Psychiatric disorders have been traditionally conceptualized as latent conditions producing observable symptoms, but recent studies suggest that psychopathology may emerge from symptoms interactions. Psychometric networking model these relations focusing on pairwise associations but overlooks higher-order dependencies arising among groups of variables. These dependencies may reflect synergistic mechanisms, where joint symptom configurations convey more information than pairwise relations, or redundancy, where information overlaps. We introduce an information-theoretic multiplex hypergraph framework to identify and compare higher-order interactions in eating disorders data, across diagnostic groups (e.g., anorexia nervosa). Higher-order structures are quantified using $Ω$-information, a measure that captures the balance between redundancy and synergy. To address the combinatorial growth of candidate subsets, multiple testing and estimation instability, we propose a structured pipeline comprising: (i) targeted candidate selection based on dyadic network topology and theory-driven subscale information; (ii) a three-stage inferential procedure combining null-model testing with bootstrap robustness assessment; and (iii) the construction and analysis of diagnosis-layered, synergistic and redundant multiplex hypergraphs. Results highlight how synergy captures the emergent, higher-order organization of diagnoses, revealing both a stable transdiagnostic core and diagnosis-specific ways in which these domains combine. By contrast, redundancy is confined to eating and body-image related content, marking reinforcement rather than broader symptom integration.

We propose an accurate thermometry approach for Rydberg atom tweezer arrays combining data from correlation and local susceptibility measurements with a theoretical high-temperature expansion method for dynamic spin correlations. We apply our approach to a recent quantum simulation experiment [Bornet et al., arXiv 2602.14323] realizing an anti-ferromagnetic dipolar spin-1/2 XY model on the Kagome lattice. We obtain T=0.55J and S/N=0.67 ln2 for temperature and entropy respectively, showing that further experimental efforts are required to reach the putative quantum spin liquid regime.

We develop an analytical framework for Boolean Promise Constraint Satisfaction Problems (PCSPs) that studies polymorphisms through the notion of influence from Fourier analysis of Boolean functions. Extending the work of Brakensiek, Guruswami, and Sandeep [ICALP'21] on Ordered PCSPs, we identify two general phenomena in Boolean minions indicative of hardness or tractability: (1) preservation of coordinate influence under random 2-to-1 minors and (2) the presence of sharp thresholds. We demonstrate that these phenomena occur in broader settings than previously established, yielding new hardness/tractability results for minions consisting of unate or polynomial threshold functions.

swdatatoolkit is a Python-based scientific software library designed to support the acquisition, preprocessing, and analysis of solar and space weather data. The toolkit consolidates functionality across multiple domains, including data downloading from established heliophysics sources, image preprocessing, edge detection, image texture quantification, magnetic field analysis, and the derivation of higher-level parameters commonly used in solar physics research. Its modular structure reflects the heterogeneous nature of space weather data and enables reproducible, extensible workflows for both exploratory analysis and machine-learning-driven studies. This paper presents an overview of the library's available capabilities, its scientific motivations, and its role in the broader space weather research ecosystem.

We investigate the problem of jointly testing a pair of composite hypotheses and, depending on the test result, estimating a random parameter under distributional uncertainties. Specifically, it is assumed that the distribution of the data given the parameter of interest, is subject to uncertainty. Both, a Bayesian formulation and a Neyman-Pearson-like formulation, are considered. It is shown that the optimal policy induces an $f$-similarity that must be maximized to identify the least favorable distributions. Besides the general results, the implementation is investigated using a band-type uncertainty model. For designing the minimax procedures, existing algorithms are modified to increase convergence speed while maintaining numerical stability. The proposed theory is supplemented by numerical results for both formulations.

Although the spin parameter of dark matter halos is well known to follow a log-normal distribution at fixed epoch, its quantitative redshift evolution - encompassing both the mean and the dispersion - remains only partially explored. Prior studies either lack the mass resolution required to establish reliable evolutionary trends or do not provide analytical relations that enable forward modelling. Using a suite of LCDM N-body simulations with controlled resolution across the redshift range 0 < z < 5, we characterise the evolution of the mean and dispersion of the Peebles (lambda) and Bullock (lambda') definitions of spin. We find a mild but statistically robust linear evolution for ln(lambda) and a non-monotonic trend with a turnover at z ~ 1 - 2 for ln lambda', which we verify are unaffected by mass resolution of choice of halo definition. We provide closed-form fitting functions for these trends that allow modellers to draw physically motivated spin values at any redshift within our range of validity. This is a practical, redshift-dependent alternative to the common assumption of a constant spin distribution, and provides a useful input to semi-empirical and semi-analytic models of galaxy formation.

In this paper, we present the Electric Mobility Dial-a-Ride Problem (EM-DARP), which extends the Electric Vehicle Dial-a-Ride Problem (EV-DARP) to better accommodate human-focused mobility services. The problem involves utilizing a fleet of heterogeneous Electric Vehicles (EVs) to fulfill a set of customer requests with DARP and mobility-related specifications, while incorporating visits to charging stations amid requests. The problem is formulated as a Mixed-Integer Linear Program (MILP) and subsequently solved for a number of curated evaluation scenarios to demonstrate its practical applicability.

Since their rediscovery in the 1990s, multiple zeta values have become ubiquitous in many areas of mathematics and physics. Their standard integral and sum representations can usually be traced back to a single source, namely the iterated integrals on the Riemann sphere with three punctures. We refer to such representations as the \emph{linear} geometry of multiple zeta values, since the denominators of the corresponding integrands factor completely into linear terms. However, there also exist equally important and entirely distinct integral representations for multiple zeta values arising in mathematics and physics, in which matrix determinants appear in the denominator of the integrand. We call this the \emph{non-linear} geometry of multiple zeta values. These lectures trace the origins of this non-linear geometry and provide an introductory journey through a range of topics including tropical geometry, the moduli spaces of tropical curves, Feynman integrals in quantum field theory, the general linear group of integer matrices, and the reduction theory of quadratic forms. In doing so, we propose a geometric framework for multiple zeta values based on such non-linear, determinantal representations and set out a number of open questions for future research.

Several theoretical and astrophysical problems - including gravitational-wave modeling for extreme mass-ratio inspirals - require accurate time-domain solutions of the spin-weight $s=-2$ Teukolsky equation in Boyer-Lindquist coordinates. Because such simulations are performed on finite computational domains, they typically introduce an artificial outer boundary where nontrivial boundary conditions must be imposed. If these conditions are inaccurate, then spurious reflections and slowly-growing unphysical modes may corrupt long-time evolutions. We develop and implement exact radiation outer boundary conditions for the Bardeen-Press equation (a harmonic moment of the $a=0$ Teukolsky equation), making the artificial boundary transparent at any finite radius. We also construct near-to-far field teleportation kernels that map field data recorded at finite radius $r_1$ to the data reaching $r_2 > r_1$. The possible choice $r_2 = \infty$ corresponds to asymptotic waveform evaluation, that is propagation of the data to future null infinity. We show that both boundary and teleportation kernels are well approximated by exponential sums, with associated error bounds. Implemented in a time-domain solver, our kernel-based boundary conditions eliminate unphysical late-time growth and give the correct late-time decay rates, affording efficient long-duration simulations for waveform modeling and related blackhole perturbation calculations.

The set of matrix tuples with invariant subspaces whose dimensions sum up to the dimension of the space, but which do not span the whole space form an algebraic hypersurface. We found the equation of this hypersurface. This generalizes previous joint work.

Component Mode Synthesis methods, such as the Craig-Bampton (CB) approach, are widely used in structural dynamics due to their modularity and compatibility with substructuring workflows. While highly effective for linear systems, extending these methods to geometrically nonlinear structures remains a significant challenge. In this work, we propose a nonlinear extension of the CB method tailored to such contexts. The approach is based on the construction of a quadratic reduction manifold, derived via perturbation analysis, in which high-frequency fixed-interface modes are statically condensed onto a reduced set of low-frequency modes and interface coordinates. This formulation enables the representation of geometric nonlinear effects without increasing the number of reduced degrees of freedom.The resulting Nonlinear Craig-Bampton (NL-CB) reduced-order model is obtained through Galerkin projection onto the tangent space of the manifold and admits a polynomial structure that is efficient for time integration. The formulation preserves the Lagrangian structure of the underlying finite element model, ensuring consistent energetic behavior and numerical stability.The proposed method is demonstrated on representative nonlinear structural systems of increasing complexity. The results show that the NL-CB model captures the essential nonlinear dynamic response while retaining the modularity and computational efficiency of classical substructuring approaches.

We investigate how multi-band gravitational wave (GW) observations can constrain the uncertainties in the Hubble parameter ($H_0$) using primordial black holes (PBHs) as possible sources. Our framework combines scalar-induced and merger-induced GWs from PBHs, and forecasts on a combination of two future detectors Square Kilometre Array (SKA) and the Einstein Telescope (ET), enabling a multi-band analysis. We perform a statistical forecast of the PBH parameters, $M_{\rm PBH}$ and $f_{\rm PBH}$, using signal-to-noise ratio (SNR) estimates and Fisher matrix analysis. Imposing $\mathrm{SNR} \geq 1$, we identify the accessible PBH parameter space and propagate these uncertainties to estimate the corresponding uncertainties in $H_0$. For $δθ_i/θ_i \leq 0.1$, with $θ_i \equiv M_{\rm PBH}(f_{\rm PBH})$, we find $δH_0 \lesssim 2~{\rm km\,s^{-1}\,Mpc^{-1}}$ in a conservative approach, improving to $δH_0 \lesssim \mathcal{O}(0.1)~{\rm km\,s^{-1}\,Mpc^{-1}}$ for $δθ_i/θ_i \leq 0.01$ for an optimistic approach of precision measurement. The results are further found to be largely insensitive to the fiducial choice of the $H_0$, with only moderate dependence on the PBH collapse efficiency. These findings demonstrate that multi-band GW observations provide an independent and complementary approach to constraining the uncertainties in $H_0$, with the potential to provide a novel, cosmic distance ladder-independent measure of the Hubble parameter.

The consequences of a protoplanetary disk collision with a gas stream are being studied using three-dimensional numerical gas-dynamic simulation. The influence of orbital parameters and the stream mass on the accretion activity of the star is examined. It is shown that the orbital inclination and the initial mass of the infalling material are the most influential parameters in determining the accretion rate. The obtained accretion rate dependencies are compared with actual observational data for two FU~Ori type stars. It turns out that not only is the maximum accretion rate consistent with observational estimates, but the behavior of the accretion rate over time is very similar to available long-term light curves.

Recent advances in high-temperature-superconductor technology have made substantially higher toroidal magnetic fields technologically accessible, reopening the design space for compact, high-field tokamak reactors. Because reactor performance projections remain anchored to empirical confinement scalings, the recent update to the ITPA global H-mode confinement database raises an important question: what does the present experimental record and its uncertainty imply for the path to reactor-grade fusion performance? In this work, we revisit confinement extrapolation from an explicitly extrapolation-oriented perspective and, to complement its implications in terms of a direct reactor performance measure, present a cross-machine empirical scaling for fusion power. We systematically search for a minimally complex confinement scaling that optimizes the tradeoff between variance capture and extrapolative robustness. We find that low-order models centered near $N=3$ to $N=4$ optimize this tradeoff, with plasma current, machine size, heating power, and elongation emerging as the dominant engineering levers, together with an empirically inferred confinement penalty associated with metallic walls. Recast in reactor-performance terms, the results indicate that both the fusion triple product and fusion power are governed primarily by plasma current: the triple product scales approximately as $I_p^2$, and the empirical fusion power scaling exhibits a similarly near-quadratic dependence over a survey of the highest performing discharges across several machines. Projecting to reactors, these results suggest that high-field devices with metal walls may require higher plasma current than standard IPB98$(y,2)$-based expectations imply, and that gigawatt-class tokamak performance likely demands operation at $I_p \gtrsim 20\mathrm{MA}$.

We present converged ab initio calculations of short-range neutrinoless double-beta ($0νββ$) decay nuclear matrix elements for the key experimental isotopes $^{76}$Ge, $^{82}$Se, $^{130}$Te and $^{136}$Xe. Starting from different nuclear forces derived from chiral effective field theory, we apply the in-medium similarity renormalization group to obtain an effective valence-space Hamiltonian along with consistently transformed $0νββ$-decay operators. We then obtain a range of values for the matrix elements that is consistent with, but generally smaller than, those from phenomenology. Finally, we combine our results with current limits from $0νββ$-decay searches to obtain constraints for the sterile-neutrino mixing-mass parameter space when considering the inclusion of a fourth sterile neutrino.

The mechanical scaling laws of dealloyed nanoporous metals depart from classical Gibson-Ashby predictions for open-cell foams due to a decreased connectivity in their solid network. However, these scaling relations have been established almost exclusively on nanoporous gold produced by electrochemical dealloying, and it is an outstanding question whether the relations apply to nanoporous networks fabricated by other dealloying methods. Here, we investigate the mechanical response of single-crystalline nanoporous tantalum (np-Ta) produced by liquid metal dealloying (LMD) a TiTa alloy in molten CuBi. Nanoindentation of individual microparticles yields an elastic modulus of 10-30 GPa and a hardness of 0.3-1.1 GPa, both scaling with the solid volume fraction in agreement with Gibson-Ashby predictions. This stiffness-density response of np-Ta departs from previous reports on nanoporous gold and is attributed to enhanced ligament connectivity enabled by the thermodynamics of the CuBi metal bath. Molecular dynamics simulations reveal dislocation-dominated plasticity during indentation of np-Ta, consistent with scanning electron microscopy observations of limited densification beneath the indents, ruling out unusual deformation mechanisms as an origin of the observed scaling. These findings identify solvent chemistry in LMD as a tunable lever for ligament connectivity, and thus for the mechanical response of nanoporous metals.

This paper presents a method for computing local mean, variance, standard deviation, and effective sample count on incomplete gridded data using boundary-aware spectral operators. The framework combines normalized convolution with explicit boundary-condition modeling: reflective Discrete Cosine Transform (DCT) for non-periodic Cartesian axes and periodic Real Fast Fourier Transform (RFFT) for circular azimuth processing in polar geometry. Stability safeguards (denominator floor, prefill fallback, and variance clamp) are specified for under-supported regions. We evaluate the framework across three targeted scenarios: a Cartesian boundary-condition check demonstrating the mitigation of wrap-around artifacts, a synthetic 3D outlier-identification test, and a real-radar polar application. Results establish bounded, support-aware interpretation of local statistics while preserving a concise reproducibility path through the open-source 'dct\_toolkit' implementation.

We analyse approximation algorithms (greedy heuristics) for the classical domination number and two multiple domination numbers in simple graphs. First, we present a short self-contained proof of the known result that the minimum domination problem in any graph $G$ with maximum degree $Δ$ can be solved within the approximation ratio of ${\ln(Δ+1)+1}$. The proof is based on an analysis of a simple greedy heuristic. Then, by analysing more advanced greedy heuristic techniques and using ideas from our self-contained proof for the classical domination number, we fix a gap in the existing proof of a similar result for the $k$-tuple domination number. That is, we prove that the minimum $k$-tuple domination problem indeed can be approximated within the ratio of $\ln(Δ+1)+1$. The proof of this result is self-contained, direct, and much shorter than the existing proof, which contains the gap. Finally, we show that the known approximation ratio of $\ln(2Δ)+1$ for the minimum $k$-domination problem can be improved to a better ratio.

We present an amplitude analysis of high-energy polarized photoproduction of $π^-Δ^{++}$ within a Regge exchange framework. A Regge amplitude model incorporating $π$, $ρ$, $b_1$, and $a_2$ trajectory exchanges is fit simultaneously to spin density matrix elements measured by the GlueX experiment at photon energies of $E_γ= 8.2$--$8.8$ GeV and differential cross section data from SLAC. By including SDME data, the fit constrains not only the magnitudes but also the relative phases of the helicity amplitudes. The results confirm the dominance of pion exchange at small momentum transfer, while natural parity exchanges become significant at larger $t$. We analytically continue the $s$-channel amplitude to the $t$-channel, taking care of the kinematical singularities, and isolate the dynamical residues at the meson poles. The extracted $πNΔ$ coupling constant is found to be consistent with the value obtained from the decay width of the $Δ(1232)$. For the $ρNΔ$, $b_1 NΔ$, and $a_2 NΔ$ vertices, first extractions of the relevant coupling constants are provided.

We describe a numerical many-body technique that is based on both tensor networks and quantum Monte Carlo. The variational ansatz is a tensor network that can harvest volume-law entanglement. It is constructed from a tensor train to which one applies a set of non-local operators that force several indices of the tensor train to represent the same physical index, hence its name -- replica tensor train (RTT). From the tensor network toolbox, it inherits the possibility to make linear combinations of these states and apply a certain class of operators. We can therefore find the ground-state of a local Hamiltonian in a purely algebraic way as in standard tensor network algorithms -- i.e. without using gradient descent methods. On the other hand, the volume-law structure forbids calculating physical observables directly. In much the same way as on a quantum computer where one can prepare a state but can only sample it at the end, here we have to use Markov Chain Monte Carlo to compute the observables. We further show that the approach can be extended to build Krylov-subspace ground-state methods within the variational manifold. We illustrate the different algorithms on a two-dimensional spin model with a transverse magnetic field, which can be solved by this approach at low computational cost.

Although Hubel and Wiesel established decades ago how individual V1 neurons transform retinal inputs, functions of V1 as a whole are being discovered only recently. First, V1 acts as a motor cortex for exogenously guiding saccades by constructing a bottom-up saliency map of the visual field. Second, V1 initiates a processing bottleneck: a massive reduction of visual information begins at its output to downstream areas. Third, downstream recognition is limited by impoverished information, V1 supports ongoing recognition by providing additional information queried by top-down feedback from downstream areas, directed predominantly to central visual field representations. These V1 functions underpin a framework in which vision is mainly looking and seeing through the bottleneck. Looking selects a fraction of visual information into the bottleneck, largely by saccades that center selected contents at gaze. Seeing recognizes the selected contents. Looking and seeing rely mainly on processing in the peripheral and central visual fields.

Purpose: Widespread adoption and methodological advancement of Magnetic Resonance Fingerprinting (MRF) are limited by the lack of unified, reproducible implementation frameworks and fragmented open-source tools. To address these barriers, we introduce OpenMRF - a comprehensive Pulseq-based solution - designed to enable consistent, reproducible, and transferable MRF research across vendors, sites, and field strengths. Methods: OpenMRF integrates modular Pulseq-based sequence design, Bloch-simulation-based dictionary creation directly from .seq files, and iterative low-rank subspace reconstruction. The framework was evaluated through digital phantom simulations, a multi-site ISMRM/NIST phantom study on Siemens MRI systems at 0.55 T, 1.5 T, and 3 T as well as GE and United Imaging 3 T platforms, and representative in vivo acquisitions in the liver (0.55 T), myocardium (1.5 T), and brain (3 T). Results: Simulations demonstrated high mapping accuracy in an ISMRM/NIST-like digital phantom, with low-rank reconstruction yielding deviations of 0.03+/-0.32 % (T1) and 0.12+/-1.94 % (T2). The multi-site phantom study yielded relaxation times consistent with reference values at all field strengths, with mean deviations of -0.1+/-2.9 % (T1), -1.5+/-8.7 % (T2), and -4.0+/-7.2 % (T1rho). In vivo acquisitions produced high-quality parameter maps across platforms and field strengths. Conclusion: OpenMRF provides a robust, open-source, end-to-end Pulseq-based solution for MRF that enables reproducible sequence implementation, physics-accurate dictionary simulation, and advanced reconstruction across vendors and field strengths. By providing a unified platform for method development, comparison, and multi-site validation, OpenMRF aims to accelerate reproducible and harmonized quantitative MRI research within the community.

These lecture notes introduce the statistical analysis of continuous-time generative models built from Markov dynamics. We begin with the stochastic-calculus foundations of score-based diffusion models, including time reversal, score matching, and sampling from learned scores. We then present the broader framework of generator matching, which describes flows, diffusions, jump processes, and discrete generative models through their infinitesimal generators. We then focus on finite-sample guarantees. We explain how errors in the learned drift or generator propagate to the final generated distribution, why stability and regularity properties are essential, and how time-adaptive neural network classes can achieve optimal Wasserstein rates for smooth target distributions. Overall, the notes aim to connect modern generative modeling algorithms with the probabilistic, analytic, and statistical tools needed to understand their worst-case performance.

We extend the refined asymptotics of analytic torsion associated to congruence subgroups of $\operatorname{SL}(n)$ in previous work, to congruence subgroups in a large family of reductive groups. This is applied to give new asymptotics and bounds on the growth of torsion in the cohomology of congruence subgroups of $\operatorname{SL}(2,\mathcal{O}_F)$ for $F$ a number field, and of congruence subgroups in $\operatorname{SO}(n,1)$ with $n$ odd.

Spatial domain exploitation through 3D beamforming serves as a critical technology enabler for performance enhancement in the Fifth Generation New Radio (5G NR) specification. This is realized at the gNodeB (gNB) through the integration of massive antenna element arrays that facilitates 3D spatial multiplexing. However, these systems with high-directional transmissions also represent a threat to incumbent services such as radar and satellites. These incumbents already operate in midband spectrum\textemdash{}including the 4.4-4.9 GHz and 7.125-7.4 GHz bands\textemdash{}that are currently being evaluated for future cellular deployments. Here, we present the first work that evaluates the transmitted Effective Isotropic Radiated Power (EIRP) of a gNB in 3D space, using the 3GPP Release-18 standard for FR-1 instead of theoretical analyses of beam nulling, which can be simplistic. We shed light on the problems requiring attention with the EIRP profile in 3D space for existing codebook designs predefined in 3GPP: i) interference from a gNB does not depend only on the worst-case beamforming direction, but on a variety of beamforming directions due to side-lobes; ii) advanced antenna systems (AAS) architecture and antenna port configurations play a crucial role in average 3D EIRP, which are implementation dependent, and iii) we introduce two beam nulling methods, which achieve a 11 dB power reduction toward a target direction, with 3.5-4.5 dB SNR loss in UE link performance at a 10^{-4} bit error rate (BER) across modulation schemes under ideal and practical channel estimation, a higher loss compared to predictions from theoretical analyses.

Failure attribution, i.e., identifying the responsible agent and decisive step of a failure, is particularly challenging in LLM-based multi-agent systems (MAS) due to their natural-language reasoning, nondeterministic outputs, and intricate interaction dynamics. A reliable benchmark is therefore essential to guide and evaluate attribution techniques. Yet existing benchmarks rely on partially observable traces that capture only agent outputs, omitting the inputs and context that developers actually use when debugging. We argue that failure attribution should be studied under full execution observability, aligning with real-world developer-facing scenarios where complete traces, rather than only outputs, are accessible for diagnosis. To this end, we introduce TraceElephant, a benchmark designed for failure attribution with full execution traces and reproducible environments. We then systematically evaluate failure attribution techniques across various configurations. Specifically, full traces improve attribution accuracy by up to 76\% over a partial-observation counterpart, confirming that missing inputs obscure many failure causes. TraceElephant provides a foundation for follow-up failure attribution research, promoting evaluation practices that reflect real-world debugging and supporting the development of more transparent MASs.

We prove a sufficient condition for the existence of a $T$-periodic solution for the planar system $\dot z=F(t,z)$, characterized by the growth to infinity of the rotations made in one period by solutions starting at increasingly large initial values. Our result applies in particular to superquadratic Hamiltonian systems satisfying the Ambrosetti--Rabinowitz condition. The key novelty of the paper is that we do not require any growth condition on the flow to ensure global existence of solutions, allowing finite-time blow-up. Our method is based on a fixed-point theorem which exploits the rotational properties of the dynamics. To conclude, we discuss a family of examples of Hamiltonian systems showing finite-time blow-up.

The Terahertz (THz) band (0.1-10 THz) has emerged as a critical frontier for future communication systems, offering ultra-wide bandwidths that enable Terabits-per-second (Tbps) wireless links and high-precision sensing and imaging. However, practical deployment of THz systems is hindered by unique challenges, including intricate channel characteristics, high-dimensional and large-scale optimization problems, and highly dynamic network environments. Artificial Intelligence (AI) serves as a transformative enabler to address these challenges, providing robust capabilities for precise modeling, advanced signal processing, complex optimization, real-time decision-making, and prediction, among others. Reciprocally, the unprecedented bandwidth and high-resolution sensing capabilities of THz networks provide a promising physical infrastructure for AI, facilitating training, inference, and data collection. This survey presents a systematic and comprehensive overview of AI-driven solutions across the entire THz communication network and the symbiosis of AI and THz networks. To begin with, a foundational overview of AI technologies tailored for wireless communications is presented. Subsequently, AI-based innovations are investigated, spanning from hardware design, channel modeling, physical layer optimization, up to higher-layer network protocols and advanced THz services, including mobile edge computing and sensing-empowered applications. In parallel, the capacity of THz networks to serve AI is examined, underscoring a profound paradigm shift towards a mutual symbiosis where AI and THz co-evolve and empower each other. Finally, by synthesizing these state-of-the-art advancements and identifying open research directions, this survey highlights the potential of AI in copilot with development of THz communication systems.

We conclude an investigation of Abrishami, Esperet, Giocanti, Hamman, Knappe and Möller studying the existence of periodic colourings of locally finite graphs. A colouring of a graph $Γ$ is periodic if the resulting coloured graph has a finite number of orbits under its colour-preserving automorphisms, as such it is natural to consider those quasi-transitive graphs with finite quotient. In the case that the graph is planar and has 1-end we prove that it always permits a periodic proper vertex colouring. This is shown by constructing isometry respecting embedded maps into the Euclidean and hyperbolic planes and leveraging known properties of Euclidean and hyperbolic isometry groups. Moreover, in the case that a graph is Euclidean we show that this can always be done in 5 colours.

Precise and autonomous clocks are of fundamental interest and central importance to both foundational studies and practical applications. Here, we construct a blueprint for a quantum clock governed by time-independent interactions. By carefully-engineered coherent transport in dissipative spin chains, we achieve a scaling exponent at the precision-resolution trade-off fundamental bound, bringing this within reach of physically realistic and experimentally accessible systems. We further introduce a sudden-quench protocol that enables repeated operation through a simple initialization and detachment mechanism. Remarkably, the protocol is robust to imprecise detachment timing, implying that high-precision timekeeping can be achieved even when driven by a clock with much lower precision.