Measure of convergence of interated sequences

For the function $f(x)=x^3-2x+2$, let $g(x)=x-\frac{f(x)}{f'(x)}$ and $M\subseteq \mathbb R$ be the set of points $c$ for which the sequence $(x_n)_{n\in\omega}$ with $x_0=c$ and $x_{n+1}=g(x_n)$ for $n\in\omega$ is convergent.

Problem. What is $\lim\limits_{R\to\infty}\frac{\lambda(M\cap[-R,R])}{2R}$?

Here $\lambda$ is the Lebesgue measure on the real line.

Prize: A brick of Pu'er tea.

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This problem was posed on 7 August 2025 by Yevhen Azarov on page 174 of Volume 3 of Lviv Scottish Book.

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Diameter-Triameter Problem

Let $G$ be a connected simple graph on $n\ge 3$ vertices. The triameter of $G$ is defined as $$tr(G)=\max\{d(u,v)+d(v,w)+d(u,w):u,v,w\in V\}.$$

Consider the following properties:

(A) Any triametric triple contains a diametric pair;

(B) Any diametric pair can be extended to a triametric triple.

(A') Any triametric triple contains a peripheral vertex (i.e., a vertex that can be completed to a diametric pair).

(B') Any peripheral vertex can be extended to a triametric triple.

Problem. Does (B) (or at least (B')) hold for median graphs?

Prize: 1 kg of pistachio.

Notes:

1. The property (A) does not hold for median graphs.
2. Both (A) and (B) hold for trees and block graphs.
3. Neither modular nor distance-hereditary graphs satisfy any of those properties (even their weak versions).

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This problem was posed on 10 February 2025 by Artem Hak and Sergiy Kozerenko on page 154 of Volume 3 of Lviv Scottish Book.

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Generalised Hermite polynomials

I am interested in the derivatives ${d^m\over dx^m} e^{x^n}$ for all $m,n \in \mathbb N$. I understand that when $n=2$, these can be understood in terms of the (physicist's) Hermite polynomials $H_m(x)$, which are polynomials in $x$ such that $${d^m\over d x^m}e^{x^2}=H_m(x)e^{x^2}$$ for all $ m \in \mathbb N$. These have a nice closed formula: $$\label{eq:Hermite} H_m(x) = \sum_{i=0}^{\lfloor m / 2 \rfloor} {(-1)^im!\over(m-2i)!i!}(2x)^{m-2i}.$$ One sees easily that there are polynomials $H_m^{(n)}(x)$ for any $ n \in \mathbb N$ such that $${d^m\over dx^m}e^{x^n} = H_m^{(n)}(x) e^{x^n}.$$ Here are my questions:

1. Is there a known closed formula for $H_m^{(n)}(x)$ which generalises the closed formula for $H_m(x)$?
2. Is it possible to make the closed formula for $H_m(x)$ (and generalisation for $n>2$) more combinatorial? I perhaps want to see some Stirling numbers or Lah numbers appearing here, possible coming from the combinatorial Faà di Bruno formula.

Click here for the corresponding StackExchange post.

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Can the self-similarity of the Collatz difference function $D(n) = |T(n)-n|$ lead to a proof of $M(n) = O(n^{1.085})$?

I have been investigating the Syracuse function $T(n) = (3n+1)/2^k$ where $k = v_2(3n+1)$, and the difference function $D(n) = |T(n)-n|$.

Through computational experiments, I discovered a striking self-similarity pattern when plotting $D(n)$ within blocks $I_m = [2^{m-1}, 2^m - 1]$. Specifically, for $n = 2^{m-1} + t$ with $0 \le t < 2^{m-1}$, the behavior of $D(n)$ depends only on $t \bmod 8$:

• If $t \equiv 0,2,4,6,3,7 \pmod{8}$, then $D(n) = D(t) + 2^{m-2}$
• If $t \equiv 1 \pmod{8}$, then $D(n) = D(t) + 2^{m-3}$
• If $t \equiv 5 \pmod{8}$, then $D(n) = D(t) + 2^{m-4}$ (for $m \ge 4$)

This self-similarity is visually clear when plotting $D(n)$ for consecutive blocks.

From this structure, I have proven two results:

1. Bounded difference for class C: For all numbers not in special classes (A: powers of 2, B: $3n+1=2^k$), we have $D(n) \le 0.75n$.
2. No consecutive growth steps: There is no $n$ such that $T(n) > n$ and $T(T(n)) > T(n)$ simultaneously My question: Can this self-similarity structure and the prohibition of consecutive growth be leveraged to prove a sub-exponential bound on the maximum of Collatz trajectories? Specifically, I conjecture that $$M(n) = \max_{k \ge 0} T^{(k)}(n) = O(n^\alpha) \quad \text{with} \quad \alpha = 1 + \frac{\log_2 1.125}{2} \approx 1.085.$$

The heuristic argument: The worst-case allowed growth factor over two steps is approximately $9/8 = 1.125$. Since blocks $I_m$ have size $2^{m-1}$, and the self-similarity suggests a "compression" effect keeping trajectories from expanding too rapidly, traversing $O(\log n)$ blocks would give the exponent $\alpha = \log_2 \sqrt{1.125} + 1 \approx 1.085$.

What I'm looking for:

1. Is there a known approach to convert such self-similarity observations into rigorous bounds on $M(n)$?
2. Are there similar results in arithmetic dynamical systems that might apply here?
3. Could the self-similarity be expressed in terms of transfer operators or generating functions to obtain growth estimates?

References to related work or suggestions for making the heuristic argument rigorous would be greatly appreciated. enter image description here Question: I've discovered that $D(n)$ exhibits exact self-similarity on dyadic intervals $[2^{m-1}, 2^m-1]$ with explicit recurrence formulas depending on $n \bmod 8$. From this I proved $D(n) \le 0.75n$ for generic numbers and no consecutive growth steps. Can this structure lead to a rigorous proof that the maximum trajectory value satisfies $M(n) = O(n^{1.085})$? What techniques from dynamical systems might convert the observed self-similarity into growth bounds?

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Fields $\ (\mathbb R\,\ \oplus\, \cdot\,\ 0\ \ 1) $

Let operation $\, \ \oplus:\mathbb R^2\to\mathbb R\,\ $ be such that $$ (\mathbb R\,\ \oplus\, \cdot\,\ 0\ \ 1) $$ is a field, where $\ \cdot\ $ is the ordinary multiplication of the real numbers. There are infinitely many of different operations $\ \oplus\ $ like this -- call them real additions.

Question:   What is the cardinality of the set of all real additions?

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Reference request: Variance decomposition in Markov-Switching models with heavy-tailed (Lévy) components

I am analyzing the dispersion properties of a viral vector mobility model governed by a Markov-Switching Hidden Markov Model (MS-HMM). The system switches between a local diffusion state and a heavy-tailed transport state (Lévy flight).Using the Law of Total Variance logic, I have derived a decomposition for the dispersion (interpreted as the Mean Squared Displacement MSD(t) over a finite time or truncated variance):$$\text{MSD}(t) \approx \underbrace{ (\pi_1 D_{\text{local}} + \pi_2 D_{\text{Lévy}}) }_{\text{Diffusive Component}} + \underbrace{ \text{Var}(v \cdot \tau) }_{\text{Ballistic/Transport Component}}$$Where:$S_t \in \{E_{\text{local}}, E_{\text{Lévy}}\}$ is the latent state.The Lévy component has index $\alpha < 2$.It appears that under these conditions, the second term leads to super-diffusive scaling $\sim t^{2/\alpha}$, overriding the additivity usually seen in Gaussian MS-HMMs.My Question:Is this explicit decomposition for hybrid Gaussian/Lévy MS-HMMs a known result in the probability or statistical physics literature? I am looking for references that treat the MSD decomposition when one of the switching states lacks a finite second moment.

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Relating $\dim H^1_f(\mathbb{Q},V_p(E)(n))$ to zeros of $L_p(E)$

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime of good reduction. Put $V:=V_p(E)=T_p(E)\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$, and for $n\in\mathbb{Z}$ let $H^1_f(\mathbb{Q},V(n))$ be the Bloch--Kato Selmer group of the Tate twist.

Let $L_p(E)$ be the (cyclotomic) $p$-adic $L$-function of $E$. I am looking for a reference for a theorem of the following type:

Under suitable hypotheses on $E$ and $p$, one has $$ \dim_{\mathbb{Q}_p} H^1_f(\mathbb{Q},V(n)) = \operatorname{ord}_{s=s(n)} L_p(E,s) $$ for some point $s(n)$ determined explicitly by $n$ (including the correct normalization of the $s$-variable).

In particular, in analytic rank $0$ (i.e. $L(E,1)\neq 0$), is there a reference which explicitly implies $\dim_{\mathbb{Q}_p} H^1_f(\mathbb{Q},V(1))\le 1$?

Any precise pointers (with hypotheses and normalizations) would be greatly appreciated.

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How to write down a crepant resolution of this variety?

We work over $\mathbb{C}$. I have been studying crepant resolutions of threefolds and am trying to write down explicit examples. Let $X\subset \mathbb{A}^{4}$ be the hypersurface defined by $y^{2}=x^{3}+u^{4}x+u^{5}v$. This defines an elliptic fibration over $\mathbb{A}^{2}_{u,v}$. Does $X$ admit a crepant resolution, and if so, how can one describe it explicitly?

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Classification of finite simple groups of some particular order

This is a continuation of an earlier post: link

Question 1. Apart from $A_5$ and $\operatorname{PSL}(2,p)$ where $p>3$ is a prime with $p\equiv \pm ~3~(\operatorname{mod}8)$ and both $\frac{p-1}{2},\frac{p+1}{2}$ are square-free, are there any other simple group of order $2^n\cdot m$ where $n\geq 2$ and $m$ is an odd square-free integer. (The original post deals with the cases $n=2$ and $3$.)

Question 2. What are simple groups of order $2^n\cdot p^2\cdot m$, where $p$ is an odd prime and $m$ is an odd square-free integer such that $p\nmid m$?

P.S. Context of the question: I am trying to prove a theorem for a finite group, where I need to rule out only the simple groups of these two particular orders to prove it in general.

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What is the strength of the reals being a countable union of countable sets?

Let $\sf ZF^- + R$ means $\sf ZF$ minus Powersets axiom, plus the existence of a powerset for every countable set. So the set $\sf R$ of all reals is definable.

Now, what is the consistency strength of $\sf ZF^- + R$ is a countable union of countable sets?

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Torsion and parallel transport

There's a close relationship between curvature and the holonomy group; the holonomy theorem of Ambrose and Singer, for example. It seems to me that there should be an analogous result for torsion. I believe that torsion measures the extent to which certain geodesic parallelograms don't close. Is there a theorem that makes this more precise?

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What are the main structure theorems on finitely generated commutative monoids?

I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's what I've heard so far:

• If a commutative monoid is finitely generated it is finitely presented.

• Finitely generated commutative monoids have decidable word problems, the isomorphism problem for them is decidable, and indeed the first-order theory of finitely generated commutative monoids is decidable. (From Kharlampovich and Sapir's Algorithmic problems in varieties.)

• If a finitely generated commutative monoid is cancellative ($a + b = a' + b \Rightarrow a = a'$) then it embeds in a finitely generated abelian group.

• If a finitely generated commutative monoid is cancellative and torsion-free (for any natural number $n \ge 1,$ $n a = n b \Rightarrow a = b$) then it embeds in a finitely generated free abelian group. (This follows easily from the previous claim.)

• If a commutative monoid is a submonoid of $(\mathbb{N},+,0)$ it is called a numerical monoid and of course it is cancellative. A lot is known about numerical monoids, though I don't believe they have been "classified" in any useful sense.

If we drop the property of being cancellative we get an enormous wilderness of finitely generated commutative monoids, so there shouldn't be any simple 'classification theorem'. But there still might be interesting structure theorems which help us understand this wilderness, just as there are for (say) compact topological abelian groups. What are they?

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Definite integral $\int_0^1 \ln\sin\left(\pi e^{\pi i x}\right) dx$

By following a somehow indirect way I can show that \begin{equation*} \int_0^1 \ln\sin\left(\pi e^{\pi i x}\right) dx = \ln \pi + \frac{\pi i}{2}, \end{equation*} where $\ln(\cdot)$ denotes the principal branch of the logarithm.

I wonder whether there is a direct derivation of this integral.

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How are constants/functions named after their discoverer?

In general, when a paper references an object discovered/defined in another paper by author X, it goes something along the lines of:

"Let $\tau$ be the constant defined by X in 1999 [1]$\ldots$",

or

"Let $f_{\mu}$ denote the function that generalizes the case $\ldots$ (X, [1])".

At what point does the literature start talking about "X's constant" or "The X function"?

Who/what determines that an object discovered by somebody deserves to take his name?

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On a partial sum with Moebius function and Dirichlet characters

Considering a Dirichlet Character $\chi(n)$, I would like to study the convergence of following sums in the critical strip:

$S=\sum_{n=1}^{\infty} \sum_{r|n} \mu(\frac{n}{r}) \chi(r) r^{-s}$

Any reference on the subject ? What is the method to study this type of sums ?

Formally the sum would be :

$S=\sum_{n=1}^{\infty} \mu(n). \sum_{r=1}^{\infty} \chi(r) r^{-s} $

but $\sum_{n=1}^{\infty} \mu(n)$ does not converge...

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The probability of the existence of a function of ordering prime numbers

The possibility of the existence of an order function for prime numbers is one of the biggest problems in mathematics that has not been solved to this day.

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Is this representation of $e^\gamma$ in terms of zeta values known?

I recently encountered the following identity involving the Euler--Mascheroni constant exponentiated and values of the Riemann zeta function:

$$ e^{\gamma}=1+\frac{1}{2}+\frac{\zeta(2)-1}{6} +\frac{(\zeta(2)-1)(\zeta(2)-4)+2\zeta(3)}{24} +\frac{(\zeta(2)-1)(\zeta(2)-4)(\zeta(2)-9)+2(13-3\zeta(2))\zeta(3)+6\zeta(4)}{120} +\cdots $$

More generally, the series can be written as $$ e^{\gamma}=\sum_{r=1}^{\infty}\rho_r, $$ where $$ \rho_r=\frac{1}{r!}\sum_{k=0}^{r-1}(-1)^{r-1-k}\,s(r-1,k)\,\phi_k\bigl(\zeta(2),\dots,\zeta(k+1)\bigr), $$ $s(n,k)$ are the (unsigned) Stirling numbers of the first kind, and $\phi_k$ are polynomials in $\zeta(2),\dots,\zeta(k+1)$. The first few $\phi_k$ are: $$ \begin{aligned} \phi_0&=1,\\ \phi_1&=\zeta(2)-1,\\ \phi_2&=(\zeta(2)-1)(\zeta(2)-4)+2\zeta(3),\\ \phi_3&=(\zeta(2)-1)(\zeta(2)-4)(\zeta(2)-9)+2(13-3\zeta(2))\zeta(3)+6\zeta(4). \end{aligned} $$

The series converges factorially because $\rho_r=O(1/r!)$.

Question: Has this explicit representation of $e^\gamma$ as a series in zeta values appeared previously in the literature? If so, where?

Context: The identity arose in a combinatorial setting related to expectations of $\log(C+a)$ where $C$ is the number of cycles of a uniform permutation. It resembles expansions involving cumulants and Bell polynomials, but I have not found it in standard references (e.g., Mertens’ product formula, series for $\gamma$ involving $\zeta(k)$, etc.).

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Regularity inequality in $ W^{2,p}(\mathbb{R}^N) $

Let $ N \geq 3 $ and $ 1 < p < + \infty $ is rational number. Let $ f \in L^p( \mathbb{R}^N) $ and $ u \in W^{2,p}( \mathbb{R}^N) $ such that $ - \Delta u + u = f $ in the distributions sense (also in $ L^{p}( \mathbb{R}^N)). $ My question is to know if it is true that $$ \left| \Delta u\right|_{L^p( \mathbb{R}^N)} \leq \left| f\right|_{L^p( \mathbb{R}^N)} $$ (the constant needs to be exactly 1). I tried to write $ u = L_2 * f $ where $ L_2 $ is the Bessel kenel but I think we have have $ -\Delta L_2 = L_2 + \delta_0 $ and we donot reach our result using the identity $ u = L_2*f!! $

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Grothendiek duality and dualizing sheaf

Let $i:Z\to X$ be a closed immersion and quasi smooth maps between derived algebraic stacks. Let $\omega$ be the dualizing sheaf on $X$. Then does the following isomorphism is true? Where $i^!$ is right adjoint of $i_*$.

$i^!\omega = \omega \otimes O_Z$

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Upper bound for the complex Beta function

The question is almost the same as here.

What is the upper bound for a complex Beta function $$\DeclareMathOperator{\Im}{Im}\DeclareMathOperator{\Re}{Re} \displaystyle B(s,z)=\frac{\Gamma(s) \Gamma(z)}{\Gamma(s+z)} $$ where $0<\Re(s)<1$, $0<\Re(z)<1$, $\Im(s)>10$ and $\Im(z)>10$ ?

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