Joint communication and sensing applications require devices that can analyze multiple electromagnetic waves and process them in real time directly in the analog domain. In optics, the growing maturity of photonic integrated platforms allows the fabrication of complex circuits that can perform such operations, but their large number of sensors and actuators requires scalable control strategies to efficiently monitor and actively stabilize their functionality at runtime. In this work, we report on a multi-aperture silicon photonic programmable circuit that operates both as a coherent beam adder and a multi-aperture beam analyzer. The circuit consists of a reconfigurable mesh of Mach-Zehnder interferometers controlled through monolithically integrated electronic circuits, which are used to serialize/deserialize the readout of integrated sensors and the driving of actuators with a time-multiplexed addressing scheme. The circuit operation is validated in a communication and sensing scenario, where the photonic chip is used to simultaneously receive a 25 Gbit/s high-speed transmission and to measure the phase difference between the input light beams.

Scattering between dark matter (DM) and protons leads to suppressed small-scale fluctuations, with implications for a variety of cosmological observables. In this work, we search for evidence of DM-proton scattering with an interaction cross section $σ\!=\!σ_0 (\frac{v}{c})^n$ for $n=0,2$ and $4$, corresponding e.g. to velocity-independent contact interactions from heavy mediators, velocity-dependent pseudoscalar-mediated scattering, and higher-order dipole interactions, respectively, using high-redshift ($z \sim4-10$) ultraviolet galaxy luminosity functions (UVLFs) observed by Hubble Space Telescope (HST). We employ an adjusted implementation of GALLUMI combined with the modified Boltzmann solver CLASS DMeff that accounts for interacting DM, and incorporate UVLF data from both blank and lensed HST fields, alongside Planck CMB data and the Pantheon supernova catalog in a Bayesian analysis framework to set constraints on $σ_0$. Our results show that including lensed UVLF data, which probe fainter galaxies than the blank HST fields and thus smaller scales, leads to a substantial improvement in the constraints on $σ_0$ for $n>0$, surpassing existing bounds from Milky-Way (MW) satellite abundance and CMB anisotropies. For $m_χ = 1\,\rm MeV $, for example, we set the upper bounds at $1.1\times 10^{-25} \, \rm cm^2$ for $n=2$ and $2.1\times 10^{-22} \, \rm cm^2$ for $n=4$. For $n=0$, our bound is within an order of magnitude of those from the Lyman-$α$ forest and MW satellites.

In this paper, we study a class $\mathcal{C}$ of squarefree monomial ideals $I\subseteq R=\mathbb{K}[x_1,\dots,x_n]$ over a field $\mathbb{K}$, defined by the condition that $\dim R/I$ equals the maximum degree of the minimal generators of $I$ minus one. We show that the Stanley-Reisner ideal of every $i$-skeleton of a simplicial complex $Δ$ belongs to $\mathcal{C}$ for all $-1\le i<\dimΔ$. To investigate their homological properties, we introduce the notion of a degree resolution and prove that every ideal in $\mathcal{C}$ possesses this property. Moreover, we show that every squarefree monomial ideal admits a truncation whose regularity coincides with that of the original ideal, thereby reducing the study of degree resolutions to that of linear resolutions. Finally, we provide an explicit formula describing the relationship between the graded Betti numbers of a simplicial complex and those of its skeletons.

This Review consolidates publicly available aerial wireless measurement datasets collected using AERPAW. We organize signal-level, power-level, and KPI-level datasets under a unified taxonomy, harmonize metadata, and provide verified access with reproducible post-processing scripts. The curated catalog supports propagation modeling, machine learning, localization, and system-level evaluation for 5G-Advanced and emerging 6G aerial networks.

In this paper, we consider the resilient multi-dimensional consensus and distributed optimization problems of multi-agent systems (MASs) in the presence of both agent-based and denial-of-service (DoS) attacks. The considered agent-based attacks can cover malicious, Byzantine, and stubborn agents. The links between agents in the network can be blocked by DoS attacks, which may lead the digraph to be time-varying and even disconnected. The objective is to ensure that the remaining benign agents achieve consensus. To this end, an "auxiliary point"-based resilient control algorithm is proposed for MASs. Under the proposed algorithm, each healthy agent constructs a "safe kernel" utilizing the states of its in-neighbors and updates its state toward a specific point within this kernel at each iteration. If an agent cannot receive its neighbors' states owing to DoS attacks, it will use the states received immediately before the DoS period. Moreover, a resilient multi-dimensional distributed optimization (RMDO) algorithm is also proposed. Theoretical proofs and numerical examples are presented to demonstrate the effectiveness of the proposed algorithms.

A qubit, or quantum bit, is conventionally defined as "a physical system for storing information that is capable of existing in either of two quantum states or in a superposition of both". In this paper, we examine the simple question of whether two distinct levels, each consisting of multiply degenerate sub-states, could serve as a practical quantum bit. We explore this idea using a well-characterized atomic system of the kind employed in several quantum computing implementations. We approximate the atom as a two-level system without degeneracy lifting in the magnetic quantum number while using the angular momentum addition rules to select the desired state transition. We find that, in the continuous presence of the field, the atom still undergoes Rabi oscillations, which are suitable for quantum gate construction. In addition, we compute the average fidelity in quantum gate performance for a single degenerate atom and postulate the required form of two-atom interaction to construct a controlled Z gate.

Magnetic hyperthermia with metallic nanoparticles is a therapeutic strategy that relies on heating cancer cells to levels sufficient to damage or destroy them. After injection, the nanoparticles accumulate in tumor tissues, where they transfer energy from the applied time-dependent magnetic field to the surrounding medium, thereby increasing the local temperature. This heating effect can be spatially focused (superlocalized) by combining AC and DC magnetic fields. Heat generation arises either from the rotation of the particle or from the rotation of its magnetic moment. The theoretical framework is provided by the Martsenyuk-Raikher-Shliomis (MRSh) equation for the former and the stochastic Landau-Lifshitz-Gilbert (sLLG) equation for the latter. However, by using the concept of magnetic and ordinary viscosity, the results of these approaches can be directly compared, which is our goal in this work, with special emphasis on their ability to achieve spatial localization. On the basis of this comparison, we propose the use of perpendicular AC and DC magnetic fields for image-guided thermal therapy with magnetic particle imaging.

Understanding the formation and evolutionary history of the Milky Way requires detailed mapping of its stellar components, which preserve fossil records of the Galaxy's assembly through cosmic time. RR Lyrae stars are particularly well-suited for this endeavor, as they are old, standard candle variables that probe the Galaxy's earliest formation epochs. In this work, we employ a hierarchical Bayesian Gaussian Mixture Model (GMM) to characterize the three-dimensional density distribution of RR Lyrae stars in the Milky Way over the galactocentric radius ($R$) of $\approx 0.2-120~\rm{kpc}$. This approach provides a flexible framework for modeling complex stellar distributions, particularly in the inner Galaxy where the bulge, disk, and halo components overlap. Our analysis reveals that the inner Galaxy ($R\lesssim10~\rm{kpc}$) is dominated by a distinct prolate stellar population with axis ratio $q$=1.31. Consistent with previous work, we find the halo follows a $r^{-4}$ power-law profile that flattens within 12 kpc of the Galactic center. We also confirm the outer halo ($R\gtrsim10~\rm{kpc}$) is oblate $q$=0.70 with a tilt angle of $18^{\circ}$. We report for the first time that this tilt aligns the halo major axis in the direction of the Sagittarius dwarf galaxy. These results establish GMMs as an effective and flexible tool for modeling Galactic structure and provide new constraints on the distribution of old stars in the inner Galaxy.

We systematically investigate the S-wave singly heavy tetraquark systems containing two or three strange quarks, $Qs\bar{s}\bar{s}$, $Qn\bar{s}\bar{s}$ and $Qs\bar{s}\bar{n}\left( Q=c,b,n=u,d \right) $, within the constituent quark potential model. We solve the four-body Schrödinger equation using the Gaussian expansion method (GEM) and identify resonances via the complex scaling method (CSM). There are no bound states below the lowest two-meson thresholds. We obtain several compact resonances with $J^P=0^+,2^+$ in $Qs\bar{s}\bar{s}$, and $J^P=2^+$ in $Qn\bar{s}\bar{s}$ and $Qs\bar{s}\bar{n}$. The pole positions are mainly distributed around $7.0-7.2$ GeV (bottom) and $3.7-3.9$ GeV (charm), with widths from a few to several tens of MeV. These resonances decay into $D_sη^\prime ,{D_{(s)}^*}ϕ,{D_s}^*K^*$ and $D_s^*\bar{K}^*$ (and their bottom counterparts), providing targets for future experimental searches.

Searches for intrinsic electric dipole moments (EDMs) of nucleons, atoms and molecules are precision flavor-diagonal probes of new $CP$-odd physics, as motivated by the need to explain the matter-antimatter asymmetry in the universe. We review and summarise the effective field theory analysis of the observable EDMs in terms of a general set of $CP$-odd operators at 1~GeV, and the ensuing model-independent constraints on new physics. We also review and discuss the EDMs induced by $CP$-violation in the Standard Model, and the implications of EDM limits for various models of physics beyond the Standard Model.

Modeling and simulating how oxygen supply shapes neuronal excitability is crucial for advancing the understanding of brain function in pathological scenarios, such as ischemia. This condition is caused by a reduced blood supply, leading to the deprivation of oxygen and other metabolites; this energy deficit impairs ionic pumps and causes cellular swelling. In this work, this phenomenon is modeled through a volumetric variation law that links cell swelling to local oxygen concentration and the percentage of blood flow reduction. The swelling law links volume changes to local oxygen and the degree of blood-flow depression, providing a simple mechanistic pathway from hypoxia to tortuosity-driven transport impairment. The interplay between oxygen supply and excitability in brain tissue is described by coupling the monodomain model with specific neuronal ionic and metabolic models that characterize ion and metabolite concentration dynamics. The numerical approximation of this coupled multiscale problem is particularly challenging, owing to the presence of sharp and fast-propagating wavefronts and complex geometrical domains. To address these challenges, suitable space- and time-adaptive schemes are employed to capture the action potential dynamics accurately. This multiscale model is discretized in space with the high-order p-adaptive discontinuous Galerkin method on polygonal and polyhedral grids (PolyDG) and integrated in time with a Crank-Nicolson scheme. We numerically investigate different pathological scenarios on a two-dimensional idealized square domain and on a realistic geometry, both discretized with a polygonal grid, analyzing how subclinical and severe ischemia can affect brain electrophysiology and metabolic concentrations.

We study the dynamics of electrically charged black-hole binaries and their gravitational-wave emission during the inspiral phase. Within the post-Newtonian framework, we derive the conservative and dissipative dynamics up to second order (2PN), combining Effective Field Theory and classical methods. We compute the NNLO conservative Lagrangian, LO dissipative effects in harmonic and Lorenz gauges, and provide the equations of motion, center-of-mass transformations, and the Lagrangian/Hamiltonian in ADM-type coordinates. We also obtain gauge-invariant expressions for the binding energy, periastron advance in quasi-circular orbits, and the scattering angle in unbound orbits. Our results extend previous analyses and are fully consistent with recent post-Minkowskian findings.

Understanding the emergent macroscopic behavior of dynamical systems on networks is a crucial but challenging task. One of the simplest and most effective methods to construct a reduced macroscopic model is given by mean-field theory. The resulting approximations perform well on dense and homogeneous networks but poorly on scale-free networks, which, however, are more realistic in many applications. In this paper, we introduce a modified version of the mean-field approximation for voter model dynamics on scale-free networks. The two main deviations from classical theory are that we use degree-weighted shares as coarse variables and that we introduce a correlation factor that can be interpreted as slowing down dynamics induced by interactions. We observe that, for moderate noise and comparable interaction rates, the correlation factor is only a property of the network and not of the state or of parameters of the process. This approach achieves a significantly smaller approximation error than standard methods without increasing dimensionality.

Very high energy electrons (VHEE) in the 50-250 MeV range, delivered in short pulses at ultra-high dose rates, are proposed for clinical FLASH radiotherapy (RT) targeting deep-seated tumors. The clinical implementation of VHEE-FLASH RT requires online verification to optimize dose delivery. In this study we propose a novel online dose verification technique based on the detection of bremsstrahlung photons during VHEE interactions with matter. A polymethyl methacrylate (PMMA) phantom was simulated to evaluate the dose deposited by a VHEE beam and to optimise the system design to detect the bremsstralung radiation. Experimental validation was performed at the Beam Test Facility at Laboratori Nazionali di Frascati (Isituto Nazionale di Fisica Nucleare-INFN). A deep learning pipeline was developed to reconstruct the dose distribution in the phantom based on the bremsstrahlung radiation profile. Experimental results demonstrated successful detection of bremsstrahlung radiation emitted orthogonally to the beam axis. The deep learning model achieved accurate dose reconstruction based on the bremsstrahlung radiation profile with a discrepancy of less than 2 percent compared to the simulated dose distribution. This study confirms that bremsstrahlung detection provides a viable online verification for VHEE RT.

Accurate phase estimation -- the process of assigning phase values between $0$ and $2π$ to repetitive or periodic signals -- is a cornerstone in the analysis of oscillatory signals across diverse fields, from neuroscience to robotics, where it is fundamental, e.g., to understanding coordination in neural networks, cardiorespiratory coupling, and human-robot interaction. However, existing methods are often limited to offline processing and/or constrained to one-dimensional signals. In this paper, we introduce ROPE, which, to the best of our knowledge, is the first phase-estimation algorithm capable of (i) handling signals of arbitrary dimension and (ii) operating in real-time, with minimal error. ROPE identifies repetitions within the signal to segment it into (pseudo-)periods and assigns phase values by performing efficient, tractable searches over previous signal segments. We extensively validate the algorithm on a variety of signal types, including trajectories from chaotic dynamical systems, human motion-capture data, and electrocardiographic recordings. Our results demonstrate that ROPE is robust against noise and signal drift, and achieves significantly superior performance compared to state-of-the-art phase estimation methods. This advancement enables real-time analysis of complex biological rhythms, opening new pathways, for example, for early diagnosis of pathological rhythm disruptions and developing rhythm-based therapeutic interventions in neurological and cardiovascular disorders.

We identify images of the determinant morphism of trace-free co-Higgs bundles modeled on rank 2 Schwarzenberger bundles.

Recent studies have shown that quantum gravity introduces important corrections to the process of spherically symmetric gravitational collapse expected from general relativity. In particular, instead of falling into a central singularity, the collapsing body undergoes a bounce and eventually exits its Schwarzschild radius, and this entire process of collapse and rebound can occur in a single asymptotic region. In this paper, particle creation during such non-singular gravitational collapse is studied. It is shown that the probability of spontaneous emission of particles differs from the well-known probability of Hawking radiation from classical gravitational collapse. It is argued that the different result implies a deviation from thermality. Some arguments are also adduced concerning how the Hawking process during non-singular dust collapse could potentially remove shell crossing singularities.

A joint effect of the density-stratified turbulence (or inhomogeneous turbulence) and a large-scale shear for arbitrary Mach numbers results in the $α$ effect and mean-field dynamo action. These effects also produce the effective pumping velocity of a large-scale magnetic field. Compressibility of the turbulent velocity field (i.e., finite Mach number effects) does not affect the contributions to the $α$ tensor caused by the joint effect of inhomogeneity of turbulence and a large-scale shear. The isotropic part of the $α$ tensor is independent of the exponent of the turbulent kinetic energy spectrum. There is also an additional contribution to the effective pumping velocity of the mean magnetic field that is proportional to the product of the fluid density gradient and the divergence of the mean fluid velocity caused, e.g., by collapsing (or expanding) astrophysical clouds. Applications of these effects to protoplanetary discs, proto-galactic and proto-stellar clouds are discussed.

A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts that no elusive group is $2$-closed. Existing constructions of elusive groups mostly arise from split extensions. In this paper, we initiate the construction of elusive groups via non-split extensions. As a demonstration, we construct elusive groups of new degrees, namely $p^{3k-4}(p+1)/2$ for each Mersenne prime $p\geq7$ and integer $k\geq2$. We also construct the first examples of elusive groups with odd degree, namely $3^{k+1}\cdot5^2$, and twice odd degree, namely $2\cdot3^{k + 1}\cdot5^2$ for each $k\geq1$. We conclude by proposing further problems to advance this new direction of research.

We present a systematic statistical analysis of an informal astrophysical phenomenon known as Renzo's rule (or Sancisi's law), which states that "for any feature in a galaxy's luminosity profile, there is a corresponding feature in the rotation curve, and vice versa." This is often posed as a challenge for the standard LCDM model while supporting alternative theories such as MOND. Indeed, we identify clear features in the dwarf spiral NGC 1560 -- a prime example for Renzo's rule -- and find correlation statistics which support Renzo's rule with a slight preference for MOND over LCDM halo fits. However, a broader analysis on galaxies in the SPARC database reveals an excess of features in rotation curves that lack clear baryonic counterparts, with correlation statistics deviating up to $3σ$ on average from that predicted by both MOND and LCDM haloes, challenging the validity of Renzo's rule. Thus we do not find clear evidence for Renzo's rule in present galaxy data overall. We additionally perform mock tests, which show that a definitive test of Renzo's rule is primarily limited by the lack of clearly resolved baryonic features in current galaxy data.

We study a dirty two-dimensional superconductor with Rashba spin-orbit coupling and in-plane Zeeman fields described by the nonlinear sigma model that includes the Cooper and long-range Coulomb interactions. The renormalized Ginzburg-Landau theory, which includes the weak localization effects at the one-loop level, is constructed using the Keldysh functional formalism. It is shown that the transition temperature and magnetic field, as well as the tricritical point appearing in the phase diagram, are suppressed by the interactions. Nevertheless, we have found a universal behavior in the high-transition-temperature regime that demonstrates the robustness of the superconducting diode effect against the interactions. The conductivity of the resistive states emerging after the superconducting states are destroyed by the critical current is also calculated, and localization behaviors are demonstrated.

Charged-hadron identification (PID) is a critical requirement for the physics program of the Circular Electron-Positron Collider (CEPC). The baseline detector relies on ionization measurements from a time projection chamber (TPC), which provides strong PID capability at low momenta but becomes less effective at higher momenta. In this work, we investigate an extended PID strategy that combines dN/dx information from the TPC with time-of-flight (ToF) measurements from a silicon-based outer tracker (OTK) and a timing-upgraded inner tracker (ITK) equipped with AC-LGAD sensors. A unified discriminant is constructed to exploit the complementary sensitivity of ionization and timing observables. The performance is evaluated using simulated $Z \to q\bar{q}$ events, focusing on kaon identification in the presence of dominant pion backgrounds. The combined configuration significantly improves both efficiency and purity over a broad kinematic range, extending PID capability to both sub-GeV and multi-GeV regions. These results highlight the impact of precision timing on tracking detectors and demonstrate a viable path to enhanced PID performance for future lepton colliders.

We generalise the Aichelburg-Sexl Stress-Energy-Momentum tensor for massless particles to include motion with angular velocity, introducing tensorial spherical harmonics. Further, we make steps towards the concept of self-force for photons which could manifest as frequency shift.

Semiclassical descriptions of a few-level system coupled to an electromagnetic field mode reduce the field to a time-dependent driving term. Although such methods are widely used, the underlying quantum character of the field generates corrections to the dynamics that can become significant. Here we develop a general approach for systematically calculating quantum corrections to the time-dependent Floquet dynamics that emerges in the semiclassical limit. Taking the Rabi model of a spin-field system as an example, we show how approximate analytical expressions for short-time quantum corrections to the semiclassical dynamics can be obtained. This framework helps explain the emergence of semiclassical behavior, sheds new light on collapse and revival dynamics in the Rabi model, and provides a tool for assessing corrections to semiclassical control techniques.

Quantum computers represent a new computational paradigm with steadily improving hardware capabilities. In this article, we present the first study exploring how current quantum computers can be used to classify different neutrino event types observed in neutrino telescopes. We investigate two quantum machine learning approaches, Neural Projected Quantum Kernels (NPQKs) and Quantum Convolutional Neural Networks (QCNNs), and find that both achieve classification performance comparable to classical machine learning methods across a wide energy range. By introducing a moment-of-inertia-based encoding scheme and a novel preprocessing approach, we enable efficient and scalable learning with large neutrino astronomy datasets. Tested on both simulators and the IBM Strasbourg quantum processor, the NPQK achieves a testing accuracy near 80%, with robust results above 1 TeV and close agreement between simulation and hardware performance. A simulated QCNN achieves a ~70% accuracy over the same energy range. These results underscore the promise of quantum machine learning for neutrino astronomy, paving the way for future advances as quantum hardware matures.

Computing volumetric correspondences between 3D shapes is a prominent tool for medical and industrial applications. In this work, we pave the way for spectral volume mapping, extending for the first time the surface-based functional maps framework. We show that the eigenfunctions of the volumetric Laplace operator define a functional space that is suitable for high-quality signal transfer. We also experiment with various techniques that edit this functional space, porting them to volume domains. We validate our method on novel volumetric datasets and on tetrahedralizations of well established surface datasets, also showcasing practical applications involving both discrete and continuous signal mapping, for segmentation transfer, mesh connectivity transfer and solid texturing. Finally, we show that the volumetric spectrum greatly improves the accuracy for classical shape matching tasks among surfaces, consistently outperforming surface-only spectral methods.

The analysis of local minima in time series data and random landscapes is essential across numerous scientific disciplines, offering critical insights into system dynamics. Recently, Kundu, Majumdar, and Schehr derived the exact distribution of the number of local minima for a broad class of Markovian symmetric walks [Phys. Rev. E \textbf{110}, 024137 (2024)]; however, many real-world systems are non-Markovian, typically due to interactions with possibly hidden degrees of freedom. This work investigates the statistical properties of local minima in discrete-time samples of fractional Brownian motion (fBm), a non-Markovian Gaussian process with stationary increments, widely used to model complex, anomalous diffusion phenomena. We derive a complete asymptotic characterization of the fluctuations of the number of local minima $m_N$ in an $N$-step discrete-time fBm. We show that the fluctuations of $m_N$ exhibit a sharp transition at the Hurst exponent $H=3/4$: for $H\le 3/4$ they satisfy a central limit theorem with Gaussian limiting law, whereas for $H>3/4$ they converge to a non-Gaussian Rosenblatt process. The convergence at the process level gives us full statistical description at all times. We exemplify it on the covariance of the rescaled minima process, which displays two qualitatively distinct regimes matching Brownian and Rosenblatt covariances on either side of this threshold. Our analysis relies on a Hermite/Wick decomposition of the local-minimum indicator, which isolates a quadratic functional of an effective long-memory mode as the unique driver of the anomalous statistics. These results identify the count of local minima as a simple and robust diagnostic of long-range dependence in non-Markovian Gaussian processes, a conclusion supported by numerical simulations.

The main purpose of this paper is using a very simple constructive method to study an old number theory problem related to the Legendre symbol modulo p, and completely solved it. The proving method of the result is purely elementary and has been desired in the literature at least since 1927.

The growing use of Generative AI (GenAI) conversational search tools has raised concerns about their effects on people's metacognitive engagement, critical thinking, and learning. As people increasingly rely on GenAI to perform tasks such as analyzing and applying information, they may become less actively engaged in thinking and learning. This study examines whether metacognitive prompts - designed to encourage people to pause, reflect, assess their understanding, and consider multiple perspectives - can support critical thinking during GenAI-based search. We conducted a user study (N=40) with university students to investigate the impact of metacognitive prompts on their thought processes and search behaviors while searching with a GenAI tool. We found that these prompts led to more active engagement, prompting students to explore a broader range of topics and engage in deeper inquiry through follow-up queries. Students reported that the prompts were especially helpful for considering overlooked perspectives, promoting evaluation of AI responses, and identifying key takeaways. Additionally, the effectiveness of these prompts was influenced by students' metacognitive flexibility. Our findings highlight the potential of metacognitive prompts to foster critical thinking and provide insights for designing and implementing metacognitive support in human-AI interactions.

We explore the notion of an adjusted connection for principal 3-bundles. We first derive the explicit form of an adjustment datum for 3-term $L_\infty$-algebras, which allows us to give a local description of such adjusted connections and their infinitesimal symmetries. We then integrate the corresponding action Lie 3-algebroid to an action Lie 3-groupoid, encoding local connection forms with finite (higher) symmetries. This also yields the notion of an adjusted 2-crossed module of Lie groups. Stackifying the action Lie 3-groupoid then gives us the explicit description of principal 3-bundles with adjusted connections in terms of differential cohomology. These connections appear in a number of contexts within high-energy physics, and we list local examples arising in gauged supergravity as well as a global example arising in various contexts in string/M-theory. Our primary motivation, however, stems from U-duality, and we also define a notion of categorified torus that forms an adjusted 2-crossed module, which we hope to be useful in lifting T-duality to M-theory.