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A system's internal dynamics and its interaction with the environment can be determined by tracking how external perturbations affect its transition rates between states. However, accurately measuring these rates poses a significant challenge, especially when they span a wide range of time scales. Here we introduce a broadband measurement method, called stochastic resonance spectroscopy (SRS), that operates in the frequency domain and quantifies the stochastic dynamics of atomic-scale quantum systems. We apply this method to determine spin switching rates in a scanning tunneling microscope over an extremely wide frequency range, from 199 ms^-1 for few-atom structures to 1.73 ns^-1 for individual atoms. SRS relies on the universal phenomenon of stochastic resonance which synchronizes stochastic dynamics to an oscillating perturbation. We develop an analytical theory that extracts quantitative transition rates from scanning tunneling microscopy measurements of the frequency-dependent tunnel current and show that the signal is dominated by homodyne detection. Our theory indicates that SRS is not limited to spin dynamics and we corroborate this by measuring transport dynamics through a bound state in a superconductor. We anticipate that the ability to characterize broadband stochastic dynamics at the atomic scale will enable insights into excited-state dynamics of quantum systems and even non-Markovian processes emerging from correlations with the environment., Comment: 12 pages, 5 figures

We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation \begin{equation*} \partial_t u-\Delta u =f(u), \end{equation*} for nonlinearities which are genuinely non scale invariant, unlike in the standard case $f(u)=u^p$. Indeed, our results apply to a large class of nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function at infinity (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). More precisely, denoting by $\psi$ the unique positive solution of the corresponding ODE $y'(t)=f(y(t))$ which blows up at the same time $T$, we show that if $a\in\Omega$ is a blowup point of $u$, then \begin{equation*} \lim_{t\to T}\frac{u(a+y\sqrt{T-t},t)}{\psi(t)}= 1,\quad \text{uniformly for $y$ bounded.} \end{equation*} Additional blow-up properties are obtained, including the compactness of the blow-up set for the Cauchy problem with decaying initial data.

Generative AI models have recently blossomed, significantly impacting artistic and musical traditions. Research investigating how humans interact with and deem these models is therefore crucial. Through a listening and reflection study, we explore participants' perspectives on AI- vs human-generated progressive metal, in symbolic format, using rock music as a control group. AI-generated examples were produced by ProgGP, a Transformer-based model. We propose a mixed methods approach to assess the effects of generation type (human vs. AI), genre (progressive metal vs. rock), and curation process (random vs. cherry-picked). This combines quantitative feedback on genre congruence, preference, creativity, consistency, playability, humanness, and repeatability, and qualitative feedback to provide insights into listeners' experiences. A total of 32 progressive metal fans completed the study. Our findings validate the use of fine-tuning to achieve genre-specific specialization in AI music generation, as listeners could distinguish between AI-generated rock and progressive metal. Despite some AI-generated excerpts receiving similar ratings to human music, listeners exhibited a preference for human compositions. Thematic analysis identified key features for genre and AI vs. human distinctions. Finally, we consider the ethical implications of our work in promoting musical data diversity within MIR research by focusing on an under-explored genre., Comment: Reviewed pre-print accepted for publication at ISMIR 2024

We consider the semilinear heat equation $$u_t-\Delta u=f(u) $$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions). For any positive radial decreasing blow-up solution, we obtain the sharp, global blow-up profile in the scale of the original variables $(x, t)$, which takes the form: $$u(x,t)=(1+o(1))\,G^{-1}\bigg(T-t+\frac{p-1}{8p}\frac{|x|^2}{|\log |x||}\bigg), \ \hbox{as $(x,t)\to (0,T)$, \quad where } G(X)=\int_{X}^{\infty}\frac{ ds}{f(s)}.$$ This estimate in particular provides the sharp final space profile and the refined space-time profile. As a remarkable fact and completely new observation, our results reveal a {\it structural universality} of the global blow-up profile, being given by the "resolvent" $G^{-1}$ of the ODE, composed with a universal, time-space building block, which is the same as in the pure power case., Comment: 30 pages; minor corrections with respect to v1

Fake news significantly influence our society. They impact consumers, voters, and many other societal groups. While Fake News exist for a centuries, Generative AI brings fake news on a new level. It is now possible to automate the creation of masses of high-quality individually targeted Fake News. On the other end, Generative AI can also help detecting Fake News. Both fields are young but developing fast. This survey provides a comprehensive examination of the research and practical use of Generative AI for Fake News detection and creation in 2024. Following the Structured Literature Survey approach, the paper synthesizes current results in the following topic clusters 1) enabling technologies, 2) creation of Fake News, 3) case study social media as most relevant distribution channel, 4) detection of Fake News, and 5) deepfakes as upcoming technology. The article also identifies current challenges and open issues., Comment: 16 pages, 2 figures. Submitted to ACM Transactions on Intelligent Systems and Technology (ACM TIST). Added references

A factorisation problem in the symmetric group is central if any two permutations in the same conjugacy class have the same number of factorisations. We give the first fully combinatorial proof of the centrality of transitive star factorisations that is valid in all genera, which answers a natural question of Goulden and Jackson from 2009. Our proof is through a bijection to a class of monotone double Hurwitz factorisations. These latter factorisations are also not obviously central, so we also give a combinatorial proof of their centrality. As a corollary we obtain new formulae for some monotone double Hurwitz numbers, and a new relation between Hurwitz and monotone Hurwitz numbers. We also generalise a theorem of Goulden and Jackson from 2009 that states that the transitive power of Jucys-Murphy elements are central. Our theorem states that the transitive image of any symmetric function evaluated at Jucys-Murphy elements is central, which gives a transitive version of Jucys' original result from 1974.

Designing terahertz sensors with high sensitivity to detect nanoscale thin films and single biomolecule presents a significant challenge, and addressing these obstacles is crucial for unlocking their full potential in scientific research and advanced applications. This work presents a strategy for the design optimization of metamaterial sensors employed in the detection of small amounts of dielectric materials. The sensors usually utilize the shift of the resonance frequency as an indicator of the presence of the analyte. The amount of shifting depends on intrinsic properties (electric field distribution, quality factor, and mode volume) of the bare cavity, as well as the overlap volume of its high-electric-field zone(s) and the analyte. Guided by the simplified dielectric perturbation theory, interdigitated electric split-ring resonators (ID-eSRR) are devised to significantly enhance the detection sensitivity for thin-film analytes compared to eSRRs without interdigitated fingers in the SRR gap region. The fingers of the ID-eSRR metamaterial sensor redistribute the electric field, creating strongly localized field enhancements that substantially boost the interaction with the analyte. Additionally, the periodic change of the orientation of the inherent anti-phase electric field in the interdigitated structure reduces radiation loss, leading to a higher Q-factor. Experiments with e-beam-fabricated ID-eSRR sensors operating at around 300 GHz demonstrate a remarkable frequency shift of 33.5 GHz upon deposition of a SiO2 layer with a thickness of 150 nm as an analyte simulant. The figure of merit (FOM) improves by over 50 times compared to structures without interdigitated fingers. This rational design option opens a promising avenue for highly sensitive detection of thin films and trace biomolecules., Comment: 16pages, 6 figures

We show that for the product of two fixed point free conjugacy classes, the average number of cycles is always very similar. Specifically, our main result is that for a randomly chosen pair of fixed point free permutations of cycle types $\alpha$ and $\beta$, the average number of cycles in their product is between $H_n-3$ and $H_n+1$, where $H_n$ is the harmonic number.

Recent work in the field of symbolic music generation has shown value in using a tokenization based on the GuitarPro format, a symbolic representation supporting guitar expressive attributes, as an input and output representation. We extend this work by fine-tuning a pre-trained Transformer model on ProgGP, a custom dataset of 173 progressive metal songs, for the purposes of creating compositions from that genre through a human-AI partnership. Our model is able to generate multiple guitar, bass guitar, drums, piano and orchestral parts. We examine the validity of the generated music using a mixed methods approach by combining quantitative analyses following a computational musicology paradigm and qualitative analyses following a practice-based research paradigm. Finally, we demonstrate the value of the model by using it as a tool to create a progressive metal song, fully produced and mixed by a human metal producer based on AI-generated music., Comment: Pre-print accepted for publication at CMMR2023

In this paper, we consider the moments of statistics on conjugacy classes of the colored permutation groups $\mathfrak{S}_{n,r}=\mathbb{Z}_r\wr \mathfrak{S}_n$. We first show that any fixed moment coincides on all conjugacy classes where all cycles have sufficiently long length. Additionally, for permutation statistics that can be realized via a process we call symmetric extensions, these moments are polynomials in $n$. Finally, for the descent statistic on the hyperoctahedral group $B_n\cong \mathfrak{S}_{n,2}$, we show that its distribution on conjugacy classes without short cycles satisfies a central limit theorem. Our results build on and generalize previous work of Fulman (\textit{J. Comb. Theory Ser. A.}, 1998), Hamaker and Rhoades (arXiv, 2022), and Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin (arXiv, 2023). In particular, our techniques utilize the combinatorial framework introduced by Campion Loth, Levet, Liu, Stucky, Sundaram, and Yin.

Machine-learning methods rely on sufficiently large dataset to learn data distributions. They are widely used in research in X-Ray Computed Tomography, from low-dose scan denoising to optimisation of the reconstruction process. The lack of datasets prevents the scalability of these methods to realistic 3D problems. We develop a 3D procedural dataset in order to produce samples for data-driven algorithms. It is made of a meshed model of a left hand and a script to randomly change its anatomic properties and pose whilst conserving realistic features. This open-source solution relies on the freeware Blender and its Python core. Blender handles the modelling, the mesh and the generation of the hand's pose, whilst Python processes file format conversion from obj file to matrix and functions to scale and center the volume for further processing. Dataset availability and quality drives research in machine-learning. We design a dataset that weighs few megabytes, provides truthful samples and proposes continuous enhancements using version control. We anticipate this work to be a starting point for anatomically accurate procedural datasets. For instance, by adding more internal features and fine tuning their X-Ray attenuation properties.

We introduce the notion of a weighted inversion statistic on the symmetric group, and examine its distribution on each conjugacy class. Our work generalizes the study of several common permutation statistics, including the number of inversions, the number of descents, the major index, and the number of excedances. As a consequence, we obtain explicit formulas for the first moments of several statistics by conjugacy class. We also show that when the cycle lengths are sufficiently large, the higher moments of arbitrary permutation statistics are independent of the conjugacy class. Fulman (J. Comb. Theory Ser. A., 1998) previously established this result for major index and descents. We obtain these results, in part, by generalizing the techniques of Fulman (ibid.), and introducing the notion of permutation constraints. For permutation statistics that can be realized via symmetric constraints, we show that each moment is a polynomial in the degree of the symmetric group., Comment: We would also like to express our gratitude to Yan Zhuang for kindly alerting us to the arXiv paper of Hamaker and Rhoades (arXiv:2206.06567), after seeing the first version of the present paper. We also thank Zach Hamaker for taking the time to explain the results of the Hamaker--Rhoades paper and its overlap with the present work

We are interested in the distribution of the number of faces across all the $2-$cell embeddings of a graph, which is equivalent to the distribution of genus by Euler's formula. In order to study this distribution, we consider the local distribution of faces at a single vertex. We show an asymptotic uniformity on this local face distribution which holds for any graph with large vertex degrees. We use this to study the usual face distribution of the complete graph. We show that in this case, the local face distribution determines the face distribution for almost all of the whole graph. We use this result to show that a portion of the complete graph of size $(1-o(1))|K_n|$ has the same face distribution as the set of all permutations, up to parity. Along the way, we prove new character bounds and an asymptotic uniformity on conjugacy class products., Comment: To appear in European Journal of Combinatorics

A random 2-cell embedding of a connected graph $G$ in some orientable surface is obtained by choosing a random local rotation around each vertex. Under this setup, the number of faces or the genus of the corresponding 2-cell embedding becomes a random variable. Random embeddings of two particular graph classes, those of a bouquet of $n$ loops and those of $n$ parallel edges connecting two vertices, have been extensively studied and are well-understood. However, little is known about more general graphs. The results of this paper explain why Monte Carlo methods cannot work for approximating the minimum genus of graphs. In his breakthrough work [Permutation-partition pairs, JCTB 1991], Stahl developed the foundation of "random topological graph theory". Most of his results have been unsurpassed until today. In our work, we analyze the expected number of faces of random embeddings (equivalently, the average genus) of a graph $G$. It was very recently shown that for any graph $G$, the expected number of faces is at most linear. We show that the actual expected number of faces $F(G)$ is almost always much smaller. In particular, we prove: 1) $\frac{1}{2}\ln n - 2 < \mathbb{E}[F(K_n)] \le 3.65 \ln n +o(1)$. 2) For random graphs $G(n,p)$ ($p=p(n)$), we have $\mathbb{E}[F(G(n,p))] \le \ln^2 n+\frac{1}{p}$. 3) For random models $B(n,\Delta)$ containing only graphs, whose maximum degree is at most $\Delta$, we get stronger bounds $\mathbb{E}[F(B(n,\Delta))]=\Theta(\ln n)$., Comment: Accepted at the 35th ACM-SIAM Symposium on Discrete Algorithms (SODA 2024). 55 pages, 10 figures

A random 2-cell embedding of a given graph $G$ is obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces of such an embedding, which is equivalent to studying its average genus. In 1991, Stahl proved that the expected number of faces in a random embedding of an arbitrary graph of order $n$ is at most $n\log(n)$. While there are many families of graphs whose expected number of faces is $\Theta(n)$, none are known where the expected number would be super-linear. This lead to the conjecture that there is a linear upper bound. In this note we confirm the conjecture by proving that for any $n$-vertex multigraph, the expected number of faces in a random 2-cell embedding is at most $n(1+H_m)$, where $m$ is the maximum edge-multiplicity and $H_m$ denotes the $m$th harmonic number. This bound is best possible up to a constant factor.

An increasing number of large-scale multi-modal research initiatives has been conducted in the typically developing population, as well as in psychiatric cohorts. Missing data is a common problem in such datasets due to the difficulty of assessing multiple measures on a large number of participants. The consequences of missing data accumulate when researchers aim to explore relationships between multiple measures. Here we aim to evaluate different imputation strategies to fill in missing values in clinical data from a large (total N=764) and deeply characterised (i.e. range of clinical and cognitive instruments administered) sample of N=453 autistic individuals and N=311 control individuals recruited as part of the EU-AIMS Longitudinal European Autism Project (LEAP) consortium. In particular we consider a total of 160 clinical measures divided in 15 overlapping subsets of participants. We use two simple but common univariate strategies, mean and median imputation, as well as a Round Robin regression approach involving four independent multivariate regression models including a linear model, Bayesian Ridge regression, as well as several non-linear models, Decision Trees, Extra Trees and K-Neighbours regression. We evaluate the models using the traditional mean square error towards removed available data, and consider in addition the KL divergence between the observed and the imputed distributions. We show that all of the multivariate approaches tested provide a substantial improvement compared to typical univariate approaches. Further, our analyses reveal that across all 15 data-subsets tested, an Extra Trees regression approach provided the best global results. This allows the selection of a unique model to impute missing data for the LEAP project and deliver a fixed set of imputed clinical data to be used by researchers working with the LEAP dataset in the future., Comment: 22 pages, 3 figures, 3 tables

A metal oxide support and a catalytically active metal are the two main ingredients for complex catalysts used in heterogeneous catalysis. The gas environment can change the catalyst during the reaction, modifying its structural and electronic properties. Here, we use monochromated electron energy loss spectroscopy (EELS) to reveal hydrogen-pressure-dependent changes of the electronic structure at the Pd/rutile-TiO$_2$ interface in an environmental transmission electron microscope (ETEM). Hydrogen-induced changes are observed in rutile-TiO$_2$ within $2$~nm from the interface at $10$~Pa of hydrogen pressure, in the Ti $L_{3,2}$ EEL spectra. Lower pressures such as $1$~Pa show no changes in the EEL spectra. We attribute the observed changes in the EEL spectra to hydrogen-induced defects accumulating in the vicinity of the interface. Based on DFT calculations, we developed a thermodynamic multistate defect (TMD) model of the interface and the bulk of the rutile-TiO$_2$. This TMD model predicts high concentrations of positively charged defects accumulating at the interface. The presence of the Schottky barrier stabilizes these defects by significantly lowering their formation energy. Our findings provide important new insights into catalytic processes taking place at metal/metal oxide interfaces in hydrogen gas environments., Comment: 18 pages, 12 figures

Random 2-cell embeddings of a given graph $G$ are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, $\mathbb{E}[F_G]$, of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar with $n$ nonleaf edges lies in an interval of length $2/(n + 1)$ centered at the expected number of faces of an $n$-edge dipole. This allows us to derive bounds on $\mathbb{E}[F_G]$ for any given graph $G$ in terms of vertex degrees. We conjecture that $\mathbb{E}[F_G ] \le O(n)$ for any simple $n$-vertex graph $G$., Comment: 15 pages, 2 figures. Accepted to European conference on combinatorics, graph theory and applications (EUROCOMB) 2021