To make precision particle mass measurements in charged spectrometers detailed understanding of the influence of detector effects is critical. In this paper the influence of detector-related uncertainties on the determination of the parent particle mass in two-body decays is investigated. It is shown how the dependence of observed mass shifts on the sum and difference of the daughter particle momenta can be used to determine the physical causes of a bias more rigorously than the \textit{ad hoc} rules that are often adopted. The approach is illustrated using the case of measuring the $Λ$ hyperon mass. This observable is of interest because our current knowledge relies on information from a single experiment that has not been updated to account for changes in the value of the $\textrm{K}_{\textrm{s}}^0$ mass used for calibration. With the approach developed in the paper it shown that the LHCb experiment has the capability to make a measurement of the $Λ$ mass with systematic uncertainties from the tracking system controlled to $0.7\,$keV/$c^2$. This allows a total precision of $2.2\,$keV/$c^2$ to be achieved, dominated by the knowledge of the $\textrm{K}_{\textrm{s}}^0$ mass used for calibration. This would improve the current knowledge of the $Λ$ hyperon mass by a factor of three.

We study higher-order weighted Dirichlet-type spaces on the unit disc associated with a class of poly-superharmonic weights. A higher-order Littlewood Paley formula is established enabling the computation of higher-order weighted Dirichlet integrals and allowing us to relate iterates of the Laplacian of the weight to higher-order defect operators of the shift operator on these spaces. This leads to the introduction of Dirichlet-type operators of finite order, a class containing $m$-isometries as well as completely hyperexpansive and completely hypercontractive operators of finite order. We prove that every cyclic operator in this class admits a functional model as the shift on a suitable higher-order weighted Dirichlet-type space, thereby providing a unified extension of the model theories for cyclic completely hyperexpansive operators and cyclic $m$-isometries.

Conventional femtosecond coherent laser spectroscopy predominantly focuses on the temporal phase coherence through time- or frequency-resolved methods. In this work, we suggest an alternative experimental framework based on spatial phase coherence. The intrinsic spectral dispersion of wavevectors in femtosecond pulses and sample dimensions exceeding the laser wavelength create a compelling basis to establish spatial phase coherence as a novel and robust foundation for femtosecond laser spectroscopy. Using rotational Raman coherence in gas molecules as a case study, we analyze the transverse spatial distribution of the third-order signal generated by a rotational wave-packet. Our findings reveal apparent temporal shifts and distortions in time-resolved signals that arise in conventional measurements lacking sensitivity to spatial phase coherence. Moreover, we demonstrate that spatial phase coherence can serve as a useful tool for thermometric applications, showcasing its sensitivity to temperature variations. These discoveries open new avenues in femtosecond laser spectroscopy, including an alternative single-shot detection scheme, a new form of Raman coherence imaging and molecular species quantification during overlapping fractional revivals.

The recently proposed first-order viscous relativistic hydrodynamics formulation by Bemfica, Disconzi, Noronha, and Kovtun (commonly known as the BDNK formulation) has been shown to be causal, stable, strongly hyperbolic, and thus locally well-posed. It is now a viable new option for modelling out-of-equilibrium effects in fluids, and has attracted wide attention in its potential applications to astrophysical systems. In this work, we present the first non-linear numerical simulation of spherically symmetric neutron stars using the BDNK formulation under the Cowling approximation. Using a simplified equation of state, we show that stable evolutions can be constructed within a restricted parameter space up to the simulation time we explored. From these simulations, we analyse the frequency content of the quasi-normal modes and the decay rate of the fundamental mode. This analysis serves as a first step towards constructing a fully consistent model of neutron stars using the BDNK formulation.

Most previous studies on relativistic quantum information have primarily focused on the vacuum state $|0\rangle$ and the first excited state $|1\rangle$ in two-mode entangled systems. In this work, we go beyond these limitations by considering arbitrary $q$-th excited states $|q\rangle$, aiming to investigate their role in preserving quantum resources. We analyze the influence of the Hawking effect on multipartite quantum states in the Schwarzschild spacetime, with particular attention to quantum entanglement and coherence. Our results show that, under the influence of the Hawking effect, increasing the excitation number $q$ leads to a reduction in quantum entanglement and mutual information, while enhancing quantum coherence. This indicates that the Hawking effect on excited multipartite states tends to degrade quantum correlations but simultaneously protects quantum coherence in curved spacetime. Therefore, when implementing quantum information protocols in gravitational settings, reducing the excitation number $q$ is favorable for maintaining entanglement, whereas increasing $q$ may be advantageous for tasks that rely on quantum coherence in relativistic quantum information processing.

A systematic comparison was carried out to assess the influence of representative thermostat methods in constant-temperature molecular dynamics simulations. The thermostat schemes considered include the Nosé--Hoover thermostat and its chain generalisation, the Bussi velocity rescaling method, and several implementations of the Langevin dynamics. Using a binary Lennard-Jones liquid as a model glass former, we investigated how the sampling of physical observables, such as particle velocities and potential energy, responds to changes in time step across these thermostats. While the Nosé--Hoover chain and Bussi thermostats provide reliable temperature control, a pronounced time-step dependence was observed in the potential energy. Amongst the Langevin methods, the Grønbech-Jensen--Farago scheme provided the most consistent sampling of both temperature and potential energy. Nonetheless, Langevin dynamics typically incurs approximately twice the computational cost due to the overhead of random number generation, and exhibits a systematic decrease in diffusion coefficients with increasing friction. This study presents a broad comparison of thermostat methods using a binary Lennard-Jones glass-former model, offering practical guidance for the choice of thermostats in classical molecular dynamics simulations. These findings provide useful insights for diverse applications, including glass transition, phase separation, and nucleation.

We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove \emph{a posteriori} error bounds in the $L^\infty(L^2)$-norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically.

Quantum machine learning (QML) has the potential to achieve quantum advantage for specific tasks by combining quantum computation with classical machine learning (ML). In classical ML, a significant challenge is membership-privacy leakage, whereby an attacker can infer from model outputs whether specific data were used in training. When specific data are required to be withdrawn, removing their influence from the trained model becomes necessary. Machine unlearning (MU) addresses this issue by enabling the model to forget the withdrawn data, thereby preventing membership-privacy leakage. However, this leakage remains underexplored in QML. This raises two research questions: do QML models leak membership privacy about their training data, and can MU methods efficiently mitigate such leakage in QML models? We investigate these questions using two quantum neural network (QNN) architectures, a basic QNN and a hybrid QNN, evaluated in noiseless simulations and cloud quantum device demonstrations. To answer the first question, we analyze how quantum constraints shape membership-privacy leakage in QML and then formalize a realistic gray-box threat model accordingly. Based on this, we design a membership inference attack (MIA) tailored to QNN outputs, and our results provide clear evidence of membership leakage in both QNNs. To answer the second question, we propose a quantum machine unlearning (QMU) framework, comprising three MU mechanisms. Evaluations on two QNN architectures show that QMU removes the influence of the withdrawn data while preserving accuracy for retained data. A comparative analysis further characterizes the three MU mechanisms with respect to data dependence, computational cost, and robustness.

Quantum geometry is universally bounded from below by Wilson-loop windings. In this work, we define an isolated set of bands to be Wilson-loop-ideal, if their quantum metric saturates the Wilson-loop lower bound. The definition naturally incorporates the known Chern-ideal and Euler-ideal bands, and allows us to define other types of ideal bands, such as Kane-Mele $Z_2$-ideal and inversion-fragile-ideal bands. In particular, we find that in the case of zero total Chern number, an isolated WL-ideal set of two bands with non-singular nonabelian Berry curvature and nontrivial normal Wilson-loop winding always admits a Chern-ideal gauge, without the need of a global good quantum number (such as spin). This enables the direct construction of new topologically ordered states, such as fractional topological insulator wavefunctions. We further propose a general framework of constructing monotonic flows that achieve Wilson-loop-ideal states starting from non-ideal bands through band mixing, where Wilson-loop-ideal states are not energy eigenstates but have smooth projectors similar to isolated bands. We apply the constructed flows to the realistic model of $3.89^\circ$ twisted bilayer MoTe$_2$, a moiré Rashba model and another moiré time-reversal-breaking models, and numerically find Chern-ideal, $Z_2$-ideal and inversion-fragile states, respectively, with relative error in the integrated quantum metric below $5\times 10^{-3}$. Our exact-diagonalization calculations on the numerically ideal states demonstrate the potential of our general definition of Wilson-loop-ideal bands and general procedure of constructing Wilson-loop-ideal states for future study of novel correlated physics.

In randomized controlled trials (RCTs) of infectious disease interventions, it is well recognized that unmeasured individual heterogeneity at baseline can induce selection bias over time, thereby complicating the interpretation of the estimated hazard ratio. The present study examines a simplified setting: RCTs consisting of homogeneous participants, with no individual heterogeneity at baseline. However, even in such an apparently ideal setting, selection bias can emerge over time due to temporal dependence in exposure, a realistic feature of infectious disease transmission. In this study, we mathematically characterize the mechanism underlying this bias and quantitatively evaluate its magnitude. Our results show that this bias should be recognized as an issue in both the design and interpretation of RCTs of infectious disease interventions.

Multi-invariants are local-unitary invariants of state replicas introduced as potential probes of multipartite entanglement and correlations in quantum many-body systems. In this paper, we investigate two multi-invariants for tripartite pure states, namely multientropy and dihedral invariant. We compute the (genuine) multientropy for Lifshitz groundstates, and obtain its analytical continuation to noninteger values of Rényi index. We show that the genuine multientropy can be expressed in terms of mutual information and logarithmic negativity, a relation that also holds for stabilizer states. For general tripartite pure states, we demonstrate that dihedral invariants are related to Rényi reflected entropies. In particular, we show that the dihedral permutations of replicas are equivalent to the reflected construction, or alternatively to the realignment of density matrices.

We prove that the signed counting (with respect to the parity of the ``$\operatorname{inv}$'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we introduce an interesting class of partition matrices called improper partition matrices. We further show that a subset of improper partition matrices is equinumerous with the set of Motzkin paths. Such an equidistribution is established both analytically and bijectively.

This study investigates the relationship between phase retardation and strain rates in combined extensional-shear flows using the Jeffery-Hamel flow formalism, which yields an analytical velocity solution. Flow birefringence was measured in a 1.0 wt$\%$ cellulose nanocrystal (CNC) suspension using a high-speed polarization camera. The velocity field was validated via particle image velocimetry (PIV), which showed good agreement with the analytical solution. In regions dominated by either shear or extensional components, the birefringence behavior was consistent with prior theoretical and experimental findings. In the combined extensional-shear regions of the Jeffery-Hamel flow, the birefringence magnitude followed the root-sum-square (RSS) of the shear- and extension-induced contributions. This observation aligns with the principal stress formulation derived from Mohr's circle, in which the principal stress is expressed as the RSS of extensional and shear stresses. This finding provides a basis for extending stress-birefringence analysis to flows with coexisting deformation modes.

We compute the second-harmonic response of two-dimensional topological Dirac semimetals subjected to an external in-plane magnetic field. The quantum kinetic equation for the Wigner distribution function is derived and then solved to evaluate the second-order electric-field contributions to the current density. Both the Berry curvature dipole and the field-induced terms in the current are analyzed across a broad range of model parameters. We propose that our theory can be tested experimentally by measuring the dependence of the anomalous Hall resistivity on the in-plane magnetic field in the surface states of the topological insulator SnTe, in WTe$_2$ and WSe$_2$ monolayers, as well as in the Kondo lattice material Ce$_3$Bi$_4$Pd$_3$ at very low temperatures.

This paper studies the non-preemptive single-machine scheduling problem with heterogeneous release times and processing times, with the objective of minimizing total waiting time. The problem is known to be NP-hard. By modeling machine idle time as negative waiting time, we establish a unified waiting-time formulation that compresses the original four-dimensional description into a two-dimensional representation depending only on processing times and waiting times, thereby substantially simplifying the structural analysis of the problem. Based on this representation, ideal directions are defined from the structure induced by an optimal solution, and it is shown that every non-optimal schedule admits an ideal improvement path. It is further proved that improvable ideal directions require no coordination and can therefore be treated as independent improvement units. The theoretical analysis also shows that the realizability of every improvable ideal direction can be determined in finitely many steps, thereby laying the foundation for an overall solution framework. For each improvable ideal direction, by analyzing the propagation of the decrease flow and the increase flow under local adjustments, we prove that queue discontinuity is the unique structural obstacle to the realization of ideal improvement paths. A complete characterization of the resulting local optima is then established, and corresponding repair operators are designed. On this basis, an iterative repair algorithm under the improvement-path framework is developed. It is proved to terminate in finitely many steps, return a globally optimal schedule, and admit an explicit upper bound on its time complexity. This work provides a new analytical perspective and solution framework for this class of NP-hard scheduling problems.

We develop a general mathematical framework to study mixtures of different physical systems brought together on a discrete interface. Adapting work by Măntoiu et al., we use an operator algebraic framework such that the bulk systems at infinity of the mixture are recovered via the spatial asymptotics of the operators on the interface. Fixing an asymptotics and interface algebra, we show how the essential spectrum and topological properties can be inferred from the bulk systems at infinity. By working with Hilbert $C^*$-modules, we can further refine these results with respect to an ambient algebra of observables.

We use simultaneous Padé approximations to $_3F_2$ hypergeometric functions to estimate from below linear forms in $1$, $π\sqrt d$, $Ω_D/π$ and $π/Ω_D$ with integral coefficients, for certain choices of positive integer $d$ and negative integer $D$, where $Ω_D$ is (the square of) a Chowla--Selberg period attached to the imaginary quadratic field $Q(\sqrt{D})$.

In an earlier letter [Ducharme \textit{et al.} Phys. Rev. Lett. \textbf{126}, 134803 (2021)], a solution to the Dirac equation for a relativistic Gaussian electron beam showed that for a diverging beam the spin of each electron is the sum of fractional contributions from both the spin angular momentum (SAM) and orbital angular momentum (OAM) operators. Fractional angular momenta emerge when eigenstates of the Dirac equation can be decomposed into two terms of opposite spin. Each of these terms being eigenstates of both the SAM and OAM operators. Building on this understanding, the same method used to calculate fractional angular momenta in beams is applied here to solutions of the Dirac equation for hydrogen-like atoms. The results strengthen the idea that factorization of the Klein-Gordon equation using Dirac matrices equation does more than introduce spin, it also produces a specific mixing of the angular momentum states leading to the presence of fractional angular momenta and related effects such as fractional Gouy phase in the case of beams and fractional wave-particle duality in both atoms and beams.

We present the first targeted searches for continuous gravitational waves (CWs) from 114 active galactic nuclei (AGN) that may host supermassive black hole binaries, using the NANOGrav 15 yr data set. By incorporating electromagnetic priors on sky location, distance, redshift, and CW frequency, our strain and chirp mass upper limits are typically improved by a factor of $\sim 2$ (median 2.2) relative to all-sky limits at the same frequency. Bayesian comparisons against a model including only a Hellings-Downs correlated background disfavors a CW signal for all targets, with a mean Bayes factor of $0.73 \pm 0.32$. Two targets have Bayes factors slightly above unity, but coherence tests, random targeting experiments, and a conservative accounting of the 114-target trials factor all indicate that they are consistent with noise. We use these two candidates as worked examples to illustrate an end-to-end targeted CW search analysis and a suite of follow up tests that future promising candidates would need to pass. We find that the electromagnetic interpretations of both candidates are ambiguous, and we update the constraints on a putative binary in 3C 66B, ruling out part of its previously allowed parameter space. Ultimately, our results demonstrate the current sensitivity of targeted pulsar timing array searches for CWs and define a roadmap for future multimessenger CW detections.

We generalize the framework of spurion analysis to a class of selection rules arising from non-invertible fusion algebras in perturbation theory. As a first step toward systematic applications to particle physics, we analyze the near-group fusion algebras, defined by fusion rules built from a finite Abelian group $G$ extended by a single non-invertible element. Notable examples include the Fibonacci and Ising fusion rules. We introduce a systematic scheme for labeling coupling constants at the level of the non-invertible fusion algebra, enabling consistent tracking of couplings when constructing composite amplitudes from simpler building blocks. Our labeling provides a clear interpretation of why the tree-level exact non-invertible selection rules are violated through radiative corrections, a unique phenomenon essential to ``loop-induced groupification''. We also identify the limit where the near-group fusion algebra is lifted to a $G\times \mathbb{Z}_2$ group, which provides an alternative scheme of spurion analysis consistent with the original one based on the near-group algebra. Meanwhile, we highlight the distinctions between the selection rules imposed by the near-group fusion algebra and those from breaking the $G\times \mathbb{Z}_2$ group.

This study investigates the identification power gained by combining experimental data, in which treatment is randomized, with observational data, in which treatment is self-selected, for distributional treatment effect (DTE) parameters. While experimental data identify average treatment effects, many DTE parameters, such as the distribution of individual treatment effects, are only partially identified. We examine whether and how combining these two data sources tightens the identified set for such parameters. For broad classes of DTE parameters, we derive nonparametric sharp bounds under the combined data and clarify the mechanism through which data combination improves identification relative to using experimental data alone. Our analysis highlights that self-selection in observational data is a key source of identification power. We establish necessary and sufficient conditions under which the combined data strictly shrink the identified set, and show that such gains arise generically unless selection-on-observables holds in the observational data. We also propose a linear programming approach to compute sharp bounds that can incorporate additional structural restrictions, such as positive dependence between potential outcomes and the generalized Roy selection model. An empirical application using data on negative campaign advertisements in the 2008 U.S. presidential election illustrates the practical relevance of the proposed approach.

Finite mixture models provide a flexible framework for approximating and estimating multivariate probability densities. We study mixtures formed from translated and rescaled copies of a fixed density kernel and obtain explicit results for both approximation and least-squares estimation. Our main deterministic result is a quantisation theorem showing that, after smoothing the target density at a fixed resolution, the resulting convolution can be compressed into a finite location mixture with controlled error. Combining this with the smoothing bias yields approximation rates in $\mathcal{L}_{p}$ over Sobolev classes. For estimation, we analyse least-squares $\varepsilon$-minimisers over suitably tuned mixture sieves. Under exponential decay of the Fourier transform of the kernel, a matching moment condition, and bounded Sobolev targets, the estimator attains a squared $\mathcal{L}_{2}$ risk bound whose rate matches the Sobolev minimax benchmark up to a logarithmic factor. If, in addition, the kernel is bandlimited, then the same theorem recovers the Sobolev rate $n^{-2s/\left(2s+d\right)}$. We further report a slower convergence rate under weaker VC-type assumptions. At fixed scale, the Fourier-based approach also gives a nearly parametric risk bound for the associated location-mixture class, and the same bandlimited simplification removes the logarithmic correction. In the Gaussian case, this recovers the known Gaussian location-mixture rate. We also prove matching lower bounds on Gaussian convolution submodels, including strict submodels of the Gaussian location-mixture class, and on the tensor-product odd-degree Student-$t$ location-mixture family.

We propose a new method for detecting dark matter axions using a resonant cavity coupled with a quantum Hall system. When a small sample exhibiting quantum Hall effect is placed inside the cavity and the cavity is tuned to resonance, two-dimensional electrons absorb the amplified radiation, leading to a rise in the sample's temperature. By monitoring this temperature increase, the mass $m_a$ of the axion can be inferred. As an example, consider a GaAs sample with surface area $S=0.01\text{cm}^2$ and small thickness $d = 1\,μ\mathrm{m}$ and its heat capacity $C_s$ at temperature $T = 20\,\mathrm{mK}$. Because the energy flux of the incoming radiation is $P_{ra}\sim 5.9\times10^{-20}\text{W}\,(S/0.01\text{cm}^2)\,(g_{aγγ}/10^{-14}\text{GeV}^{-1})^2\,(σ/10^7\text{eV})\, (10^{-5}\mbox{eV}/m_a)^3(B/15\text{T})^2 (ρ_d/0.3\rm GeV cm^{-3})$ at the resonance with electrical conductivity $σ$ of the cavity wall, the temperature increase is $P_{ra}t_{ob}/C_s \simeq 4.8\mbox{mK}(t_{ob}/1\text{s})(g_{aγγ}/10^{-14}\text{GeV}^{-1})^2(20\mbox{mK}/T)^3 (10^{-5}\mbox{eV}/m_a)^3(σ/10^7\text{eV})(1μ\text{m}/d) (B/15\text{T})^2$ with $1\text{T}=10^4$ Gauss where $t_{ob}=1$s is the observation time. It must be smaller than a time constant $τ>1$s associated with the heat dissipation into thermal bath. Such a large time constant can be realized using superconducting nanowire lead and thin film pedestal supporting the sample dilution refrigerator. The temperature increase $ΔT\sim 5$mK is detectable using quantum point contact thermometer.

In antiferromagnetic spintronics, accessing the spin degree of freedom is essential for generating spin currents and manipulating magnetic order, which generally requires lifting spin degeneracy. This is typically achieved through relativistic spin-orbit coupling or non-relativistic spin splitting in altermagnets. Here, we propose an alternative approach: a dynamical spin splitting induced by an optical field in antiferromagnets. By coupling the driven system to a thermal bath, we demonstrate the emergence of steady-state pure spin currents, as well as linear-response longitudinal and transverse spin currents. Crucially, thermal bath engineering enables a nonrelativistic Edelstein effect--the generation of a net spin accumulation--without relying on spin-orbit coupling. Our results provide a broadly applicable and experimentally tunable route to control spins in antiferromagnets, offering new opportunities for spin generation and manipulation in antiferromagnetic spintronics.

Exploring potential empirical manifestations of quantum gravity is a challenging pursuit. In this study, we utilise a lattice representation of a (2+1)D massive gravity toy model interacting with Dirac fermions that can support specific spacetime fluctuations. We focus on the evolution of the fermion's spin due to its coupling to spacetime fluctuations. To monitor these dynamics, a minimal model is required that comprises two bosonic modes describing spacetime geometry fluctuations coupled to the spin of the fermion. A possible emulation of this system involves encoding spin degrees of freedom in the electronic states of an atom coupled to a bimodal optical cavity that provides the two bosonic modes. Our proposal introduces a novel approach for modelling the effect of interactions between quantum gravity and matter that can be probed with current technology.

Gravitational wave signals from binary neutron star (BNS) mergers and binary low-mass black hole (BLMBH) mergers are highly similar in the early inspiral phase. Consequently, the astrophysical origin of recently detected low-mass compact binary coalescences has remained ambiguous, particularly in the absence of electromagnetic counterparts. In this work, we demonstrate that proposed detectors with increased high-frequency sensitivity $-$ including NEMO, Cosmic Explorer, and the Einstein Telescope $-$ will reliably distinguish these two source classes in the late inspiral and postmerger regimes. We further show how these detections can be used to disentangle the individual contributions of BNS and BLMBH systems to the compact binary merger rate, while accounting for misclassification probabilities. Finally, we show this can lead to constraints on the interaction of heavy, non-annihilating dark matter with nucleons. This is achieved by noting that capture of such dark matter particles into neutron stars would lead to transmuted black holes (TBHs), formed via neutron star collapse, which would contribute to the BLMBH rate.

We study the problem of recovering a function from the magnitude of its Gabor transform sampled on a discrete set. While it is known that uniqueness fails for general square integrable functions, we show that phase retrieval is possible for a dense class of signals: specifically, those whose Bargmann transforms are entire functions of exponential type. Our main result characterises when such functions can be uniquely recovered (up to a global phase) from magnitude only data sampled on uniformly discrete sets of sufficient lower Beurling density. In particular, we prove that every entire function of exponential type is uniquely determined (up to a global phase) among all second order entire functions by its modulus on a sufficiently dense shifted lattice with suitable structure.

Magnetic reconnection is a fundamental plasma process that alters the magnetic field topology and releases magnetic energy. Most numerical simulations and spacecraft observations assume a two-dimensional diffusion region, with the electron diffusion region (EDR) embedded in the same plane as the ion diffusion region (IDR) and a uniform guide field throughout. Using observations from Magnetospheric Multiscale (MMS) mission, we report a non-coplanar, knotted EDR in Earth's magnetotail current sheet. The reconnection plane of the knotted EDR deviates by approximately 38° from that of the IDR, with the guide field exhibiting both a 38° directional shift and a twofold increase in amplitude. Moreover, the Hall magnetic field is bipolar in the EDR but quadrupolar in the IDR, indicating different Hall current structures at electron and ion scales. These observations highlight the importance of three-dimensional effects and illustrate the complexity of multiscale coupling between the EDR and IDR during reconnection studies.1

Previous generations of cellular communication, such as 5G, have been designed with the objective of improving key performance indicators (KPIs) such as throughput, latency, etc. However, to meet the evolving KPI demands and the ambitious sustainability targets for the Information and Communication Technology (ICT) industry, 6G will need to be designed differently. 6G will need to consider both the performance and sustainability targets for the various use cases it will serve. In addition, 6G will have various candidate technological enablers, making the design space of the system even more complex. Furthermore, due to the subjective nature of sustainability indicators, especially social sustainability, the literature still lacks clear methods to link them with technical enablers and 6G system design. Hence, in this article a novel method for 6G end-to-end (E2E) system design based on Knowledge graphs (KG) has been introduced. It considers as its input: the use case KPIs, use case sustainability requirements expressed as Key Values (KV) and KV Indicators (KVIs), the ability of the technological enablers to satisfy these KPIs and KVIs, the 6G system design principles defined in Hexa-X-II project, the maturity of a technological enabler and the dependencies between the various enablers. The KG method also introduces a novel approach for determining the key values addressed by a technological enabler. The effectiveness of the KG method was demonstrated by its application in designing the 6G E2E system for the cooperating mobile robot use case defined in the Hexa-X-II project, where 82 enablers were selected. Lastly, results from proof-of-concept demonstrations for a subset of the selected enablers have also been provided, which reinforce the efficacy of the KG method for designing a sustainable 6G system.

We offer the first operational interpretation of the $α$-z relative entropies, a measure of distinguishability between two quantum states introduced by Jakšić et al. and Audenaert and Datta. We show that these relative entropies appear when formulating conditions for large-sample or catalytic relative majorization of pairs of flat states and certain generalizations of them. Indeed, we show that such transformations exist if and only if all the $α$-z relative entropies for $α$<1 of the two pairs are ordered. In this setting, the $α$ and z parameters are truly independent from each other. These results also yield an expression for the optimal rate of converting one flat state pair into another. Our methods use real-algebraic techniques involving preordered semirings and certain monotone homomorphisms and derivations on them.