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1. Tree Height and the Asymptotic Mean of the Colijn–Plazzotta Rank of Unlabeled Binary Rooted Trees

   https://doi.org/10.1007/s11538-025-01538-7  

   Devroye, Luc; Doboli, Michael R; Rosenberg, Noah A; Wagner, Stephan (December 2025, Bulletin of Mathematical Biology)

   Abstract The Colijn–Plazzotta ranking is a bijective encoding of the unlabeled binary rooted trees with positive integers. We show that the rankf(t) of a treetis closely related to its heighth, the maximal path length from a leaf to the root. We consider the rank$$f(\tau _n)$$ $f\left({\tau }_n\right)$ of a randomn-leaf tree$$\tau _n$$ ${\tau }_n$ under each of three models: (i) uniformly random unlabeled unordered binary rooted trees, or unlabeled topologies; (ii) uniformly random leaf-labeled binary trees, or labeled topologies under the uniform model; and (iii) random binary search trees, or labeled topologies under the Yule–Harding model. Relying on the close relationship between tree rank and tree height, we obtain results concerning the asymptotic properties of$$\log \log f(\tau _n)$$ $loglogf\left({\tau }_n\right)$ . In particular, we find$${\mathbb {E}}\{\log _2 \log f(\tau _n)\} \sim 2 \sqrt{\pi n}$$ $E\{{log}_{2}logf\left({\tau }_n\right)\}\sim 2\sqrt{\pi n}$ for uniformly random unlabeled ordered binary rooted trees and uniformly random leaf-labeled binary trees, and for a constant$$\alpha \approx 4.31107$$ $\alpha \approx 4.31107$ ,$${\mathbb {E}}\{\log _2 \log f(\tau _n)\} \sim \alpha \log n $$ $E\{{log}_{2}logf\left({\tau }_n\right)\}\sim \alpha logn$ for leaf-labeled binary trees under the Yule–Harding model. We show that the mean of$$f(\tau _n)$$ $f\left({\tau }_n\right)$ itself under the three models is largely determined by the rank$$c_{n-1}$$ $c_{n-1}$ of the highest-ranked tree—the caterpillar—obtaining an asymptotic relationship with$$\pi _n c_{n-1}$$ ${\pi }_n{c}_{n-1}$ , where$$\pi _n$$ ${\pi }_n$ is a model-specific function ofn. The results resolve open problems, providing a new class of results on an encoding useful in mathematical phylogenetics. 

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2. Restricted spaces of holomorphic sections vanishing along subvarieties

   https://doi.org/10.1007/s00209-025-03706-w  

   Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh (May 2025, Mathematische Zeitschrift)

   Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ $\Sigma =({\Sigma }_1,\dots ,{\Sigma }_{\ell })$ is an$$\ell $$ $\ell$ -tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ $\tau =({\tau }_1,\dots ,{\tau }_{\ell })$ is an$$\ell $$ $\ell$ -tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$ be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ $L^p:=L^{\otimes p}$ that vanish to order at least$$\tau _jp$$ ${\tau }_{j}p$ along$$\Sigma _j$$ ${\Sigma }_j$ ,$$1\le j\le \ell $$ $1\le j\le \ell$ . If$$Y\subset X$$ $Y\subset X$ is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ $H_0^0\left(X\right|Y,L^p)$ of holomorphic sections of$$L^p|_Y$$ $L^p{|}_Y$ that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ $H_0^0(X,L^p)$ . Assuming that the triplet$$(L,\Sigma ,\tau )$$ $(L,\Sigma ,\tau )$ is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ $dim{H}_0^0(X,L^p)\sim p^n$ , we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ $dim{H}_0^0\left(X\right|Y,L^p)\sim p^m$ . WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$ converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ $H_0^0\left(X\right|Y,L^p)$ as$$p\rightarrow \infty $$ $p\to \infty$ . 

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3. A case study of SMEFT $$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ effects in diboson processes: pp → W±(ℓ±ν)γ

   https://doi.org/10.1007/JHEP05(2024)223  

   Martin, Adam (May 2024, Journal of High Energy Physics)

   A<sc>bstract</sc> In this paper we explorepp→W^±(ℓ^±ν)γto$$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ $O\left(1/{\Lambda }^4\right)$ in the SMEFT expansion. Calculations to this order are necessary to properly capture SMEFT contributions that grow with energy, as the interference between energy-enhanced SMEFT effects at$$ \mathcal{O}\left(1/{\Lambda}^2\right) $$ $O\left(1/{\Lambda }^2\right)$ and the Standard Model is suppressed. We find that there are several dimension eight operators that interfere with the Standard Model and lead to the same energy growth, ~$$ \mathcal{O}\left({E}^4/{\Lambda}^4\right) $$ $O\left(E^4/{\Lambda }^4\right)$ , as dimension six squared. While energy-enhanced SMEFT contributions are a main focus, our calculation includes the complete set of$$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ $O\left(1/{\Lambda }^4\right)$ SMEFT effects consistent with U(3)⁵flavor symmetry. Additionally, we include the decay of theW^±→ ℓ^±ν, making the calculation actually$$ \overline{q}{q}^{\prime}\to {\ell}^{\pm}\nu \gamma $$ $\overline{q}{q}^{\prime }\to {\ell }^{\pm }\mathrm{\nu \gamma }$ . As such, we are able to study the impact of non-resonant SMEFT operators, such as$$ \left({L}^{\dagger }{\overline{\sigma}}^{\mu }{\tau}^IL\right)\left({Q}^{\dagger }{\overline{\sigma}}^{\nu }{\tau}^IQ\right) $$ $\left(L^{†}{\overline{\sigma }}^{\mu }{\tau }^{I}L\right)\left(Q^{†}{\overline{\sigma }}^{\nu }{\tau }^{I}Q\right)$ B_μν, which contribute to$$ \overline{q}{q}^{\prime}\to {\ell}^{\pm}\nu \gamma $$ $\overline{q}{q}^{\prime }\to {\ell }^{\pm }\mathrm{\nu \gamma }$ directly and not to$$ \overline{q}{q}^{\prime}\to {W}^{\pm}\gamma $$ $\overline{q}{q}^{\prime }\to W^{\pm }\gamma$ . We show several distributions to illustrate the shape differences of the different contributions. 

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4. Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

   https://doi.org/10.1007/s00453-023-01102-6  

   Akmal, Shyan; Chen, Lijie; Jin, Ce; Raj, Malvika; Williams, Ryan (August 2023, Algorithmica)

   Abstract In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:Certifying that a list ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$ $\tilde{O}\left(n\right)$ . Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$ time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$ and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$ time).Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$ $\tilde{O}\left(n^{\lceil k/2\rceil }\right)$ (where$$k\ge 3$$ $k\ge 3$ ). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$ $\tilde{O}\left(m\right)$ . Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$ $\tilde{O}\left(n^2\right)$ . Note this is optimal, as the matrix can have$$\Omega (n^2)$$ $\Omega \left(n^2\right)$ nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$ $\tilde{O}\left(n^{2.94}\right)$ nondeterministic time algorithm.Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$ $2^{n/2-n/O\left(k\right)}$ . We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$ $2^{n/2}·\text{poly}\left(n\right)$ -time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$ $\#$ SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$ $2^{n/2}/n^{\omega \left(1\right)}$ time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time$$2^{4n/5}\cdot \textrm{poly}(n)$$ $2^{4n/5}·\text{poly}\left(n\right)$ . Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{2n/3}\cdot \textrm{poly}(n)$$ $2^{2n/3}·\text{poly}\left(n\right)$ time.Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$ $2^{n/3}·\text{poly}\left(n\right)$ , improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$ $2^{0.49991n}·\text{poly}\left(n\right)$ time. 

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5. JUNO sensitivity to invisible decay modes of neutrons

   https://doi.org/10.1140/epjc/s10052-024-13638-0  

   Abusleme, Angel; Adam, Thomas; Adamowicz, Kai; Ahmad, Shakeel; Ahmed, Rizwan; Aiello, Sebastiano; An, Fengpeng; An, Qi; Andronico, Giuseppe; Anfimov, Nikolay; et al (January 2025, The European Physical Journal C)

   Abstract We explore the decay of bound neutrons in the JUNO liquid scintillator detector into invisible particles (e.g.,$$n\rightarrow 3 \nu $$ $n\to 3\nu$ or$$nn \rightarrow 2 \nu $$ $nn\to 2\nu$ ), which do not produce an observable signal. The invisible decay includes two decay modes:$$ n \rightarrow { inv} $$ $n\to \mathrm{inv}$ and$$ nn \rightarrow { inv} $$ $nn\to \mathrm{inv}$ . The invisible decays ofs-shell neutrons in$$^{12}\textrm{C}$$ $_{}^{12}\text{C}$ will leave a highly excited residual nucleus. Subsequently, some de-excitation modes of the excited residual nuclei can produce a time- and space-correlated triple coincidence signal in the JUNO detector. Based on a full Monte Carlo simulation informed with the latest available data, we estimate all backgrounds, including inverse beta decay events of the reactor antineutrino$${\bar{\nu }}_e$$ ${\overline{\nu }}_e$ , natural radioactivity, cosmogenic isotopes and neutral current interactions of atmospheric neutrinos. Pulse shape discrimination and multivariate analysis techniques are employed to further suppress backgrounds. With two years of exposure, JUNO is expected to give an order of magnitude improvement compared to the current best limits. After 10 years of data taking, the JUNO expected sensitivities at a 90% confidence level are$$\tau /B( n \rightarrow { inv} ) > 5.0 \times 10^{31} \, \textrm{years}$$ $\tau /B(n\to \mathrm{inv})>5.0\times {10}^{31}\text{years}$ and$$\tau /B( nn \rightarrow { inv} ) > 1.4 \times 10^{32} \, \textrm{years}$$ $\tau /B(nn\to \mathrm{inv})>1.4\times {10}^{32}\text{years}$ . 

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