AutoRegressive Conditional Heteroscedasticity (ARCH) models are standard for modeling time series exhibiting volatility, with a rich literature in univariate and multivariate settings. In recent years, these models have been extended to function spaces. However, functional ARCH and generalized ARCH (GARCH) processes established in the literature have thus far been restricted to model ``pointwise'' variances. In this paper, we propose a new ARCH framework for data residing in general separable Hilbert spaces that accounts for the full evolution of the conditional covariance operator. We define a general operator-level ARCH model. For a simplified Constant Conditional Correlation version of the model, we establish conditions under which such models admit strictly and weakly stationary solutions, finite moments, and weak serial dependence. Additionally, we derive consistent Yule--Walker-type estimators of the infinite-dimensional model parameters. The practical relevance of the model is illustrated through simulations and a data application to high-frequency cumulative intraday returns.

Dark matter (DM) could be subject to non-gravitational self-interactions which is relevant to resolve potential problems of cold DM on small scales. Their impact on astrophysical objects such as galaxies and galaxy clusters allows for constraining the strength of this scattering and eventually further properties of the cross-section. To model self-interacting dark matter (SIDM), N-body simulations are a crucial tool widely employed by the SIDM community. In this paper, we describe the SIDM implementation in the cosmological hydrodynamical N-body code OpenGadget3 and release it to the public. It is capable of simulating elastic scattering for various differential cross-sections, including strongly anisotropic cross-sections. Beyond single-species models, the code also allows simulating a two-species model with cross-species interactions. In addition to describing the numerical schemes for modelling various flavours of SIDM, we discuss the technical challenges of implementing them. Moreover, we demonstrate through several test problems that OpenGadget3 can accurately simulate DM self-interactions. Furthermore, we assess the performance of the code and provide scaling tests. Lastly, we highlight remaining challenges in the context of SIDM and describe directions for improving the current state of the art.

We study linear $α_p$-actions on affine spaces and the associated quotient singularities, using explicit stacky resolutions. We describe when the quotient singularities are log canonical, canonical or terminal, and we compute their stringy motivic invariants. The second author and Fabio Tonini conjectured that these invariants coincide with those of linear $\mathbb{Z}/p$-quotients: our approach reduces this conjecture to an equality of explicit multi-sets, which we check for a large number of primes using a computer software. A general proof of the equality of multi-sets is given in the appendix written by Linus Rösler.

People experiencing mental health crises frequently turn to open-ended generative AI (GenAI) chatbots for support. However, rather than providing immediate assistance, some GenAI chatbots are designed to respond to crisis situations in ways that minimize their developers' liability, primarily through avoidance (e.g., refusing to engage beyond templated referrals to crisis hotlines). Withholding crisis support in these cases may harm users who have no viable alternatives and reduce their motivation to seek further help. At scale, this avoidant design could undermine population mental health. We propose empowerment-oriented design principles for AI crisis support, informed by community helper models. As an initial touchpoint in help-seeking, AI chatbots can act as a supportive bridge to de-escalate crises and connect users to more reliable care. Coordination between AI developers and regulators can enable a better balance of risk mitigation and user empowerment in AI crisis support.

We refine the combinatorial 1-cocycle $\mathbb{L}R_{reg}$ for regular isotopies of long knots to a 1-cocycle with values in the free $\mathbb{Z}[x,x^{-1}]$-module generated by regular isotopy classes of oriented tangles with exactly one signed ordinary double point. We use it to define the refined tangle equations for couples of knot diagrams, where the coefficients are now Laurent polynomials instead of integers. A solution of the tangle equations gives quantitative information about any knot isotopy which relates two given knot diagrams. If the tangle equations have no solution, then the diagrams represent different knots.

The AND gate is not reversible$\unicode{x2014}$on qubits. However, it is reversible on qutrits, making it a building block for efficient simulation of qubit computation using qutrits. We first observe that there are multiple two-qutrit Clifford+T unitaries that realize the AND gate with T-count 3, and its generalizations to $n$ qubits with T-count $3n-3$. Our main result is the construction of a novel qutrit $\mathopen{[\![} 6,2,2 \mathclose{]\!]}$ quantum error-correcting code with a transversal implementation of the AND gate. The key insight in our approach is that a symmetric T-depth one circuit decomposition$\unicode{x2014}$composed of a CX circuit, T and T dagger gates, followed by the CX circuit in reverse$\unicode{x2014}$of a given unitary can be interpreted as a CSS code. We can increase the code distance by augmenting the code circuit with additional stabilizers while preserving the logical gate. This results in a code with a "built-in" transversal implementation of the original unitary, which can be further concatenated to attain a $\mathopen{[\![} 48,2,4 \mathclose{]\!]}$ code with the same transversal logical gate. Furthermore, we present several protocols for mixed qubit-qutrit codes which we call Qubit Subspace Codes, and for magic state distillation and injection.

We present a new method for computation of the index of completely mixed equilibria in finite games, based on the work of Eisenbud et al.(1977). We apply this method to solving two questions about the relation of the index of equilibria and the index of fixed points, and the index of equilibria and payoff-robustness: any integer can be the index of an isolated completely mixed equilibrium of a finite game. In a particular class of isolated completely mixed equilibria, called monogenic, the index can be $0$, $+1$ or $-1$ only. In this class non-zero index is equivalent to payoff-robustness. We also discuss extensions of the method of computation to extensive-form games, and cases where the equilibria might be located on the boundary of the strategy set.

Water is a critical resource for data centers and an efficient means of cooling. However, meeting the growing water demand of data centers requires substantial peak water withdrawals, which many communities in the United States cannot supply, especially during the hottest days of the year. This largely overlooked water capacity constraint is emerging as a bottleneck for data centers and can force operators to rely on less efficient dry cooling, further stressing the power grid during summer peaks. In this paper, we focus on the direct water withdrawal of U.S. data centers for cooling and examine their impacts on public water systems. Our analysis indicates that, if the 2024 water use intensity persists, U.S. data centers could collectively require 697-1,451 million gallons per day (MGD) of new water capacity through 2030, comparable to New York City's average daily supply of roughly 1,000 MGD. Under an optimistic scenario with a compound annual water use intensity reduction by 10%, the water capacity demand decreases to 227-604 MGD, although high-growth IT loads could still require enough capacity to hypothetically supply about half of New York City for most of the year. The total valuation of the new water capacity is on the order of \$10 billion, reaching up to \$58 billion in the high-growth case. These impacts are highly concentrated on communities hosting data centers. Finally, we provide recommendations to address the growing water capacity demand of U.S. data centers, including reporting peak water use, developing corporate-community partnerships, adopting a Water Capacity Neutral approach (colloquially "Pipe Neutral") to allow host communities to retain limited water capacity resources, and implementing coordinated water-power planning to responsibly leverage water for peak power reduction and opportunistically utilize surplus power to mitigate impacts on public water systems.

The computation of the Log-determinant of large, sparse, symmetric positive definite (SPD) matrices is essential in many scientific computational fields such as numerical linear algebra and machine learning. In low dimensions, Cholesky is preferred, but in high dimensions, its computation may be prohibitive due to memory limitation. To circumvent this, Krylov subspace techniques have proven to be efficient but may be computationally expensive due to the required orthogonalization processes. In this paper, we introduce a novel technique to estimate the Log-determinant of a matrix using Léja points, where the implementation is only based on matrix multiplications and a rough estimation of eigenvalue bounds of the matrix. By coupling Léja points interpolation with a randomized algorithm called Hutch++, we achieve substantial reductions in computational complexity while preserving significant accuracy compared to the stochastic Lanczos quadrature. We establish the approximation errors of the matrix function together with multiplicative error bounds for the approximations obtained by this method. The effectiveness and scalability of the proposed method on both large sparse synthetic matrices (maximum likelihood in Gaussian Markov Random fields) and large-scale real-world matrices are confirmed through numerical experiments.

In this paper, we firstly establish weighted heat kernel comparison theorems for the weighted heat equation on complete manifolds with radial curvatures bounded, and then by mainly using this conclusion, we can obtain two eigenvalue comparison theorems for the first Dirichlet eigenvalue of the Witten-Laplacian as applications in spectral geometry.

Linear regression estimators are known to be sensitive to outliers, and one alternative to obtain a robust and efficient estimator of the regression parameter is to model the error with Student's $t$ distribution. In this article, we compare estimators of the degrees of freedom parameter in the $t$ distribution using frequentist and Bayesian methods, and then study properties of the corresponding estimated regression coefficient. We also include the comparison with some recommended approaches in the literature, including fixing the degrees of freedom and robust regression using the Huber loss. Our extensive simulations on both synthetic and real data demonstrate that estimating the degrees of freedom via the adjusted profile log-likelihood approach yields regression coefficient estimators with high accuracy, performing comparably to the maximum likelihood estimators where the degrees of freedom are fixed at their true values. These findings provide a detailed synthesis of $t$-based robust regression and underscore a key insight: the proper calibration of the degrees of freedom is as crucial as the choice of the robust distribution itself for achieving optimal performance. The {\tt R} package that implements our method is available at https://github.com/amanda-ng518/RobustTRegression.

Tidal interactions in close binaries play a key role in the long-term rotational and orbital evolution. The distributions of circularization across open clusters (OCs) place strong observational constraints on tidal dissipation in binaries. However, direct observational constraints on synchronization among binaries in OCs remain limited. For the 125 Myr OC Pleiades, this work combines cluster membership from Gaia Data Release 3, rotation periods from the K2 mission, and orbital solutions of the binary population from a long-term spectroscopic survey, to investigate the degree of tidal synchronization in each binary by comparing the pseudo-synchronization period to the rotation period of the primary stars. Among 42 binaries with reliable orbital periods Porb and rotation periods, we identify seven tidally synchronized systems with Porb < 8.6 days, including one early-type system and six late-type systems. For binaries with longer Porb, primaries generally are super-synchronized, and most systems are eccentric. We find a synchronization transition near Porb ~ 8.6-14 days, comparable to the known circularization period (Porb ~ 7.2 days) in the Pleiades, which suggests similar critical period scales for synchronization and circularization in this coeval population. Synchronization depends much more strongly on mass ratio than on primary mass. Most synchronized systems in Pleiades have high mass ratios and are likely to evolve into double white dwarf systems. Tides likely impose strong rotational braking on close early-type binaries, while their influence on late-type close binaries is weaker, and their spins largely follow the single-star sequence.

Current studies of stellar dynamos primarily focus on spherical stars, leaving their behavior in distorted stars largely unexplored. We utilize stars of varying distortions to examine the relation between stellar cycle periods ($P_{\rm cyc}$) and rotational periods ($P_{\rm rot}$), which are closely linked to dynamo processes. By analyzing a sample of tidally distorted M dwarfs in cataclysmic variables, we identify an anti-correlation between $P_{\rm cyc}$ and $P_{\rm rot}$, in contrast to the lack of such a relation in single M dwarfs. This means that stars with greater deformation have longer cycle periods, suggesting variations in dynamo behavior under non-spherical geometries. Our numerical simulations further reveal that, the thermal convection weakens in highly distorted stars, and subsequently, the differential rotation is also reduced. These effects may lengthen the conversion timescale between poloidal and toroidal magnetic fields, potentially explaining the newly discovered $P_{\rm cyc}$-$P_{\rm rot}$ relation in cataclysmic variables.

In this work we revisit the geometric approach to chaos in Hamiltonian dynamics, by means of the Jacobi-Levi-Civita equation (JLCE). We inspect numerically two low-dimensional dynamical systems; show that, along chaotic orbits, the exponential divergence between nearby trajectories quantified by the JLCE does not unfold in a continuous manner, rather is closer to a multiplicative discrete process: in correspondence of each turning point, where the trajectory bounces away from the boundary of the energetically allowed region, the relative separation increases sharply and abruptly. We highlight through analytical and numerical arguments that the chaotic rather than regular nature of the trajectory is determined by the details of the scattering with the boundary, and interpret these results in terms of parametric resonance theory, and specifically the Mathieu equation.

An applied magnetic field affects a superconductor in two ways -- by promoting pairing fluctuations, and by inducing topological defects called vortices that carry quantized magnetic flux. A quantitative characterization of the resultant field-induced superconducting dynamics with spatio-temporal resolution remains challenging, particularly in two-dimensional materials. In this work, we analyze magnetic noise measured by the depolarization rate of a proximate single spin qubit as a non-invasive probe of such dynamical fluctuations. We demonstrate that the temperature dependence of the magnetic noise spectrum near $T_c$ deviates from predictions based on quasiparticle excitations due to critical superconducting fluctuations, which in turn are enhanced by a weak applied field. By analyzing the magnetic noise due to vortex dynamics, we further show that noise spectroscopy is not only able to distinguish between different vortex phases, but also extract key physical quantities of interest, such as oscillation frequencies of pinned vortices, phonon dispersion of vortex lattices and vortex diffusivity in a vortex liquid. Complementing recent work on noise magnetometry of quasiparticle excitations and Berezinskii-Kosterlitz-Thouless transitions in two-dimensional superconductors without an applied field, our work highlights the ability of noise spectroscopy to reveal a wealth of superconducting dynamical phenomena in an applied field.

Longitudinal electroweak gauge bosons are the most technically delicate ingredient of electroweak parton showers: in the broken Standard Model, the gauge component of a longitudinal polarisation does not cancel diagram by diagram, but is related by Ward identities to amplitudes with an insertion of the associated would-be Goldstone field. Building on the default Herwig 7 treatment based on a subtraction-defined longitudinal current, we construct a gauge-invariant scheme in which the subtraction remainder is retained as a well-defined contribution and completed by a Ward-identity-fixed Goldstone-matching term. We derive helicity-resolved building blocks and compact quasi-collinear splitting kernels for $q\to q'V$ and $V\to V'V''$ branchings, and implement the scheme in Herwig 7 as a switchable alternative to the default basis. The completion leaves the transverse sector unchanged and modifies only longitudinal entries through controlled symmetry-breaking terms, including Yukawa-sensitive contributions in massive-fermion channels. In shower-level studies, we find that the default and gauge-invariant prescriptions coincide at high evolution scales, while differing at lower scales precisely in channels where symmetry breaking is active. Exclusive single-emission observables can moreover display non-monotonic scheme dependence once Sudakov suppression and kinematic constraints are accounted for. A first multi-emission LHC-like study confirms numerical stability and yields controlled, interpretable shifts in observables that probe propagating electroweak shower currents, whereas quantities dominated by promptly contracted, near on-shell vector-boson matrix elements remain largely unchanged.

We investigate whether the long photometric cycles observed in double-periodic variables (DPVs) can arise from nodal precession of a tilted accretion disk driven by the tidal torque of the companion. Within a simple analytical framework, we derive testable relations linking the long-to-orbital period ratio to the binary mass ratio, the normalized disk size, and the disk tilt angle $β$, which itself can be inferred from the long-cycle amplitude, orbital inclination $i$, and disk luminosity fraction. The model naturally reproduces the two observed long-cycle light-curve morphologies -- sinusoidal and double-hump -- distinguished by the geometric criterion $i+β\le 90^\circ$ versus $i+β>90^\circ$. Applying these relations to a sample of DPVs, we find that the inferred disk sizes are physically reasonable and consistent with independent light-curve modeling for a non-negligible subset of systems. Our results show that tidal nodal precession represents a viable and potentially important contributor to the long-period variability of DPVs and provide a quantitative framework for future observational and theoretical studies.

As the aging population faces a chronic care deficit, domestic care is increasingly recast as spectral governance. This paper presents a design fiction set in 2036, where the home is governed by Neural-Wave, a camera-free mmWave sensing platform that infers well-being from involuntary micro-motions. Through a set of scenarios, we illustrate how such empathic systems displace autonomy, forcing residents to perform legibility to regain basic freedoms. Our primary contribution is a diegetic artifact: The Neural-Wave Quick Escape Manual. Styled as an illicit guide for the elderly, it details adversarial tactics: structured around protocols to Comply, Degrade, and Refuse, that exploit signal processing vulnerabilities to reclaim domestic privacy. Through this artifact, we argue that in the era of empathic AIoT, privacy requires more than policy opt-outs; it demands adversarial literacy:the capacity to meaningfully obfuscate one's own data traces against an infrastructural jailer that calls itself care.

We show that in situ ion irradiation transient grating spectroscopy (I3TGS) can be used to monitor the real-time evolution of vacancy concentration generated by self-ion radiation damage in Cu-based alloys. Surface acoustic wave (SAW) frequencies are shown, using a combination of theory and experiment, to reveal vacancy concentrations and their kinetics in real-time. These results are shown to agree with corresponding kinetic Monte Carlo simulations at similar temperatures and dose rates. These results suggest the utility of TGS as a non-contact, non-destructive tool for real-time defect monitoring.

Solar intensity power spectra are usually characterised by coloured noise, with the spectral energy following a segmented power-law function of frequency, $S(f)\propto f^{-α}$, over different frequency ranges. Typically, the power-law index exceeds 1 in the low-frequency part ($α_\mathrm{lf}$) and is around 0 at high frequencies ($α_\mathrm{hf}$). This work investigates the spatial and temporal evolution of the power-law indices of coronal EUV intensity power spectra in flare-hosting active regions. The spatial distribution of the power-law index in the low-frequency domain ($α_\mathrm{lf}$) closely mirrors EUV intensity images, indicating that $α_\mathrm{lf}$ can reveal the dynamics of coronal plasma structures. Temporally, $α_\mathrm{lf}$ remains stable in quiescent active regions, but it exhibits significant variability before the flare onset. Motivated by this behaviour, we analysed 14 flare events, quantifying the temporal variation of the indices $α_\mathrm{lf}$ and $α_\mathrm{hf}$ as potential flare precursors. In all flare events considered, notable deviations of $α_\mathrm{lf}$ beyond a defined threshold consistently occurred at the flare site within a few minutes before the flare. In some cases, the change in the value of $α_\mathrm{lf} - α_\mathrm{hf}$ was detected within 30--90\, minutes before the flare. This proof-of-concept study suggests that the temporal variation of the power-law indices in coronal EUV intensity power spectra could potentially serve as short-term precursors of solar flares, which needs to be validated on a larger flare sample.

We study the fair allocation of indivisible items under relevance constraints, where each agent has a set of relevant items and can only receive items that are relevant to them. While the relevance constraint has been studied in recent years, existing work has largely focused on envy-freeness. Our work extends this study to other key fairness criteria -- such as proportionality, equitability, and their relaxations -- in settings where the items may be goods, chores, or a mixture of both. We complement the literature by presenting a picture of the existence and computational complexity of the considered criteria.

Motivated by techniques developed in recent progress on lower bounds for sublinear time algorithms (Behnezhad, Roghani and Rubinstein, STOC 2023, FOCS 2023, and STOC 2024) we introduce and study a new class of randomized algorithmic processes that we call Markov Chains with Rewinding. In this setting, an algorithm interacts with a (partially observable) Markovian random evolution by strategically rewinding the Markov chain to previous states. Depending on the application, this may lead the evolution to desired states faster, or allow the agent to efficiently learn or test properties of the underlying Markov chain that may be infeasible or inefficient with passive observation. We study the task of identifying the initial state in a given partially observable Markov chain. Analysis of this question in specific Markov chains is the central ingredient in the above cited works and we aim to systematize the analysis in our work. Our first result is that any pair of states distinguishable with any rewinding strategy can also be distinguished with a non-adaptive rewinding strategy (one whose rewinding choices are determined before observing any outcomes of the chain). Therefore, while rewinding strategies can be shown to be strictly more powerful than passive strategies (those that do not rewind back to previous states), adaptivity does not give additional power to a rewinding strategy in the absence of efficiency considerations. The difference becomes apparent however when we introduce a natural efficiency measure, namely the query complexity (i.e., the number of observations they need to identify distinguishable states). Our second main contribution is to quantify this efficiency gap. We present a non-adaptive rewinding strategy whose query complexity is within a polynomial of that of the optimal (adaptive) strategy, and show that such a polynomial loss is necessary in general.

In Neyman's original formulation, a 1-alpha confidence interval procedure is justified by its long-run coverage properties, and a single realized interval is to be described only by the slogan that it either covers the parameter or it does not. On this view, post-data probability statements about the coverage of an individual interval are taken to be conceptually out of bounds. In this paper, I present two kinds of arguments against treating that "either-or" reading as the only legitimate interpretation of confidence. The first is informal, via a set of thought experiments in which the same joint probability model is used to compute both forward-looking and backward-looking probabilities for occurred-but-unobserved events. The second is more formal, recasting the standard confidence-interval construction in terms of infinite sequences of trials and their associated 0/1 coverage indicators. In that representation, the design-level coverage probability 1-alpha and the degenerate conditional probabilities given the full data appear simply as different conditioning levels of the same model. I argue that a strict behavioristic reading that privileges only the latter is in tension with the very mathematical machinery used to define long-run error rates. I then sketch an alternative view of confidence as a predictive probability (or forecast) about the coverage indicator, together with a simple normative rule for when intermediate probabilities for single coverage events should be allowed. Keywords: confidence intervals; coverage probability; frequentist inference; single-case probability; predictive probability; Neyman. Disclaimer: The findings and conclusions in this report are those of the author and do not necessarily represent the official position of the Centers for Disease Control and Prevention.

The main focus of the present paper is the following inequality $\left( \sqrt{bc}-a\right) m_a+ \left(\sqrt{ac}-b\right)m_b+\left(\sqrt{ab}-c\right)m_c \geq 0,$ where $a,b,c$ are the sides of a non-degenerate triangle and $m_a,m_b,m_c,$ the respective medians; which was conjectured to be true but had not been proved. We provide a proof. We also show that analogous inequality is true when the medians are replaced by the altitudes or the internal angle bisectors. Finally, we conclude with an open problem regarding the Cevians which would satisfy such inequality.

Combinatorial optimization problems are central to science and engineering and specialized hardware from quantum annealers to classical Ising machines are being actively developed to address them. These systems typically sample from a fixed energy landscape defined by the problem Hamiltonian encoding the discrete optimization problem. The recently introduced Probabilistic Approximate Optimization Algorithm (PAOA) takes a different approach: it treats the optimization landscape itself as variational, iteratively learning circuit parameters from samples. Here, we demonstrate PAOA on a 64$\times$64 perimeter-gated single-photon avalanche diode (pgSPAD) array fabricated in 0.35 $μ$m CMOS, the first realization of the algorithm using intrinsically stochastic nanodevices. Each p-bit exhibits a device-specific, asymmetric (Gompertz-type) activation function due to dark-count variability. Rather than calibrating devices to enforce a uniform symmetric (logistic/tanh) activation, PAOA learns around device variations, absorbing residual activation and other mismatches into the variational parameters. On canonical 26-spin Sherrington-Kirkpatrick instances, PAOA achieves high approximation ratios with $2p$ parameters ($p$ up to 17 layers), and pgSPAD-based inference closely tracks CPU simulations. These results show that variational learning can accommodate the non-idealities inherent to nanoscale devices, suggesting a practical path toward larger-scale, CMOS-compatible probabilistic computers.

We generalize the topological description of the $\mathbb{N}$-graded Proj construction to the multigraded Proj construction for factorially graded rings that are graded by finitely generated abelian groups $D$. However, there is one big structural difference: While the classical description is given by the space of homogeneous prime ideals not containing the irrelevant ideal, we characterize the multigraded Proj setting using $D$-prime ideals, i.e.\ ideals that have the prime property, but only for homogeneous factorizations. In particular, we establish a multigraded version of the Nullstellensatz. Additionally, we present algebraic conditions for separability in terms of factorially graded rings, and observe that Proj$^D(S)$ is not separated in many cases. Finally, building on Mayeux-Riche's definition of Serre twists, we give a criterion for their freeness.

Defects in cuprous oxide (Cu$_2$O) strongly influence its performance in applications ranging from photovoltaics to emerging quantum technologies based on Rydberg excitons where microscopic crystal purity is essential. Photoluminescence (PL) spectroscopy is widely used as a diagnostic of material quality yet the origins of the sub-band-gap PL lines remain controversial despite decades of study. Using density functional theory we systematically evaluate native point defects in Cu$_2$O and identify which produce robust electronic states within the band gap. By combining supercell-size convergence criteria and cross-functional consistency checks we show that the widely accepted assignments of the 1.35\,eV, 1.5\,eV, and 1.7\,eV PL lines to copper and oxygen vacancies are unsupported. Instead we find that oxygen interstitials, copper interstitials and one form of split copper vacancy are the only native defects that consistently introduce true in-gap states. These results provide a revised framework for interpreting PL spectra and offer a more reliable basis for diagnosing crystal quality in Cu$_2$O-based quantum-device platforms.

Integrated sensing and communication (ISAC) can substantially improve spectral, hardware, and energy efficiency by unifying radar sensing and data communications. In wideband and scattering-rich environments, clutter often dominates weak target reflections and becomes a fundamental bottleneck for reliable sensing. Practical ISAC clutter includes "cold" clutter arising from environmental backscatter of the probing waveform, and "hot" clutter induced by external interference and reflections from the environment whose statistics can vary rapidly over time. In this article, we develop a unified wideband multiple-input multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) signal model that captures both clutter types across the space, time, and frequency domains. Building on this model, we review clutter characterization at multiple levels, including amplitude statistics, robust spherically invariant random vector (SIRV) modeling, and structured covariance representations suitable for limited-snapshot regimes. We then summarize receiver-side suppression methods in the temporal and spatial domains, together with extensions to space-time adaptive processing (STAP) and space-frequency-time adaptive processing (SFTAP), and we provide guidance on selecting techniques under different waveform and interference conditions. To move beyond reactive suppression, we discuss clutter-aware transceiver co-design that couples beamforming and waveform optimization with practical communication quality-of-service (QoS) constraints to enable proactive clutter avoidance. We conclude with open challenges and research directions toward environment-adaptive and clutter-resilient ISAC for next-generation networks.

Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we present the following examples: a graph with an edge that is crossed in every optimal drawing of the graph, but the edge is not in any Kuratowski subgraph of the graph; a graph with an edge that is in every Kuratowski subgraph but is not crossed in any optimal drawing of the graph; and a graph with a crossing-critical edge that is not present in any Kuratowski subgraph and is not crossed in any optimal drawing of the graph. Fáry's theorem implies that the Kuratowski subgraphs are the only obstructions to a graph having a crossing-free drawing with all edges drawn as straight lines. The three example graphs given also hold if we restrict drawings to only have straight line edges, and thus also apply to the rectilinear crossing number.

It is intuitively expected, and supported by earlier studies, that many-body quantum chaos is suppressed, or even destroyed, by dissipative effects induced by continuous monitoring. We show here that this is not always the case. For this purpose, we study the quenched dynamics of a continuously monitored Sachdev-Ye-Kitaev (SYK) model, described by the Lindblad formalism, coupled to a thermal environment modeled by another SYK maintained at constant temperature. We find that the combined effect of monitoring and the thermal bath drives the system toward a non-thermal steady state independently of the initial conditions. The corresponding retarded Green's function exhibits two stages of exponential decay, with rates that depend non-monotonously on the thermal bath coupling and the monitoring strength. In the limit of weak coupling, the late time decay of the Green's function, computed analytically, is closely related to that of the thermal bath. Strikingly, we identify a range of parameters in which continuous monitoring, despite being a source of decoherence, induces or enhances quantum chaotic dynamics suppressed by the thermal bath. For instance, in the limit of weak coupling to the thermal bath, the Lyapunov exponent increases sharply when monitoring is turned on. For intermediate values of the thermal bath coupling, the Lyapunov exponent exhibits re-entrant behavior: it vanishes at zero or sufficiently weak monitoring strength, and becomes positive again as the monitoring strength is increased. Our results offer intriguing insights on the mechanisms leading to quantum scrambling which paves the way to its experimental control and consequently to a performance enhancement of quantum information devices.