NASA's Solar Dynamics Observatory (SDO) Extreme-ultraviolet Variability Experiment (EVE) has been making solar full-disk extreme ultraviolet (EUV) spectral measurements since 2010 over the spectral range of 6nm to 106nm with 0.1nm spectral resolution and with 10-60sec cadence. A primary motivation for EVE's solar EUV irradiance observations is to provide the important energy input for various studies of Earth's upper atmosphere. For example, the solar EUV creates the ionosphere, heats the thermosphere, and drives photochemistry in Earth's upper atmosphere. In addition, EVE's observations have been a treasure trove for solar EUV flare spectra. While EVE measures the full-disk spectra, the flare spectrum is easily determined as the EVE spectrum minus the pre-flare spectrum, as long as only one flare event is happening at a time. These EVE flare observations provide EUV variability that have been used to study flare phases (including the discovery of the EUV Late Phase flare class), flare energetics (plasma temperature variations), corona heating (plasma abundance changes that support nano-flare heating mechanism), flare dynamics (downwelling and upwelling plasma flows during flares from Doppler shifts), and coronal mass ejections (CME) energetics (CME mass and velocity derived from coronal dimming in some EUV lines). We also introduce a new EVE data product called the EVE Level 4 Lines data product, which provides line profile-fit results for intensity, wavelength shift, and line width for 70 emission features. These emission features are from the chromosphere, transition region, and corona, and so Doppler measurements of those lines can reveal important plasma dynamical behavior during a flare's impulsive phase and gradual phase. With over 10,000 flares detected in the EVE observations, there is still much to study and to learn about solar flare physics using EVE solar EUV spectra.

Quantum thermodynamics defines the ideal quantum thermoelectric, with maximum possible efficiency at finite power output. However, such an ideal thermoelectric is challenging to implement experimentally. Instead, here we consider two types of thermoelectrics regularly implemented in experiments: (i) finite-height potential barriers or quantum point contacts, and (ii) double-barrier structures or single-level quantum dots. We model them with Landauer scattering theory as (i) step transmissions and(ii) Lorentzian transmissions, respectively. We optimize their thermodynamic efficiency for any given power output, when they are used as thermoelectric heat engines or refrigerators. The Lorentzian's efficiency is excellent at vanishing power, but we find that it is poor at the finite powers of practical interest. In contrast, the step transmission is remarkably close to ideal efficiency (typically within 15\%) at all power outputs. The step transmission is also close to ideal in the presence of phonons and other heat leaks, for which the Lorentzian performs very poorly. Thus, a simple nanoscale thermoelectric - made with a potential barrier or quantum point contact - is almost as efficient as an ideal thermoelectric.

We present a statistical analysis of the 175 SPARC galactic rotation curves to test the hypothesis of whether the Keplerian velocity tapering at large radii ($V(r)\propto 1/\sqrt{r}$) germane to a convergent mass distribution in typical spherical halo models agrees with observational data. The null hypothesis is Rubin's flat-rotation curve, $V(r)=\text{constant}$ -such as can be obtained from a spherical, isothermal-like density profile, or alternatively with a very prolate halo-. To decide whether we adopt the null (Rubin behaviour) or alternative (Keplerian behaviour) hypothesis, we evaluate the derivative in each galaxy of $V(r)$ with its last data points. The test is model independent inasmuch we are testing for the \emph{slope} of the dark matter rotation curve, whether it is or not compatible with zero. We conclude that the data is presently compatible with the null hypothesis -no taper off, no decline of $V(r)$ is seen. Separately, beyond SPARC, our own Milky Way galaxy, for which recent data sets have been reported, does show clear $V(r)$ fall-off at the level of 20\%.

Parameterized quantum circuits (PQCs) are pivotal components of variational quantum algorithms (VQAs), which represent a promising pathway to quantum advantage in noisy intermediate-scale quantum (NISQ) devices. PQCs enable flexible encoding of quantum information through tunable quantum gates and have been successfully applied across domains such as quantum chemistry, combinatorial optimization, and quantum machine learning. Despite their potential, PQC performance on NISQ hardware is hindered by noise, decoherence, and the presence of barren plateaus, which can impede gradient-based optimization. To address these limitations, we propose novel methods for improving gradient-free optimizers Rotosolve, Fraxis, and FQS, incorporating information from previous parameter iterations. Our approach conserves computational resources by reallocating optimization efforts toward poorly optimized gates, leading to improved convergence. The experimental results demonstrate that our techniques consistently improve the performance of various optimizers, contributing to more robust and efficient PQC optimization.

Given a countable group $G$, we initiate a systematic study of the Polish spaces of all minimal and topologically transitive actions of $G$ on the Cantor space by homeomorphisms, with a focus on the existence of comeager conjugacy classes in these spaces. We develop a general model-theoretic framework to study this and related questions, recovering on the way many existing results from the literature. A substantial part of the paper is devoted to actions of free groups. We show that in that case, there is a comeager conjugacy class in the space of minimal actions, as well as in the space of minimal, probability measure-preserving actions. The first one is the Fraïssé limit of all sofic minimal subshifts and the second, the universal profinite action. The case of the integers was already treated by Hochman and there the two actions coincide with the universal odometer. In the non-abelian case, they are substantially different and new techniques are required. In the opposite direction, if $G$ is an amenable group which is not finitely generated, we show that there is no comeager conjugacy class in the space of all actions, and if $G$ is locally finite, also in the space of minimal actions. Finally, we study the question of existence of a dense conjugacy class in the space of topologically transitive actions. We show that if $G$ is hyperbolic or virtually polycyclic, then such a dense conjugacy class exists iff $G$ is virtually cyclic, suggesting that the case of the integers may be exceptional.

As information carriers for fault-tolerant quantum computing, systems composed of anyons exhibit non-tensor product state spaces due to their distinctive fusion rules, leading to fundamentally different entanglement properties from conventional quantum systems. However, a quantitative characterization of entanglement for general anyonic states remains elusive. In this Letter, within the framework of resource theory, we propose three measures that quantify total entanglement, conventional entanglement, and anyonic charge entanglement (ACE), respectively. We demonstrate that total entanglement can be decomposed into conventional entanglement and ACE, revealing distinct entanglement structures in anyonic systems compared to those in conventional quantum systems. We further illustrate a geometric interpretation of our ACE measure and establish its equivalence to a previously proposed probe of ACE, extending the known equivalence between the geometric interpretation and operational significance of bipartite correlations. Our work broadens the understanding of entanglement.

Our understanding of complex systems rests on our ability to characterise how they perform distributed computation and integrate information. Advances in information theory have introduced several quantities to describe complex information structures, where collective patterns of coordination emerge from higher-order (i.e. beyond-pairwise) interdependencies. Unfortunately, the use of these approaches to study large complex systems is severely hindered by the poor scalability of existing techniques. Moreover, there are relatively few measures specifically designed for multivariate time series data. Here we introduce a novel measure of information about macroscopic structures, termed M-information, which quantifies the higher-order integration of information in complex dynamical systems. We show that M-information can be calculated via a convex optimisation problem, and we derive a robust and efficient algorithm that scales gracefully with system size. Our analyses show that M-information is resilient to noise, indexes critical behaviour in artificial neuronal populations, and reflects states of consciousness and task performance in real-world macaque and mouse neuroimaging data. Furthermore, M-information can be incorporated into existing information decomposition frameworks to reveal a comprehensive taxonomy of information dynamics. Taken together, these results help us unravel collective computation in large complex systems.

Pattern formation driven by mechanical stress plays a fundamental role in shaping structural organization in both natural and human-made systems. Using light as a vectorial stimulus may offer a powerful route to control stress-induced pattern formation in materials. However, achieving localized, programmable, and predictable control of individual microstructures via structured polarization fields has remained a major challenge. Here, we introduce vectorial field-guided lithography, a novel approach that leverages fully structured polarization fields as lithographic tools to enable the stress-driven reconfiguration of pre-patterned azopolymer microstructures with an unprecedented degree of flexibility, complexity, and diversity. By building on the Viscoplastic PhotoAlignment model, which describes the azopolymer deformation as stress response to structured light, we quantitatively demonstrate and predict complex surface architectures generated by programmable light-induced stress pathways using a digital polarization rotator implemented via a spatial light modulator. We model and experimentally achieve single-step formation of anisotropic, bent, and chiral microstructures from a single pre-patterned geometry. Our results reveal an exceptional control over local microstructure morphology and establish, for the first time, a comprehensive theoretical framework capable of quantitatively designing and fabricating target morphologies on azopolymers. This work moves beyond conventional intensity-based photopatterning and demonstrates that the full vectorial nature of light can dictate the mechanical reshaping of functional polymer surfaces, providing a new platform for the programmable design of complex micro-architectures with applications in photonics, microfluidics, and biology.

The use of nonstandard methods to characterize properties of weak, strong and mixed extensions of congruences to ultrafilters has been the main topic of several recent papers. We show that similar methods can be used to characterize the extensions of arbitrary realtions and their interplay.

Proximity coupling of bilayer graphene (BLG) to transition metal dichalcogenides (TMDs) offers a promising route to engineer gate-tunable spin-orbit coupling (SOC) while preserving BLG's exceptional electronic properties. This tunability arises from the layer-asymmetric electronic structure of gapped BLG, where SOC acts predominantly on the layer in contact with the TMD. Here, we present high-quality BLG/WSe$_2$ devices with a proximity-induced SOC gap and excellent electrostatic control. Operating in a quasi-ballistic regime, our double-gated heterostructures allow to form gate-defined p-n-p cavities and show clear weak anti-localization (WAL) features consistent with Rashba-type SOC. At lower hole densities, a transition to weak localization (WL) is observed, signaling transport through a single spin-split valence band. These findings - in agreement with calculations - provide direct spectroscopic evidence of proximity-induced spin-split band in BLG and underscore the potential of BLG/TMD heterostructures for spintronics and spin-based quantum technologies.

We study the interplay between critical isotropic elasticity and classical Ising criticality using a functional renormalization group (FRG) approach which is implemented such that the volume is fixed during the entire renormalization group flow. For dimensions slightly smaller than four we use a simple truncation of the FRG flow equations to recover the fixed points of the constrained Ising model: the Gaussian fixed point G, the Ising fixed point I, the renormalized Ising fixed point R, and the spherical fixed point S. We show that the fixed points R and S are both characterized by a finite anomalous dimension $y_{\ast}<0$ of strain fluctuations, implying that the energy dispersion of longitudinal acoustic phonons exhibits a non-analytic momentum dependence proportional to $k^{1-y_{\ast}/2}$ for small momentum $k$. We also derive and solve flow equations for the free energy at constant strain and compute stress-strain relations in the vicinity of the fixed points. As a result, we reaffirm that Ising criticality, controlled by the fixed point I, is preempted by a bulk instability. Beyond that, we find that the stress-strain relation at R and S remains linear to leading order (Hooke's law), as long as the interaction between strain and Ising fluctuations is sufficiently weak. However, the finite anomalous dimension of strain fluctuations $y_{\ast}$ gives rise to non-analytic corrections to Hooke's law.

Tensor networks establish an adaptable framework for the emulation of quantum circuits. By partitioning exponentially large registers and gates into smaller tensors, this unlocks fast transformations through tensor algebra, and grants fine control over memory, runtime and accuracy. Due to inherently lower spatial footprint, there is a gap in distributed-memory tensor network methods. While certain parallel techniques exist, they are usually limited to direct contraction and sampling problems, and a more general approach is needed for tensor representations like matrix product states (MPS), which efficiently approximate full quantum state evolution. In this study, we expand the MPS site tensors beyond local memory by introducing a tensor-parallel distribution scheme, where individual dense tensors are evenly scattered across a subset of indices. This is further facilitated by leveraging pivoted QR factorisation instead of slower singular value decomposition (SVD). We demonstrate the capabilities of our approach by approximately emulating the classically difficult Google's random circuit sampling (RCS) benchmark. The highest bond dimensions of 16,384 is reached, surpassing the accuracy of the state-of-the-art methods by three orders of magnitude on 32 nodes of ARCHER2. We also show how this helps advance experiments involving more practical quantum phase estimation circuits. Our approach has the potential to enhance numerous algorithms based on dense tensor networks, offering a scalable and naturally load-balanced distribution formula. It is also compatible with other types of parallelism, unlocking new opportunities to push the quantum-classical computational phase boundary.

Information seeking on mobile devices is often fragmented, trapping users in repetitive cycles of context switching and data re-entry, which increases cognitive load and disrupts workflow. Existing mobile agents provide limited cross-source integration and are largely opaque, presenting progress as a linear feed with few opportunities to intervene, steer, or take control. We present DroidRetriever, a transparent, steerable system for cross-source mobile information seeking. It accepts voice or typed input and the multi-LLM system decomposes the task, navigates to target pages, takes screenshots, and synthesizes a concise report with citation-linked screenshots. We make the process transparent through a progress dashboard combining sub-task progress and real-time exploration maps for seamless takeover. DroidRetriever also pauses on detected privacy or high-risk screens and prompts intervention. Across 35 tasks over 24 apps, experiments and user studies demonstrate improvements in coverage, transparency, and reduced workload. We release our code at https://github.com/AkimotoAyako/DroidRetriever.

Using the Hopf superalgebra structure of the enveloping algebra $U(\mathfrak g)$ of a Lie superalgebra $\mathfrak=\mathrm{Lie}(G)$, we give a purely algebraic treatment of $K$-bi-invariant functions on a Lie supergroup $G$, where $K$ is a sub-supergroup of $G$. We realize $K$-bi-invariant functions as a subalgebra $\mathcal A(\mathfrak g,\mathfrak k)$ of the dual of $U(\mathfrak g)$ whose elements vanish on the coideal $\mathcal I=\mathfrak kU(\mathfrak g)+U(\mathfrak g)\mathfrak k$, where $\mathfrak k=\mathrm{Lie}(K)$. Next, for a general class of supersymmetric pairs $(\mathfrak g,\mathfrak k)$, we define the radial restriction of elements of $\mathcal A(\mathfrak g,\mathfrak k)$ and prove that it is an injection into $S(\mathfrak a)^*$, where $\mathfrak a$ is the Cartan subspace of $(\mathfrak g,\mathfrak k)$. Finally, we compute a basis for $\mathcal I$ in the case of the pair $(\mathfrak{gl}(1|2), \mathfrak{osp}(1|2))$, and uncover a connection with the Bernoulli and Euler zigzag numbers.

Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems -- a young but already well-established theory. We introduce the Conley-Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of Conley-Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose.

High-precision corner-cube retroreflectors (CCRs) are critical for advanced lunar laser ranging (LLR) because they enable sub-millimeter-scale measurements of the Earth-Moon distance -- a level of precision essential for rigorous tests of relativistic gravitation and for advancing our understanding of lunar geophysics. In this work, we develop a comprehensive two-dimensional Fourier-optics model for single CCRs with apertures ranging from 80-110 mm. Our model incorporates realistic thermal-mechanical wavefront errors, detailed diffraction effects, and velocity aberration offsets. Our analysis reveals a strong coupling between aperture size and aberration angular offset: while larger CCRs deliver high on-axis flux under near-ideal conditions, their narrow diffraction lobes suffer significant flux loss at moderate aberration offsets, thereby favoring smaller apertures with broader main lobes. Furthermore, comparisons between solid fused-silica and hollow silicon-carbide (SiC) CCRs show that hollow designs not only achieve competitive or superior photon return -- particularly at 1064 nm, where phase errors are relatively reduced -- but also offer nearly an order-of-magnitude mass reduction for the same aperture sizes. These results establish a robust quantitative framework for optimizing CCR designs to perform at the sub-millimeter level under realistic lunar conditions and underscore the advantages of precision hollow SiC CCRs for next-generation LLR operations.

We investigate the transport dynamics of elongated microparticles in microchannel flows. While smooth-walled channels preserve the dependence of particle trajectories on initial orientation and lateral position, we show that introducing periodically textured walls can trigger robust alignment of the particle along the channel centerline. This geometry-driven alignment arises from repeated reorientations generated by spatially modulated shear gradients near the textured walls. The alignment efficiency depends on particle elongation and the relative texture wavelength, with an optimal range for maximal effect. While the observed alignment behavior is not limited to low Reynolds numbers, the characteristic alignment length scale diverges as the Reynolds number increases toward the turbulent flow regime. These findings offer a predictive framework for designing microfluidic devices that passively sort or focus anisotropic particles, with implications for soft matter transport, biophysical flows, and microfluidic engineering.

Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n\times n$ matrices over the finite field $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the nilradical of $\mathfrak b_n$. For every $\mathfrak b_n(\mathbb F_q)$-stable ideal $\mathfrak a$ of $\mathfrak u_n(\mathbb F_q)$, and every partition $μ$ of $n$, we prove two formulas for the number of elements of $\mathfrak a$ of Jordan type $μ$: the first one is the Hall scalar product of a modified Hall-Littlewood function indexed by $μ$ and a chromatic quasisymmetric function associated to $\mathfrak a$, and the second one is in terms of an explicit collection of standard tableaux. In the special case that $\mathfrak a$ is the nilradical $\mathfrak u_Λ(\mathbb F_q)$ of the parabolic subalgebra associated to a composition $Λ$ of $n$, our first formula reduces to a result of Karp and Thomas: up to an explicit polynomial factor in $q$, the number of elements in $\mathfrak u_Λ(\mathbb F_q)$ of Jordan type $μ$ is equal to the coefficient of the monomial $\mathsf x^Λ$ in the specialization of the dual Macdonald symmetric function $\mathrm Q_{μ'}(\mathsf x;q^{-1},t)$ at $t=0$. We give three applications: (1) a formula for the number of points of a nilpotent Hessenberg variety, (2) a formula for the number of $X\in \mathfrak u_Λ(\mathbb F_q)$ that satisfy $X^2=0$, which in the special case $Λ=(1^n)$ is different from the Kirillov-Melnikov-Ekhad-Zeilberger formula, and (3) a formula for the number of double cosets $\mathrm U_1\backslash\mathrm{GL}_n(\mathbb F_q)/\mathrm U_2$ where $\mathrm U_1$ and $\mathrm U_2$ are unipotent subgroups corresponding to two $\mathfrak b_n(\mathbb F_q)$-stable ideals.

We study groups definable in existentially closed geometric fields with commuting derivations. Our main result is that such a group can be definably embedded in a group interpretable in the underlying geometric field. Compared to earlier work of the first two authors toguether with K. Peterzil, the novelty is that we also deal with infinite dimensional groups.

JWST has revealed a population of 'Little Red Dots' (LRDs): compact, red objects at redshifts z=2-9 with 'v'-shaped spectral energy distributions, broad permitted lines, and, often, hydrogen Balmer absorption. We use NIRSpec/IFS data from the BlackTHUNDER survey to study the H$α$ line in the LRD Abell2744-QSO1 at z=7.04, which is a confirmed AGN due to time-variable equivalent width (EW) in its broad emission lines. The H$α$ spectral profile is non-Gaussian, requiring at least two Gaussian components. We also detect a narrow-line Gaussian component, and strong H$α$ absorption (EW relative to the continuum $\sim 22_{-7}^{+12} \mathring{\mathrm{A}}$), confirming a connection between the strong Balmer break and line absorption. The absorber is at rest with respect to broad H$α$, suggesting that the gas cannot be interpreted as an inflow or outflow, forming instead a long-lived structure. Its velocity dispersion is $σ_{\rm abs}=110_{-10}^{+20}$ km s$^{-1}$, consistent with the value inferred from the analysis of the Balmer break. Based on H$α$, we infer a black hole mass of log($M_{\rm BH}/{\rm M_\odot}$)=7.2, smaller but close to the previous estimates based on H$β$. The Eddington ratio is 0.09. Combining the high signal-to-noise ratio of the narrow H$α$ line with the spectral resolution R=3,700 of the G395H grating, we infer a narrow-line intrinsic dispersion $σ_{\rm n}=22_{-6}^{+5}$ km s$^{-1}$, which places a stringent constraint on the black-hole-to-dynamical-mass ratio of this system to be $M_{\rm BH}/M_{\rm dyn}$=0.15-1.2, confirming the overmassive nature of the black hole and potentially leaving little room for a host galaxy.

Motivated by the work of Aldroubi et al., we investigate the stability of the source term of the discrete dynamical system indexing over a non-uniform discrete set arising from spectral pairs in infinite-dimensional separable Hilbert spaces. Extending results due to Aldroubi et al., firstly, we give a necessary and sufficient condition for the recovery of the source term in finitely many iterations. Afterwards, we derive a necessary condition for the stability of the source term in finitely many iterations when it belongs to the closed subspace of an infinite-dimensional separable Hilbert space. Finally, we give a necessary and sufficient condition for the recovery of the source term in infinitely many iterations.

Generalizing previous work of the first author, we introduce and study a characteristic free analog of the $F$-threshold for non-principal ideals, BCM-thresholds. We show that this coincides with the classical $F$-threshold for weakly $F$-regular rings and that the set of BCM-thresholds coincides with the set of BCM-jumping numbers in a complete local regular ring. We obtain results on $F$-thresholds of parameter ideals analogous to results of Huneke-\mustata-Takagi-Watanabe as well as a mixed characteristic version of one of their results on multiplicity. Instead of taking ordinary powers of an ideal, our definition uses fractional integral closure in an absolute integral closure of our ambient ring.

We study S=1/2 quantum spin chains with shift-invariant and inversion-symmetric next-nearest-neighbor interaction, also known as zigzag spin chains. We completely classify the integrability and non-integrability of the above class of spin systems. We prove that in this class there are only two integrable models, a classical model and a model solvable by the Bethe ansatz, and all the remaining systems are non-integrable. Our classification theorem confirms that within this class of spin chains, there is no missing integrable model. This theorem also implies the absence of intermediate models with a finite number of local conserved quantities.

In this paper, we introduces and undertake as a systematical investigation of the class $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ of normalized harmonic mappings $f = h + \overline{g}$ in the unit disk $\mathbb{D}$, defined by the differential inequality \[ \text{Re}\left((1-α)h'(z) + αz h''(z)\right) > -M + \left|(1-α)g'(z) + αz g''(z)\right|\quad\text{for}\quad z\in\Bbb{D}, \] where $M > 0$, $α\in (0,1]$, and $g'(0) = 0$. This class extends the harmonic analogue of functions with positive real part and offers a unified framework for analyzing their geometric characteristics. We obtain sharp coefficient bounds for both the analytic and co-analytic parts, establish sharp growth bounds, and determine the radii of univalency, starlikeness, and convexity. Furthermore, we show that $\mathcal{P}_{\mathcal{H}}^{0}(α,M)$ is closed under convex combinations, and under suitable restrictions on the parameters, it is also closed under convolution. Our findings generalize and extend several known results in the theory of harmonic mappings.

Linear measurements are widely applied in sensing classical signals, e.g., gravitational wave (GW), and are developing toward joint measurement of multiple parameters. In this Letter, we aim at multiparameter linear measurements to classical monochromatic signals, and establish an ultimate tradeoff relation that tightly constrains the quantum limits on estimation precision. The tradeoff relation is fundamental since it is rooted in Heisenberg's uncertainty principle, and completely characterizes the dependence between the attainable precision limits on the estimated parameters. Eventually, we identify a necessary condition under which an optimal measurement protocol saturates the tradeoff relation, and show that the measurement phase can be regulated to implement flexible allocation of precision weights. Our finding can offer valuable guidance for detuned GW sensors in ultra-sensitive searches for post-merger remnants.

Let $N$ components be partitioned into two communities, denoted ${\cal P}_+$ and ${\cal P}_-$, possibly of different sizes. Assume that they are connected via a directed and weighted Erdös-Rényi (DWER) random graph with unknown parameter $ p \in (0, 1).$ The weights assigned to the existing connections are of mean-field-type, scaling as $N^{-1}$. At each time \modif{step}, we observe the state of each component: either it sends some signal to its successors (in the directed graph) or remains silent otherwise. In this paper, we show that it is possible to find the communities ${\cal P}_+$ and ${\cal P}_-$ based only on the activity of the $N$ components observed over $T$ time units. More specifically, we propose \modif{ two simple methods, an aggregated method and a spectral method, whose {\it misclassification rates} vanish as long as $T \gg N$ (up to log terms). This condition is proved to be near-optimal in the minimax sense. Moreover, under the stronger condition $T \gg N^2$ (up to log terms), the aggregated method is shown to achieve {\it exact recovery} with probability tending to $1$. } Interestingly, these simple \modif{methods} do not require any prior knowledge of the other model parameters (e.g. the edge probability $p$). The key step in our analysis is to derive an asymptotic approximation of the 1-lagged covariance matrix associated to the states of the $N$ components, as $N$ diverges. This asymptotic approximation relies on the study of the behavior of the solutions of a \modif{Stein-type} matrix equation satisfied by the simultaneous (0-lagged) covariance matrix associated to the states of the components. This study is challenging, especially because the simultaneous covariance matrix is random since it depends on the underlying DWER random graph.

Stability is a key property of both forward models and inverse problems, and depends on the norms considered in the relevant function spaces. For instance, stability estimates for hyperbolic partial differential equations are often based on energy conservation principles, and are therefore expressed in terms of $L^2$ norms. The focus of this paper is on stability with respect to the $L^\infty$ norm, which is more relevant to detect localized phenomena. The linear wave equation is not stable in $L^\infty$, and we design an alternative solution method based on the regularization of Fourier multipliers, which is stable in $L^\infty$. Furthermore, we show how these ideas can be extended to inverse problems, and design a regularization method for the inversion of compact operators that is stable in $L^\infty$. We also discuss the connection with the stability of deep neural networks modeled by hyperbolic PDEs.

Bridge multisections are combinatorial descriptions of surface links in 4-space using tuples of trivial tangles. They were introduced by Islambouli, Karimi, Lambert-Cole, and Meier to study curves in rational surfaces. In this paper, we prove a uniqueness result for bridge multisections of surfaces in 4-space: we give a complete set of moves relating to any two multiplane diagrams of the same surface. This is done by developing a surgery operation on multiplane diagrams called band surgery. Another application of this surgery move is that any $n$-valent graph with an $n$-edge coloring is the spine of a bridge multisection for an unknotted surface. We also prove that any multisected surface in $S^4$ can be unknotted by finitely many band surgeries.

This paper mainly investigates a class of distributed Variational Generalized Nash Equilibrium (VGNE) seeking problems for both online noncooperative games and online aggregative games with time-varying coupling inequality constraints. Two novel continuous-time distributed VGNE seeking algorithms are proposed, which realize the constant regret bound and sublinear fit bound, superior to those of the criteria for online optimization problems and online games. Furthermore, to reduce unnecessary communication among players, a dynamic event-triggered mechanism involving internal variables is introduced into the distributed VGNE seeking algorithm, while the constant regret bound and sublinear fit bound are still maintained. Also, the Zeno behavior is strictly prohibited. Moreover, we further investigate the impact of communication noise on the player's measurement of its neighbors' relative states. It is demonstrated that both the regret and fit bounds remain valid as long as the noise level is not excessively large. This result reveals, to some extent, the proposed algorithm's noise-resilient capability. Finally, an online Uncrewed Aerial Vehicle (UAV) swarm game and an online Nash-Cournot game are given to demonstrate the validity of the theoretical results.

Exactly ergodicity in boundary-driven semi-infinite cellular automata (CA) are investigated. We establish all the ergodic rules in CA with 3, 4, and 5 states. We analytically prove the ergodicity for 12 rules in 3-state CA and 118320 rules in 5-state CA with any ergodic and periodic boundary condition, and numerically confirm all the other rules non-ergodic with some boundary condition. We classify ergodic rules into several patterns, which exhibit a variety of ergodic structure.