finite codimensional subalgebra of $mathbb C[z]$

If $p_1, p_2$ are two polynomials in $mathbb C(z)$, then when $mathbb C(p_1, p_2)$ is finite codimensional in $mathbb C(z)$? Is it always or there are some sufficient conditions (or NASC) on $p_1, p_2$?

co.combinatorics – Is the number of words finite, when you don’t know how to count?

This question is inspired by this one:

Can you do math without knowing how to count?

Let $M_2$ be the set of words constructed by concatenation of the letters $a_1$ and $a_2$, with :

(*) : for any $x$ word of $M_2$ $xx = x$.

Is it true $card(M_2)=card(mathbb N) $?

If not, is it true $exists n in mathbb N, card(M_n)=card(mathbb N) $?

The condition (*) comes from the hypothesis that we assume that we do not know how to count.

arrays – Relating two vectors for finite volume problem

I’m trying to come up with a relation between two vectors such that when I successively loop through each element of one vector, the correct corresponding element from the second vector should be called upon. I’ve identified a pattern between the two vectors but am having trouble coming up with a general statement or set of statements that relate the two. I was hoping someone could provide some guidance as to how I may go about implementing this.

This is part of a finite volume problem where the domain has been split into i rows and j columns. The length of both vectors is i*j.

So for example, when I loop through vector 1 from element 1 to element i*j, the correct corresponding element from vector 2 should be called:

In this case, V1[3] = 3 corresponds to v2[3] = 21

Any advice would be very much appreciated. Thank you.

Vector 1
Rows and columns for vector 1

Vector 2
Rows and columns for vector 2

I’ve tried to come up with a general pattern and failed. I came up with a convoluted series of operations – using the modulus and if statements to perform alternate operations, but I failed to come up with a proper generalization.

Failed attempt
My failed attempt at generalizing this

finite automata – FA for when length of w is 4 or w contains the substring 01

I have been trying to create an FA for the language.

{w ∈ {0, 1}∗ | |w|= 4 ∨ w contains the substring 01}

I created one that accepts words that contain the substring 01, but I have a hard time finding a solution for the length part.

This is my attempt so far:

enter image description here

Any help is greatly appreciated.

finite automata – Does a DFA accepts all strings accepted by the Equivalent NFA?

I have been studying the conversion from an NFA to the equivalent DFA. And I came across this NFA

NFA with ∑ = {0, 1} accepts all strings with 01.

enter image description here

After I converted the NFA to the equivalent DFA.

it became
enter image description here

the issue is that the NFA accepts the string “101” but the DFA doesn’t.

Is my conversion wrong or is there something I am missing about the NFA to DFA conversion?

Turing machine without return equivalent to Finite Automaton, PushDown Automaton or Turing Machine?

Informally, a pushdown automaton has a way to store and use an infinite amount of memory (the stack). In a Turing machine, the only way to store an infinite amount of memory is to write it on the tape. But in a Turing machine without return, you cannot go back to the cells that you have already written, so that memory can never be used.

That means that a Turing machine without return cannot be “as powerful” as a pushdown automaton. The solution is a finite automaton.

If you want a more formal proof, you can start with the formal definition of a Turing machine without return, note that you can dismiss what you write on the tape (because you can’t go back), and construct a finite automaton that recognize the same language as the Turing machine.

pr.probability – Exit probability on a finite interval

I have a question about the estimate of the exit probability on a finite interval. Given a $q$ function bounded and continuous, given the following SDE
begin{cases}
dX_s=(beta-q(s))X_sds+frac{1}{2}beta^2(X_s)^2dW_s \
X_t=y
end{cases}

I should estimate the probability $mathbb{P}{exists s in (t,T) : (s,X_s) in A }$, where $A$ has this form ${(t,x) in (0,T) times (0,+infty) : 0 leq y leq L}$.
Can someone help me with this estimate?

lo.logic – Are there structures in a finite signature that are recursively categorically axiomatizable in SOL but not finitely categorically axiomatizable?

Recall that a structure $mathcal{M} = langle M, I^sigma_M rangle$ in a signature $sigma$ is categorically axiomatized by a second-order theory $T$ when, for any $sigma$-structure $mathcal{N} = langle N, I^sigma_N rangle$, $langle N, mathcal{P}(N), I^sigma_N rangle vDash T$ just in case $mathcal{N}$ is isomorphic to $mathcal{M}$.

It is fairly easy to find a structure in a finite signature that is categorically second-order axiomatizable but not finitely categorically second-order axiomatizable. Add a single function symbol $f$ to the language of second-order arithmetic, and choose a non-second-order-definable $zeta: mathbb{N} rightarrow mathbb{N}$. Then consider the theory $T$ that adds to the axioms of second-order arithmetic ($mathsf{Z}^2$) the sentence $f(bar{n}) = overline{zeta(n)}$ for each natural number $n$, where $bar{m}$ is the canonical numeral for $m$. (I owe the idea for this example to Andrew Bacon.)

This theory $T$, however, is not recursively axiomatizable. Is there a structure in a finite signature that has a recursive categorical second-order axiomatization but no finite categorical second-order axiomatization?

I believe that it is possible to find a recursively axiomatizable second-order theory $T$ whose spectrum (i.e., the set ${kappa in mathsf{Card}: exists mathcal{M} (mathcal{M} vDash T$ and $vert mathscr{M} vert = kappa)}$) is shared by no finitely axiomatizable second-order theory, using partial truth predicates. (Consider the theory with $mathsf{Z}^2$ relativized to some predicate $N$ and ${$“The cardinality of the non-$N$s is not $Sigma^1_n$-characterizable”$: n in omega}$.) But I cannot see how to turn this into a categorical theory.

gt.geometric topology – Thurston measure under finite covers

Let $S= S_{g,n}$ be a finite type orientable surface of genus $g$ and $n$ punctures and let $mathcal{ML}(S)$ denote the corresponding space of measured laminations. The Thurston measure, $mu^{Th},$ is a mapping class group invariant and locally finite Borel measure on $mathcal{ML}(S)$ which is obtained as a weak-$star$ limit of (appropriately weighted and rescaled) sums of Dirac measures supported on the set of integral multi-curves.

The Thurston measure arises in Mirzakhani’s curve counting framework. Concretely, given a hyperbolic metric $rho$ on $S_{g,n}$, let $B_{rho} subset mathcal{ML}(S)$ denote the set of measured laminations with $rho$-length at most $1$. Then $mu^{Th}(B_{rho})$ controls the top coefficient of the polynomial that counts multi-curves up to a certain $rho$-length and living in a given mapping class group orbit.

Question: Fix a hyperbolic metric $rho$ on $S$ and a finite (not necessarily regular) cover $p: Y rightarrow S$. Let $rho_{p}$ denote the hyperbolic metric on $Y$ obtained by pulling $rho$ back to $Y$ via $p$. Is there a straightforward relationship between $mu^{Th}(B_{rho_{p}})$ and $mu^{Th}(B_{rho})$? For example, is the ratio
$$ frac{mu^{Th}(B_{rho})}{mu^{Th}(B_{rho_{p}})} $$
uniformly bounded away from $0$ and $infty$? Does it equal a fixed value, independent of $rho$? If so, can it be easily related to the degree of the cover $Y rightarrow S$?

It seems hard to approach the above by thinking about counting curves on $Y$ versus $S$, because “most” simple closed curves on $Y$ project to non-simple curves on $S$. But, maybe the generalizations of curve counting for non-simple curves due to Mirzakhani (https://arxiv.org/pdf/1601.03342.pdf) or Erlandsson-Souto (https://arxiv.org/pdf/1904.05091.pdf) could be useful. Of course, both apply to counting curves in a fixed mapping class group orbit, so it’s not clear (to me) how to apply these results either since multi-curves on $Y$ can project to curves on $S$ with arbitrarily large self intersection.

Thanks for reading! I appreciate any ideas or reading suggestions.

finite automata – Build an FA that accepts only the words baa, ab, and abb and no other strings longer or shorter

I have been trying solve this problem for a while now for a university assignment. I’m required to build a DFA and an NFA for the above question. So far I have been able to solve the DFA but can not find a solution for a proper NFA after multiple tries.

The solution of my DFA for the language provided above:

My attempts for the NFA are down below. I apologize for my messy handwriting but these were just rough works that I was drawing out on the go.

My first attempt at solving the NFA

My second attempt at solving the NFA

My third attempt at solving the NFA