We give a necessary and sufficient condition so that a pair of disjoint Jordan regions in the sphere can be quasiconformally mapped to a pair of disks. As a consequence, we obtain a simple characterization that involves Lipschitz functions for the case that one of the Jordan regions is a half-plane. We apply these results to prove that all polynomial cusps are quasiconformally equivalent and that a quasisymmetric embedding of the union of two disjoint disks extends to a quasiconformal map of the sphere, quantitatively. Also, in combination with previous work of the author, we obtain a new characterization of compact sets that are quasiconformally equivalent to Schottky sets.

We study how the isospin asymmetry affects quarkonium states in QCD at near zero temperature. Using lattice Non-Relativistic QCD formalism, we calculate bottom quark correlators in the gauge field ensembles generated with $N_f = 2 + 1$ flavors of dynamical staggered quarks whose dynamics include the isospin chemical potential effect and then construct $S-$ and $P-$ wave quarkonium state correlators. From these quarkonium correlators, we consider the ratios of quarkonium correlators at non-zero isospin chemical potential to that at $μ_I a = 0.000$. Here, the gauge field ensemble with $μ_I a = 0.000, 0.048, 0.053, 0.059, 0.066, 0.080, 0.092$ and $0.106$ on a $32^3 \times 48$ lattice with non-zero isospin current strength $λa = 0.0010, 0.0018,$ and $0.0036$, where $m_π= 135$ MeV and $a = 0.1535$ fm from \cite{Brandt:2022hwy}, are used. Preliminary results suggest that for $μ_I a = 0.106$, the Upsilon mass gets heavier than the Upsilon mass in the vacuum and that below $μ_I a = 0.106$ the isospin asymmetry effect on the Upsilon mass is not monotonic.

Spectrum is an important numerical invariant of an isolated hypersurface singularity, connecting its topological and analytic structures. The well-known Hertling conjecture tells the relation of range and variance of exponents i.e. elements of spectrum. For trimodal singularities, we compute their spectra and verify Hertling conjecture for them. Jung, Kim, Saito and Yoon recently defined Tjurina spectrum, stemming from Hodge ideals. This set of numerical invariants is a subset of spectrum in Steenbrink's sense. We give an estimation of exponents not in Tjurina spectrum and propose a similar Generalized Hertling Conjecture for Tjurina Spectrum. Moreover, we prove the conjecture for singularities of modality $\leq 3$.

Supported metal in the form of single atoms, clusters, and particles can individually or jointly affect the activity of supported heterogeneous catalysts. While the individual contribution of supported metal to the overall activity of supported photocatalysts has been identified, the joint activity of mixed metal species is overlooked because of their different photoelectric properties. Here, atomically dispersed Pd (Pd1) and Pd clusters are loaded onto CdS, serving as oxidation and reduction sites, respectively, for methanol dehydrogenation. The Pd1 substitutes Cd2+, forming hole-trapping states for methanol oxidation and assisting the dispersion of photo-deposited Pd clusters. Therefore, methanol dehydrogenation on CdS with supported Pd1 and Pd clusters exhibits the highest turnover frequency of 1.14 s-1 based on Pd content, and affords H2 and HCHO with a similar apparent quantum yield of 87 +/- 1% at 452 nm under optimized reaction conditions. This work highlights the synergistic catalysis of supported metal for improved photocatalytic activity.

The combinatorial approach applied to functional oxides has enabled the production of material libraries that formally contain infinite compositions. A complete ternary diagram can be obtained by pulsed laser deposition (PLD) on 100 mm silicon wafers. However, interest in such materials libraries is only meaningful if high-throughput characterization enables the information extraction from the as-deposited library in a reasonable time. While much commercial equipment allows for XY-resolved characterization at room temperature, very few sample holders have been made available to investigate structural, chemical, and functional properties at high temperatures in controlled atmospheres. In the present work, we present a furnace that enables the study of 100 mm wafers as a function of temperature. This furnace has a dome to control the atmosphere, typically varying from nitrogen gas to pure oxygen atmosphere with external control. We present the design of such a furnace and an example of X-ray diffraction (XRD) and fluorescence (XRF) measurements performed at the DiffAbs beamline of the SOLEIL synchrotron. We apply this high-throughput approach to a combinatorial library up to 735 {\textdegree}C in nitrogen and calculate the thermal expansion coefficients (TEC) of the ternary system using custom-made MATLAB codes. The TEC analysis revealed the potential limitations of Vegard's law in predicting lattice variations for high-entropy materials.

The integral $\int_0^a f(t) t^{-s} \mathrm{d}t$ diverges for $\text{Re}(s) \geq λ+ 1$, where $λ$ is the order of the first non-vanishing derivative of $f(t)$ at the origin. With the assumption that $f(t)$ is analytic at the origin, the finite-part of the divergent integral assumes the contour integral representation of the form $\bbint{0}{a} f(t) t^{-s} \mathrm{d}t = \int_C f(z) z^{-s} G(z) \mathrm{d}z$ where $G(z)$ depends on whether $z=0$ constitutes a pole or a branch point singularity of $z^{-s}$ [E. A. Galapon, \textit{Proc. R. Soc.}, \textbf{A 473} (2017), no. 2197, 20160567.]. In this paper, we extend these representations to accommodate logarithmic singularities of arbitrary order $n \in \mathbb{N}$, specifically for $\bbint{0}{a} f(t) t^{-s} \ln^n t \, \mathrm{d}t$. We then demonstrate the utility of the representations in the numerical evaluation of finite-part integrals and their use in determining the finite parts of non-Mellin-type divergent integrals -- those which exhibit singular behavior at the origin but lack a well-defined Mellin transform. Finally, these representations provide a closed-form evaluation of the Stieltjes transform $\int_0^a k(t) \ln^n t \left( t^ν(ω^2 + t^2) \right)^{-1} \mathrm{d}t$ in terms of finite-part integrals, from which the dominant asymptotic behavior is readily extracted for vanishingly small values of the parameter $ω$.

Interfacial orbital transport remains far less understood than its bulk counterpart despite its central role in orbitronic experiments. Here, we theoretically investigate the transmission and conversion of orbital angular momentum across a metallic interface using a model Hamiltonian incorporating crystal-field effects. We show that an injected orbital dipole moment undergoes pronounced oscillations driven by the crystal field and generates characteristic quadrupole moments determined by the orbital orientation relative to the interface. Unlike spin precession, the dipole relaxes toward a finite value away from the interface. We further quantify interfacial orbital memory loss and demonstrate that orbital absorption produces a sizable mechanical torque obtained from the orbital continuity equation.

Topic modeling has evolved as an important means to identify evident or hidden topics within large collections of text documents. Topic modeling approaches are often used for analyzing and making sense of social media discussions consisting of millions of short text messages. However, assigning meaningful topic labels to document clusters remains challenging, as users are commonly presented with unstructured keyword lists that may not accurately capture the respective core topic. In this paper, we introduce Narrative Topic Labels derived with Retrieval Augmented Generation (NTLRAG), a scalable and extensible framework that generates semantically precise and human-interpretable narrative topic labels. Our narrative topic labels provide a context-rich, intuitive concept to describe topic model output. In particular, NTLRAG uses retrieval augmented generation (RAG) techniques and considers multiple retrieval strategies as well as chain-of-thought elements to provide high-quality output. NTLRAG can be combined with any standard topic model to generate, validate, and refine narratives which then serve as narrative topic labels. We evaluated NTLRAG with a user study and three real-world datasets consisting of more than 6.7 million social media messages that have been sent by more than 2.7 million users. The user study involved 16 human evaluators who found that our narrative topic labels offer superior interpretability and usability as compared to traditional keyword lists. An implementation of NTLRAG is publicly available for download.

High-quality exploratory data analysis (EDA) is essential in the data science pipeline, but remains highly dependent on analysts' expertise and effort. While recent LLM-based approaches partially reduce this burden, they struggle to generate effective analysis plans and appropriate insights and visualizations when user intent is abstract. Meanwhile, a vast collection of analysis notebooks produced across platforms and organizations contains rich analytical knowledge that can potentially guide automated EDA. Retrieval-augmented generation (RAG) provides a natural way to leverage such corpora, but general methods often treat notebooks as static documents and fail to fully exploit their potential knowledge for automating EDA. To address these limitations, we propose NotebookRAG, a method that takes user intent, datasets, and existing notebooks as input to retrieve, enhance, and reuse relevant notebook content for automated EDA generation. For retrieval, we transform code cells into context-enriched executable components, which improve retrieval quality and enable rerun with new data to generate updated visualizations and reliable insights. For generation, an agent leverages enhanced retrieval content to construct effective EDA plans, derive insights, and produce appropriate visualizations. Evidence from a user study with 24 participants confirms the superiority of our method in producing high-quality and intent-aligned EDA notebooks.

Two-dimensional semiconductors have attracted considerable interest for integration into emerging quantum photonic networks. Strain engineering of monolayer transition-metal dichalcogenides (ML-TMDs) enables the tuning of light-matter interactions and associated optoelectronic properties, and generates new functionalities, including the formation of quantum dots (QDs). Here, we combine spatially resolved micro-photoluminescence ($μ$-PL) spectroscopy from cryogenic (4$\text{-}$94 K) to room temperature with micro-Raman spectroscopy at room temperature to investigate the strain-dependent emission energies of thousands of individual QDs in ML-WS$_2$ and ML-WSe$_2$, integrated across multiple heterostructures and a piezoelectric device. Compared with delocalized excitons, QDs in both materials exhibit enhanced strain sensitivities of their emission energies $-$ approximately fourfold in WS$_2$ and twofold in WSe$_2$ $-$ leading to pronounced broadening of the ensemble emission linewidth. Temperature-dependent $μ$-PL spectroscopy combined with dynamic strain tuning experiments further reveal that the enhanced strain sensitivity of individual QDs originates from strengthened interactions with low-energy phonons induced by quantum confinement. Our results demonstrate a versatile strain-engineering approach with potential for spectral matching across solid-state, atomic, and hybrid quantum photonic networks, and provide new insights into phonon-QD interactions in two-dimensional semiconductors.

We introduce the class of bilateral parking procedures on the integer line. While cars try to park in the nearest available spot to their right in the classical case, we consider more general parking rules that allow cars to use the nearest available spot to their left. We show that for a natural subclass of local procedures, the number of corresponding parking functions of length $r$ is always equal to $(r+1)^{r-1}$. The setting can be extended to probabilistic procedures, in which the decision to park left or right is random. We finally describe how bilateral procedures can naturally be encoded by certain labeled binary forests, whose combinatorics shed light on several results from the literature.

In this work, we study a hierarchical non-terrestrial network as an edge-cloud platform for remote computing of tasks generated by remote ad-hoc healthcare facility deployments, or internet of medical things (IoMT) devices. We consider a high altitude platform station (HAPS) to provide local multiaccess edge server (MEC) services to a set of remote ground medical devices, and a low-earth orbit (LEO) satellite, serving as a bridge to a remote cloud computing server through a ground gateway (GW), providing a large amount of computing resources to the HAPS. In this hierarchical system, the HAPS and the cloud server charges the ground users and the HAPS for the use of the spectrum and the computing of their tasks respectively. Each tier seeks to maximize their own utility in a selfish manner. To encourage the prompt computation of the tasks, a local delay cost is assumed. We formulate the optimal per-task cost at each tier that influences the corresponding offloading policies, and find the corresponding optimal bandwidth allocation.

Diffuse scattering of light from disordered assemblies is traditionally viewed as an uncontrollable broadband scattering background resulting in whitish hues. Here, we demonstrate that correlated disorder enables precise engineering of light scattering from 2D arrays of emitters resulting in strong observable colors. Our analytical framework shows that introducing controlled noise, with tunable probability density functions and correlations, generates three distinct scattering components: diffraction orders, diffuse background, and correlation halos. Correlation halos, often mistaken for broadened diffraction peaks, are independent features whose positions depend on correlation range and can appear between Bragg peaks. Crucially, they persist far beyond the regime where diffraction orders vanish. The noise probability density function provides an additional control: specific diffraction orders can be selectively suppressed while preserving others. This approach reproduces the scattering signatures of natural photonic structures, e.g. found in Morpho butterflies, and reveals multiple pathways from order to disorder, each with distinct optical properties. Our work provides a practical method for inverse design -finding the disorder that produces desired scattering patterns. This establishes diffuse scattering as a designable quantity, expanding the toolkit for metasurfaces and structural color beyond periodic and hyperuniform structures.

The current-induced vibrational dynamics is a key factor determining the stability of molecular nanojunctions. Beyond conventional Joule heating, a different mechanism caused by nonconservative current-induced forces has been predicted for models with multiple vibrational modes, leading to vibrational instabilities already at low bias voltages. So far, this mechanism has only been investigated in models with harmonic nuclear potentials. Consequently, a natural question is whether this effect can also be observed in more realistic models containing anharmonic nuclear potentials, and, if so, whether it has a measurable impact on observables such as the junction dissociation probability. In this work, we apply a mixed quantum-classical approach based on electronic friction and Langevin dynamics to various anharmonic two-mode systems. By performing Langevin simulations of the vibrational dynamics, we investigate the influence of anharmonicity on instabilities arising from nonconservative forces and the corresponding dissociation dynamics of the junction, as well as steady-state observables, such as the electronic current.

We discuss an ongoing first lattice study of the doubly-charmed tetraquark $T_{cc}^+$(3875) via a three-body approach. We investigate the $DDπ$ system in the $I=0$, $C=2$ sector, where the $T_{cc}^+$ appears as a pole in the $J^P = 1^+$ $DDπ$ elastic scattering amplitude. The approach automatically incorporates two-body $DD^*$ and three-body $DDπ$ effects and treats left-hand cuts due to single $π$ exchanges. Two CLS ensembles, X252 and X253, with pion mass $M_π\approx 280$ MeV, are used, and an operator set comprised of two- and three-hadron and tetraquark operators is employed to extract finite-volume energies. Additional inputs are required for the three-body finite-volume analysis, in the form of amplitudes for the $I=1$ $DD$ and $I=1/2$ $Dπ$ two-body subsystems. We present preliminary results for these subchannels and perform exploratory three-body spectra determinations for simple choices of the three-particle K-matrix $\mathcal{K}_{\text{df}, 3}$, allowing a first comparison to the lattice spectrum.

Compressed suffix arrays (CSAs) index large repetitive collections and are key in many text applications. The r-index and its derivatives combine the run-length Burrows-Wheeler Transform (BWT) with suffix array sampling to achieve space proportional to the number of equal-symbol runs in the BWT. While effective for near-identical strings, their size grows quickly as variation increases, since the number of BWT runs is sensitive to edits. Existing approaches typically trade space for query speed, or vice versa, limiting their practicality at large scale. We introduce variable-length blocking (VLB), an encoding technique for BWT-based CSAs that adapts the amount of indexing information to local compressibility. The BWT is recursively divided into blocks of at most w runs (a parameter) and organized into a tree. Compressible regions appear near the root and store little auxiliary data, while incompressible regions lie deeper and retain additional information to speed up access. Queries traverse a short root-to-leaf path followed by a small run scan. This strategy balances space and query speed by transferring bits saved in compressible areas to accelerate access in incompressible ones. Backward search relies on rank and successor queries over the BWT. We introduce a sampling technique that guarantees correctness only along valid backward-search states, reducing space without affecting query performance. We extend VLB to encode the subsampled r-index (sr-index). Experiments show that VLB-based techniques outperform the r-index and sr-index in query time, while retaining space close to that of the sr-index. Compared to the move data structure, VLB offers a more favorable space-time tradeoff.

Excited-state contamination remains one of the leading sources of systematic uncertainty in the precise determination of hadron structure observables from lattice QCD. In this work, I present a general mechanism, motivated by meson dominance and implemented through the variational method, that identifies which excited states are enhanced by the choice of inserted current and kinematics. The argument is supported by numerical evidence and predictions from chiral perturbation theory across different hadronic channels, in particular in the nucleon sector, and provides both conceptual insight and practical guidance for controlling excited-state effects in hadron three-point function analyses.

Since its introduction in the early 90s, the web has become the largest application platform available globally. HyperText Markup Language (HTML) has been an essential part of the web since the beginning, as it allows defining webpages in a tree-like manner, including semantics and content. Although the web was never meant to be an application platform, it evolved as such, especially since the early 2000s, as web application frameworks became available. While the emergence of frameworks made it easier than ever to develop complex applications, it also put HTML on the back burner. As web standards caught up, especially with milestones such as HTML5, the gap between the web platform and frameworks was reduced. HTML First development emphasizes this shift and puts focus on literally using HTML first when possible, while encouraging minimalism familiar from the early days of the web. It seems HTML-oriented web development can provide clear benefits to developers, especially when it is combined with comple- mentary approaches, such as embracing hypermedia and moving a large part of application logic to the server side. In the context of the htmx project, it was observed that moving towards HTML can reduce the size of a codebase greatly while leading to maintenance and development benefits due to the increased conceptual simplicity. Holotype-based comparisons for content-oriented websites show performance benefits, and the same observation was confirmed by a small case study where the Yle website was converted to follow HTML First principles. In short, the HTML First approach seems to have clear advantages for web developers, while there are open questions related to the magnitude of the benefits and the alignment with the recent trend of AI-driven web development.

Remote and resource-constrained Internet-of-Things (IoT) deployments often lack terrestrial connectivity for task offloading, motivating non-terrestrial networks (NTNs) with onboard multiaccess edge computing (MEC) capabilities. Nevertheless, in the presence of malicious actors, authentication needs to be performed to avoid non-authorized nodes from draining the computing resources of the NTN nodes. As a solution, we propose a four-layer MEC-enabled NTN with unmanned aerial vehicles (UAVs) acting as access nodes, a high altitude platform station (HAPS) acting as coordinator and authenticator, and a constellation of low-Earth orbit satellites (LEOSats) acting as remote MEC servers. We consider a tag-based physical-layer authentication (PLA) scheme to authenticate legitimate users, and formulate a joint task offloading decision and resource allocation for the admitted tasks, which is solved via block coordinate descent. Numerical results show that the PLA scheme is efficient and performs better than the benchmark schemes. We also demonstrate that the proposed scheme is robust against malicious attacks even under relaxed false-alarm constraints.

We provide a new proof of Ader's characterisation of the ellipse of minimal Banach-Mazur distance to the unit circle of a normed plane in terms of contact and extremal points. Our method reveals the relation of this problem to the Chebyshev alternance.

The first measurement of the polarization of charm baryons by the LHCb experiment recorded in fixed-target mode is presented. The polarization of $Λ_c$ baryons is studied in collisions of protons, at an energy of 2.51 TeV, incident on a gaseous target of neon, at a nucleon-nucleon center-of-mass energy of $68.6$ GeV. The world's first measurement of separate-charge polarizations for $Λ^+_c$ and $\overlineΛ{}^-_c$ baryons is performed, determining $$ P_{Λ^+_c} = ( 24 \pm 9 \pm 2 \, )\% , $$ $$ P_{\overlineΛ{}^-_c} = (-8 \pm 12 \pm 3 \, ) \% , $$ where the first uncertainty is statistical and the second systematic. The polarization is also measured in intervals of baryon transverse momentum and the Feynman-$x$ variable.

We review the conditions that cause or prohibit non-relativistic spin splitting of the energy bands in antiferromagnets. We propose that the existence of spin splitting in magnetically ordered systems is the default scenario and outline the criteria that must be met to avoid it. We discuss some of the properties of those special antiferromagnets that succeed in preserving their spin degeneracy.

We study an $n$-species $t$-PushTASEP, an integrable long-range stochastic process, on a one-dimensional periodic lattice with inhomogeneities $x_1,\ldots,x_L$ and arbitrary capacity $l$ at each lattice site. The Markov matrix is identified with an alternating sum of commuting transfer matrices over all fundamental representations of $U_t(\widehat{sl}_{n+1})$. Stationary probabilities are expressed in a matrix product form involving a fusion of quantized corner transfer matrices for the strange five-vertex model introduced by Okado, Scrimshaw, and the second author. The resulting partition function, which serves as the normalization factor of the stationary probabilities, is obtained from the $l=1$ case by a finite plethystic substitution of length $l$.

We study the $L^1$-smoothing properties for a broad class of semigroups arising from the ground state transformation of Schrödinger semigroups with confining potentials associated with non-local Lévy operators, for which (asymptotic) ultracontractivity and hypercontractivity fail. Our work is inspired by Talagrand's convolution conjecture in the discrete cube setting, as well as by subsequent developments on the classical Ornstein--Uhlenbeck semigroup. The estimates we provide exhibit a clear dependence on the potential and the Lévy measure defining the kinetic term operator, and they yield a description of the semigroups' action on $L^1$ in terms of Orlicz spaces. Our framework is quite general, encompassing fractional and relativistic Laplacians as kinetic operators. The results are illustrated by numerous examples demonstrating that the $L^1$-regularizing effects become stronger as $t \uparrow \infty$.

In the present work we explore features of single and pairs of solitary waves in a fractional variant of the nonlinear Schr{ö}dinger equation. Motivated by the recent experimental realization of arbitrary fractional exponents, upon quantifying the tail properties of such coherent structures, we detail their destabilization when the fractional exponent $α$ acquires values $α<1$ and showcase how the relevant destabilization is associated with collapse type phenomena. We then turn to in- and out-of-phase pairs of such waveforms and illustrate how they generically exist for arbitrary $α$ when we cross the harmonic limit, i.e., for $α>2$. Importantly, we use the parameter $α$ as a ``bifurcation parameter'' in order to connect the harmonic ($α=2$) and biharmonic ($α=4$) limits. Remarkably, not only do we retrieve the instability of all solitonic pairs in the biharmonic case, but showcase a stabilization feature of particular branches of such multipulses that is {\it unique} to the fractional case and does not arise -- to our knowledge -- for integer multi-pulse settings. We explain systematically this stabilization via spectral analysis and expand upon the implications of our results for the potential observability of fractional multipulse solitary waves.

This note is intended to provide an overview of the implications of regulations and standards on the safety of mechanical equipment, with a focus on accelerator components. Each research facility has different internal rules and standards which are applicable to specific cases; however, the main reference legal frame in Europe is everywhere based on the applicable European Directives. After a brief introduction to the 'safety' for mechanical systems, the process of 'risk analysis' will be introduced. The majority of this note will then deal with regulations and standards for pressure and cryogenic equipment. The European Pressure Equipment Directive (PED) will be briefly described, together with the concept of 'harmonized standards' and their implications on the entire lifecycle of a pressure equipment, with some hints at the peculiarities of accelerator components. In the second part of this note, regulations and standards for machinery, load-lifting accessories and buildings will be briefly mentioned to complete the picture of the most common cases in an accelerator facility.

We consider stationary states of an effectively one-dimensional Bose-Einstein condensate in a quasiperiodic lattice. We formulate sufficient conditions for a one-to-one correspondence between the stationary states with a fixed chemical potential and the set of bi-infinite sequences over a finite alphabet. These conditions can be checked numerically. A bi-infinite sequence can be interpreted as a code of the corresponding solution. A numerical example demonstrates the coding approach using an alphabet of three symbols.

This paper presents SimulatorCoder, an agent powered by large language models (LLMs), designed to generate and optimize deep neural network (DNN) accelerator simulators based on natural language descriptions. By integrating domain-specific prompt engineering including In-Context Learning (ICL), Chain-of-Thought (CoT) reasoning, and a multi-round feedback-verification flow, SimulatorCoder systematically transforms high-level functional requirements into efficient, executable, and architecture-aligned simulator code. Experiments based on the customized SCALE-Sim benchmark demonstrate that structured prompting and feedback mechanisms substantially improve both code generation accuracy and simulator performance. The resulting simulators not only maintain cycle-level fidelity with less than 1% error compared to manually implemented counterparts, but also consistently achieve lower simulation runtimes, highlighting the effectiveness of LLM-based methods in accelerating simulator development. Our code is available at https://github.com/xiayuhuan/SimulatorCoder.

A curve $C$ defined over $\mathbb Q$ is modular of level $N$ if there exists a non-constant morphism from $X_1(N)$ onto $C$ defined over $\mathbb Q$ for some positive integer $N$. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve $C$ of genus $3$ and level $N$ such that $\mathrm{Jac}{(C)}$ is $\mathbb Q$-isogenous to a given three dimensional $\mathbb Q$-quotient of $J_1 (N)$. Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus $3$. Let $C$ be a modular curve of level $N$, we say that $C$ is new if the corresponding morphism between $J_1(N)$ and $\mathrm{Jac}{(C)}$ factors through the new part of $J_1(N)$.

This paper presents the first evidence of the simultaneous production of two $Z$ bosons and one photon with the ATLAS detector at the Large Hadron Collider. The measurement is performed using the full Run-2 dataset, recorded from 2015 to 2018, of proton-proton collisions at a center-of-mass energy of $13$ TeV, corresponding to an integrated luminosity of $140$ fb$^{-1}$. The fully leptonic final state with four leptons and one photon is analyzed, $pp\rightarrow ZZγ\rightarrow\ell^+\ell^-\ell'^+\ell'^-γ$ with $\ell, \ell' = e$ or $μ$. This final state is measured in a fiducial region where photon final-state radiation is minimized, and the photon has a transverse momentum of $p_{\mathrm{T}}^γ> 20$ GeV. Eight events are selected, with a background estimate of $0.92\pm0.15$. This results in an observed (expected) significance of the $ZZγ$ final state of $4.4σ$ ($4.4σ$). The measured cross-section for $pp\rightarrow ZZγ\rightarrow\ell^+\ell^-\ell'^+\ell'^-γ$ in the fiducial region is $σ_{ZZγ} = 0.144 _{-0.051}^{+0.064} \text{ (stat.)} _{-0.005}^{+0.007} \text{ (syst.) fb}$, in agreement with the predicted Standard Model one, $σ_{\textrm{fid.}}^{\textrm{SM}} = 0.143 _{-0.004}^{+0.007}$ fb.