is something like "string together random words from topology to generate a question of the form prove or disprove". The lower level ones I have asked it to generate brand new high school math competition problems. I probably only use about 20% of the responses, and they usually need some tweaking (e.g. the one about a holomorphic function which is real on the circle, was originally about a holomorphic function which was real on the real axis, which is, uh, less interesting). The string cutting one was also pretty different from what the bot actually generated, though I don't remember the changes I made.

I stopped making these because it seemed like no one was interested except for Infrared and fresh who had stopped showing up as much, if there is renewed interest I can make some more.

 

[fresh_42]

I stopped making these because it seemed like no one was interested except for Infrared and fresh who had stopped showing up as much, if there is renewed interest I can make some more.

I made so many ridiculous mistakes that I thought a break would be a good idea.

 

[Office_Shredder]

@Muu9 for the first one consider expanding $(2/3+1/3)^k$

 

[Fred Wright]

For a, imagine a pascal's triangle of height k+1, but instead of the standard digits we have 1 at the top, then (2/3) and (1/3) on the second row, then (2/3)^2, (2/3)(1/3), and (1/3)^2 on the third row, and so on.
The k+1th row will have be (2/3)^k*(1/3)^0, then (2/3)^(k-1)*(1/3)^1, then (2/3)^(k-2)*(1/3)^2, and so on until finishing with (2/3)^0*(1/3)^k. To adjust for the fact that the middle numbers are more likely, each of these values is multiplied by the corresponding value of the normal Pascal's triangle. The mean of these adjusted values is the expected value of the final length. Anyone know how to express this in a more traditional, concise form?

For b, do the above but square each value before multiplying by the corresponding value of the normal Pascal's triangle.

Which prompt are you using to generate these?

For brevity of notation let $a=\frac{2}{3}$, $b=\frac{1}{3}$, $C_{k,n}$ be the binomial coefficient of the k'th row and n'th column of Pascal's triangle and $l_k$ be the expected length for the k'th row of Pascal's triangle. For part a of the problem the expected length of the k'th cycle is
$$
E[l_k]=\frac{\sum_{n=0}^{k}C_{k,n} a^{k-n}b^{n}}{\sqrt{\sum_{n=0}^{k}C_{k,n}^2}}
$$
For part b of the problem the expected length is
$$
E[l_k]=\frac{\sum_{n=0}^{k}C_{k,n} a^{2(k-n)}b^{2n}}{\sqrt{\sum_{n=0}^{k}C_{k,n}^2}}
$$