Non-line-of-sight (NLOS) sensing has the potential to enable use cases like intrusion detection in occluded areas, increasing the value provided by Integrated Sensing and Commu- nications (ISAC) in future 6G cellular networks. In this paper, we present a reliable NLOS intrusion detection system based on a millimeter-wave ISAC proof-of-concept. By leveraging reflections off a large surface, the proposed system addresses the challenge of detecting moving targets in cluttered indoor industrial scenarios where the direct line-of-sight is obstructed. A signal processing pipeline including a tracking stage, comparing a Kalman Filter (KF)- and a probability hypothesis density (PHD) filter-based approach, is applied to detect targets and track movements in NLOS. Experimental validation conducted in the ARENA2036 industrial research campus demonstrates that our system can reliably detect target presence in NLOS while avoiding false alarms. Tests with synthetically generated false peaks further demonstrate the robustness of our system to false alarms. Overall, the results underline the potential of NLOS ISAC as a promising technology for enabling intrusion detection and monitoring use cases.

To clarify possible deviations in $b\to cτ\overlineν$ processes, the $b\to c$ semileptonic sum rule provides a valuable tool. This relation, derived based on heavy quark symmetry (HQS), offers a powerful consistency check among experimental results. In this work, we extend the previously proposed sum rule for $\{B\to D^{(*)} l\overlineν,\,Λ_b\to Λ_c l\overlineν\}$ to include $\{B_s\to D_s^{(*)} l\overlineν,\,Ξ_b\to Ξ_c l\overlineν\}$, thereby enabling more useful cross-checks. Although the relation is supported by HQS and SU(3) flavor symmetry, both symmetries are broken in reality, and the size of the violation needs to be quantified to assess the validity of the sum rule. While the violation is expected to be moderate based on chiral perturbation theory, we perform a numerical evaluation and compare it with future experimental sensitivities. We find that the violation remains smaller than the expected experimental uncertainty. Therefore another new physics agnostic and predictive sum rules are constructed to check the consistency.

High penetration of renewables (RES) and the retirement of thermal units aggravate flexibility scarcity in power systems. Hydrogen-based low-carbon steel production systems possess substantial demand response (DR) potential. This paper proposes a process-aware DR evaluation framework for hydrogen-integrated zero-carbon steel plants coupled with methanol production (H2-DRI-EAF-MeOH). First, a novel H2-DRI-EAF-MeOH architecture is introduced to eliminate residual emissions via methanol synthesis. Integrated energy-material flows are formulated to reflect coupling interactions governing DR potential. Second, to capture electric arc furnace (EAF) operational constraints while preserving tractability, an operating feasible region model is developed and validated using field data from a pure hydrogen direct reduced iron and EAF plant, yielding a 4.1% average relative error. Third, a process-aware DR potential evaluation model is formulated, incorporating a nonlinear asymmetric penalty and an adaptive rolling mechanism to reflect operators' aversion to process deviations and avoid myopic scheduling. Finally, dual-side evaluation metrics are established to quantify grid-side delivered DR capacity and ramping risks, making load-side unit-level regulation behaviors observable. Case studies show the proposed framework achieves an average effective delivered DR capacity of 178.3 MW, improves RES-load matching from 0.257 to 0.587, and reduces costs by 15.68% compared to the baseline. Furthermore, the exponential asymmetric penalty mitigates extreme tail risks of process deviations. Ultimately, this work provides a theoretical foundation for leveraging RES-steel-chemical synergies to mitigate flexibility scarcity.

Particles whose shapes couple translation to rotation display a rich array of behaviors as they sediment at low Reynolds number. We introduce a unifying perspective in which the possible dynamical regimes and bifurcations between them can be understood. We use experimental measurements of helical ribbons, with controlled center of mass offsets, to identify the key bifurcation from complex dynamics to a single attracting state as the magnitude of the offset increases. The sedimentation dynamics are very sensitive to small center of mass offsets, with the bifurcation occurring for offsets less than one percent of the particle length. Using mobility tensors obtained from immersed boundary method simulations, we simulate helical particle sedimentation and identify an alignment bifurcation surface, defined in the three dimensional space of center of mass offsets, that separates simple from complex sedimentation dynamics. Inside this surface we find limit cycles which emerge through Hopf and homoclinic bifurcations. Cocentered particles with coincident centers of force and mobility provide a reference case at the center of the bifurcation surface. We show how the geometric and dynamical symmetries of sedimenting cocentered particles are broken as the center of force offset moves away from the cocentered case. Three parity time-reversal (PT) symmetries exist for all cocentered particles under reflections normal to the eigenvectors of its translation-rotation coupling tensor. When a center of force offset preserves at least one of these PT symmetries, then there are closed orbits for particles inside the alignment bifurcation surface.

In this paper, we first study a class of discrete $p$-Laplacian systems with logarithmic coupling on locally finite graphs. The system is specifically designed to capture the variational interplay between nonlinear diffusion and logarithmic saturation, and takes the form $$ \begin{cases} -Δ_p u + a(x)|u|^{p-2}u=\frac{p-2}{p}|u|^{p-4}u v^2\log v^2 +\frac{2}{p}|v|^{p-2}u\log u^2+\frac{2}{p} |v|^{p-2}u,\qquad in\quad V, -Δ_p v + b(x)|v|^{p-2}v=\frac{p-2}{p}|v|^{p-4}v u^2\log u^2 +\frac{2}{p}|u|^{p-2}v\log v^2+\frac{2}{p} |u|^{p-2}v,\qquad in\quad V, \end{cases} $$ on locally finite graphs $G=(V,E)$ with $p>4$. The logarithmic coupling terms would possibly render the energy functional not well-defined on the natural Sobolev space--a fundamental obstacle that does not rise in scalar equations. We establish existence of ground states under two distinct hypotheses: via the Nehari manifold when the functional lacks regularity, and via the mountain pass theorem otherwise. The core novelty is an original exponent calibration technique, specifically devised to resolve the non-separable logarithmic singularities. Finally, we establish convergence results by analyzing the concentration behavior of the ground state solutions.

We demonstrate that the failure of $L^1$ regularity in Calderón-Zygmund theory is a universal phenomenon: every non-constant holomorphic function in $\C^n$ generates a counterexample to the Poisson equation. In order to achieve this goal, we shall establish sharp level-set estimates that link harmonic analysis to the geometry of complex structure through Hironaka's resolution of singularities and the Łojasiewicz gradient inequality.

Rotating radio transients (RRATs) represent a significant but poorly understood component of the Galactic neutron star population, characterized by sporadic emission first detectable only through single-pulse searches. We present the RRATalog, an up-to-date catalogue of 335 RRATs, and utilize a uniform sample of RRATs discovered in four Parkes telescope surveys to model their Galactic population. Accounting in detail for observational selection effects, we find a radial density profile similar to pulsars, but identify a significantly steeper luminosity function (power-law index $α\simeq -1.3$) than previously assumed. For sources beaming towards Earth, we estimate $34000 \pm 1600$ potentially observable RRATs above a peak luminosity of 30 mJy kpc$^2$. At these high luminosities, the RRAT population is comparable in size to that of canonical pulsars. Consistent with the observed distribution, the underlying period distribution is significantly shifted toward longer periods compared to canonical pulsars, suggesting RRATs represent a more evolved population. We find evidence for a turnover in the luminosity function below 30 mJy kpc$^2$, and predict that the total number of potentially observable RRATs is $\lesssim 70,000$. Applying the Tauris \& Manchester beaming model, we estimate the total Galactic RRAT population to be $\lesssim 400,000$. The implied birth rate of $\lesssim 1.4$ RRATs per century is consistent with the Galactic core-collapse supernova rate, suggesting RRATs can be reconciled with known progenitor rates without requiring a separate evolutionary origin. We provide predictions for RRAT discoveries in ongoing and future surveys.

"Hidden" phases, generated using ultrafast laser pulses (few hundred femtoseconds), with properties distinct from thermodynamic equilibrium, are appealing for technologies because they can be long-lived, with lifetimes of hours or weeks, and reversible with temperature sweeping or extra pulses. In this regard, La$_{2/3}$Ca$_{1/3}$MnO$_3$ (LCMO) stands out due to its tunability through epitaxial strain, which can drive the bulk ferromagnetic metal (FMM) into an antiferromagnetic insulator (AFI), and its susceptibility to photo-induced transitions. Indeed, AFI LCMO displays a long-lived photo-induced transition into a putative 'hidden' phase whose exact nature and excitations are still largely unknown. Here, we combine ultrafast photo-excitation in the near infrared with in situ transport, x-ray absorption (XAS), and Resonant Inelastic X-ray Scattering (RIXS) to investigate the excitations (polarons, phonons, and orbital) of the photo-excited phase of LCMO and contrast them with the thermodynamic phases achieved through strain and temperature. In the thermodynamic regime, we establish the correlation between polarons and transport, placing them in the 'strong coupling' regime of the Holstein model. Upon photo-excitation of LCMO-AFI, we uncover a long-lived phase characterized by the softening of the polaron excitations, the partial suppression of the Jahn-Teller distortion, and nearly unchanged phonons, showing the emergence of a photo-excited state absent in the equilibrium phase diagram. Finally, by varying temperature, epitaxial strain, and photo-excitation fluence, we construct a polaron phase diagram and identify the key spectroscopic signatures of each phase. Our laser-RIXS approach establishes a versatile platform for exploring photo-induced 'hidden' phases in quantum materials in non-stroboscopic conditions.

A basic question in the study of measure-once quantum finite automata is whether two distinct input words can be separated with certainty. The exact separation problem reduces to a trace-vanishing question in \(SU(2)\). The main difficulty lies in the genuinely nonabelian regime, where \(u\) and \(v\) have the same abelianization. This paper develops a slice-driven framework that converts algebraic invariants of the word -- prefix statistics, metabelian polynomials, and slope specializations -- into explicit low-dimensional families in \(SU(2)^2\) on which the trace-vanishing question can be analyzed effectively. A quadratic trace-deficit identity on a principal one-parameter family provides the main algebraic-to-geometric bridge. Building on this framework, the paper establishes three core certified slice criteria: a dihedral criterion, equivalently readable through a signed \(a\)-count; a quaternionic criterion; and a local one-row criterion. Together with a supplementary interior-point test and a binary-dihedral slice, these results sharply reduce the unresolved portion of the problem to a residual super-degenerate class, while also clarifying the limitations of certification strategies based only on finitely many finite-subgroup evaluations.

This paper proposes a new notion of distributional Input-to-State Stability (dISS) for dynamic systems evolving in probability spaces over a domain. Unlike other norm-based ISS concepts, we rely on the Wasserstein metric, which captures more precisely the effects of the disturbances on atomic and non-atomic measures. We show how dISS unifies both ISS and Noise to State Stability (NSS) over compact domains for particle dynamics, while extending the classical notions to sets of probability distributions. We then apply the dISS framework to study the robustness of various Wasserstein gradient flows with respect to perturbations. In particular, we establish dISS for gradient flows defined by a class of $l$-smooth and $λ$-convex functionals subject to bounded disturbances, such as those induced by entropy in optimal transport. Further, we study the dISS robustness of the large-scale algorithms when using Kernel and sample-based approximations. This results into a characterization of the error incurred when using a finite number of agents, which can guide the selection of the swarm size to achieve a mean-field objective with prescribed accuracy and stability guarantees.

Context: LHS 3844 b (TOI-136 b) is a ultra short-period, Earth-size exoplanet detected by TESS. It is one of the most favourable object for atmospheric characterisation and the study of its surface with the James Webb Space Telescope. However, the dynamical mass of this planet has not been measured yet. Aims: We aim to determine the mass of LHS 3844 b using high-precision radial velocity (RV) measurements and assess the robustness of the inferred signal across different noise and orbital modelling assumptions. Methods: We analyse 25 ESPRESSO RV observations within a fully Bayesian framework. We explore 15 competing RV models that differ in their treatment of correlated stellar variability (through different Gaussian Process kernels) and long-term drifts. Marginal likelihoods are computed for all models and used for Bayesian model comparison and evidence-weighted parameter estimation. Results: The RV planetary signal is robustly detected across all models, and the inferred semi-amplitude remains stable under all tested noise and drift prescriptions. From the evidence-weighted posterior samples we derive a planetary mass of $2.27 \pm 0.23$ M$_\oplus$ and a bulk density of $5.67 \pm 0.65$ gcm$^{-3}$, consistent with a predominantly rocky composition. Model comparison favours GP kernels including periodic or quasi-periodic components associated with stellar rotation and disfavors models with additional long-term drifts. Using interior-structure inference, we find that the core mass fraction is comparable to (or slightly smaller than) Earth's and only trace amounts of water are permitted, supporting a dry, terrestrial interior. We also investigate a tentative additional signal near $\sim 6.9$ days, but Bayesian model comparison does not provide conclusive support for its planetary interpretation.

Materialized views are a core construct in database systems, used to accelerate analytical queries and optimize batch pipelines for extract-transform-load (ETL) workflows. Maintaining view consistency as underlying data evolves is a fundamental challenge, especially in high-throughput and real-time settings. Incremental view maintenance (IVM) has been studied for decades and continues to attract significant investment from major database vendors. However, most industrial systems either offer limited SQL-operator coverage or require users to hand-tune refresh strategies. This paper presents Enzyme, an IVM engine developed at Databricks to power Spark Declarative Pipelines. It provides a built-in, end-to-end approach to incremental pipelines, utilizing materialized views as first-class building blocks. By automating refresh planning, Enzyme reduces total cost of ownership and lets users focus on business logic rather than MV mechanics. Validation across thousands of large-scale production pipelines spanning diverse application domains has demonstrated substantial computational efficiency gains, yielding a cumulative daily compute reduction of billions of CPU seconds. Built atop Apache Spark primitives, Enzyme adds a cost-based optimization layer that selects refresh strategies for collections of materialized views organized into pipelines. Enzyme's modular architecture is designed to generalize across data sources and query engines. We present key design decisions for incremental refresh planning and execution, including optimizations that exploit batching opportunities across materialized view sources. Experimental results on standard benchmarks demonstrate significant performance improvements at scale.

In this paper we continue the program to develop the algebraic foundations of tropical (algebraic) geometry. We give strong characterizations of prime congruences containing a given congruence on a toric semiring. We give four applications of this result. (1) We prove an analogue of the strong Nullstellensatz for congruences with finite tropical basis. This extends the existing result of Joó-Mincheva to cases, such as the bend congruence of a tropical(ized) ideal, where the congruence is not finitely generated. (2) We show that, if $I$ is the ideal of an affine variety not contained in the coordinate hyperplanes, then $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is cancellative. This result has applications to the integral closure (as per Tolliver) of $\mathbb{T}[x_1, \dots, x_n]/\operatorname{Bend}(\operatorname{trop} I)$ which we explore in a forthcoming paper. (3) We show that $\mathbb{T}[x_1, \dots, x_n]/\sqrt{\operatorname{Bend}(\operatorname{trop} I)}$ is the tropical function semiring on $\operatorname{trop} V(I)$, which creates a bridge between the algebraic approach to non-embedded tropicalization in the work of J. Song and the bend congruence approach of Giansiracusa-Giansiracusa and Maclagan-Rincón. (4) As a consequence of one of our lemmas, we describe the closure of a polyhedron in a tropical toric variety even when the polyhedron is not compatible with the fan defining the tropical toric variety.

We study Knizhnik-Zamolodchikov (KZ) connection in the presence of irregular singularities, that is, poles of higher order. We consider both the case of a universal connection and the case when it is associated with a specific simple Lie algebra, such as $\mathfrak{su}(2)$. We give some general results about the monodromies of such flat connections in the configuration spaces of points, and provide explicit examples of topological invariants of links (more generally, tangles) realized by the monodromy.

Near-field beamfocusing with extremely large aperture arrays can effectively enhance physical layer security. Nevertheless, even small estimation errors of the eavesdropper's location may cause a pronounced focal shift, resulting in a severe degradation of the secrecy rate. In this letter, we propose a physics-informed robust beamforming strategy that leverages the electromagnetic (EM) caustic effect for near-field physical layer security provisioning, which can be implemented via phase shifts only. Specifically, we partition the transmit array into caustic and focusing subarrays to simultaneously bypass the potential eavesdropping region and illuminate the legitimate user, thereby significantly improving the robustness against the localization error of eavesdroppers. Moreover, by leveraging the connection between the phase gradient and the EM wave departing angle, we derive the corresponding piece-wise closed-form array phase profile for the subarrays. Simulation results demonstrate that the proposed scheme achieves up to an 80% reduction of the worst-case eavesdropping rate for a localization error of 0.25 m, highlighting its superiority for providing robust and secure communication.

This paper explores a mechanism for mitigating sovereign risk in emerging markets without risks mutualization. The mechanism involves pooling diversified portfolios of sovereign bonds and issuing them in tranches, with the senior tranche offering low-risk payoffs protected by the subordination of the junior tranches. We argue that this mechanism is feasible for emerging markets. The senior bonds issued by the securitization vehicle attain the properties of a safe asset. The risk level of the junior bonds depends on the structure of the underlying sovereign bonds portfolio. Nevertheless, the properties of the synthetic bonds are, arguably, acceptable for the practical application of the proposed mechanism in promoting the development of financial markets in emerging markets and for practical tasks such as intergovernmental aid or lending.

Based on abundant data from Insight-HXMT, we conducted a detailed analysis of type-C quasi-periodic oscillations (QPOs) in the black hole X-ray binary MAXI J1348-630. Type-C QPOs were intensively detected over a broad energy band, with frequencies ranging from 0.24 to 10.3 Hz, and several new evolutionary features were identified. First, although type-C QPOs reappear intermittently, they show a stable characteristic frequency around 7 Hz. This implies a characteristic spatial scale for the QPO emission region, despite large variations in outburst intensity. Second, from the hard state to the hard-intermediate state, type-C QPOs display a harder fractional rms spectrum, with the rms peak shifting toward high energies (>20 keV) and an amplitude exceeding 10 %. This hard rms spectrum favors a high-energy origin for type-C QPOs. The spectral hardening occurs simultaneously with the weakening of the compact jet, suggesting a physical connection between these two processes. Finally, we observed hysteresis in the QPO frequency-flux relation, with the hysteresis loop evolving in opposite directions between the main and mini-outbursts. This offers a new perspective on the physical differences between the two outburst types, which may arise from variations in initial magnetic field conditions.

We develop a practical framework for livelock analysis in self-disabling unidirectional ring protocols. Klinkhamer and Ebnenasir established that livelock detection for parameterized rings is $Σ^0_1$-complete and livelock-freedom verification is $Π^0_1$-complete, via reduction from the periodic domino problem. We observe that lifting the analysis from the transition space to an \emph{equivariant product space} -- the space of transition-witness pairs -- reveals structure that supports effective verification. We construct a \emph{product transition graph} (at most $|T|^2$ nodes) that captures all livelocks: every livelock maps into this graph as a witness-closed subgraph. The maximal such subgraph $G^*(T)$ is computable in polynomial time ($O(|T|^8)$ worst case) via monotone fixed-point iteration. When $G^*(T) = \emptyset$, the protocol is \emph{provably livelock-free} for all ring sizes -- a sound and complete livelock-freedom verifier. When $G^*(T) \neq \emptyset$, we apply a backtracking search that backward-propagates each simple cycle through $G^*$ until the chain either closes into a torus (confirming a livelock) or dies (no livelock from that cycle). This two-phase algorithm -- polynomial-time pruning followed by finite combinatorial verification -- produces three outcomes: Free, Livelock, or Inconclusive. Across 4{,}349 protocols tested (including an adversarial protocol derived from Klinkhamer and Ebnenasir's tiling construction and Kari's 14-tile aperiodic set converted via their SE gadget), the algorithm is conclusive in every case with zero errors. We further demonstrate that the algorithm extends to non-self-disabling protocols via a protocol transformation. This extends the algorithm's applicability to all parameterized unidirectional ring protocols. Python implementation and usage instructions are at URL: https://github.com/cosmoparadox/mathematical-tools.

We consider capacity (fuzzy measure, non-additive probability) on a compactum as a monotone cooperative normed game. We introduce topological analogues of well known class of exact games and show that these classes form subfunctors of the capacity functor which lie between known subfunctors of convex capacities and balanced capacities. It is natural to consider probability measures as elements of core of such games. We describe exact capacities as envelopes of the convex closed sets of probability measures. Using such representation we prove openness of the functor of exact capacities. We also consider strongly exact capacities and pose the problem of coincidence of these two classes.

Any representational enterprise must omit variation in order to function. NASA still uses Newtonian mechanics, though Einstein superseded Newton, and the standard picture of scientific progress cannot explain how. A description that omitted nothing would be identical to its subject and would explain nothing. This paper argues that omission is not a defect but the central structural feature of any enterprise that builds representations from incomplete information. The key concept is projection: a principled mapping from underlying complexity to a descriptive space that partitions states into equivalence classes, omits within-class variation, and makes patterns visible that would otherwise be lost. Projection is simultaneously revelatory and constitutive: it makes genuine invariants tractably accessible while bringing into being the concepts through which they become expressible. The paper distinguishes vertical cases, in which earlier projections survive as limiting cases of more refined successors with recoverable omission, from horizontal cases, in which omission is constitutive, and invariants are accessible only at the level of the projection that defines them. The framework accounts for persistent pluralism in mature sciences, treats the renormalization group as a systematic implementation of the invariant-tracking criterion, and defends a level-relative realism on which higher-level projections reveal genuine structural features of the world. The deepest claim is an inversion of the standard picture: perspectival structure is not a concession to complexity but the condition for invariant detection. A world rich in invariants cannot be exhausted by a single projection.

Interstellar dust has been detected in situ flowing through the heliosphere. However, our ability to derive the density and size distribution of the interstellar dust in the local interstellar medium from this directly detected dust requires modeling the transport of the grains as they interact with the solar wind magnetic field. The magnetic field in the sector region that contains the heliospheric current sheet has rapid polarity flips which can present an effectively very low averaged field strength to dust grains that have gyroradii tens of au in size. We present new calculations of dust transport through the heliosphere using models that include the sector region to assess the effects on dust transport. We show that the sector region can act as a window allowing even relatively small grains to penetrate deep into the heliosphere. We find the sector region reduces the variation in dust density with the solar cycle (as compared to models without the sector region), with very little concentration or dilution of the dust for grains larger than \sim 0.1$ $μ$m for most of the solar cycle. We still find a substantial concentration of the dust in the ecliptic plane for a focusing overall polarity of the field at solar minimum. These models do not include the time dependence of the magnetic field during transport of grains through the heliosphere. Nevertheless, our results imply that observations of interstellar dust grains, even near Earth, could be fairly accurate in determining their size distribution in the surrounding interstellar medium.

We explore how matrix bootstrap techniques can be used to constrain matrix and tensor models at finite $N$, where $N$ is the dimension of the matrix/tensor, taking a Gaussian model with a quartic interaction as example. For matrix models, we find further evidence that bounds do not depend explicitly on $N$, but rather on properties of multi-trace expectation values. For tensor models, the structure of the Schwinger-Dyson equations allow for bounds that vary as a function of $N$, admitting a broader scan of the parameter space of the theory. In the latter case, we find novel bounds on the two-point function as a function of the quartic coupling of the theory.

In this paper, we construct rotating frames for curves, including plane curves, space curves and curves on surfaces. Hence, the behaviour of an arbitrary moving point on a curve can be seen as the composite of linear motion and rotation. Conversely, it can also be proved that a curve can be determined by the two motions of a moving point on it, namely, linear motion and rotation. Thus, we obtain a new binary mathematical formation mechanism for curves based on the aforementioned two motions. Finally, we apply this rotating frame method to the study of the behaviour of moving points on ellipses.

We establish lower and upper bounds for the identification capacity of discrete-time Gaussian channels subject to inter-symbol interference (ISI), a canonical model in wireless communication. Our analysis accounts for deterministic encoders under peak power constraint. A principal finding is that, even when the number of ISI taps scales sub-linearly with the codeword length $n$ as $\sim n^κ$ with $κ\in [0,1/2),$ the number of messages that can be reliably identified grows super-exponentially in $n$ as $\sim 2^{(n \log n)R},$ where $R$ is the coding rate.

We study the one-dimensional stochastic partial differential equation \begin{equation*} \frac{\partial u}{\partial t}(t,x) = -κ\frac{\partial^4 u}{\partial x^4}(t,x) + ρ\frac{\partial^2 u}{\partial x^2}(t,x) + b(u(t,x)) + σ(u(t,x))\, \dot W(t,x), \end{equation*} posed on a bounded spatial domain, where $u$ is understood in the random field sense and $\dot W(t,x)$ is space-time white noise. Depending on the value of $κ$, this equation includes the nonlinear stochastic heat equation with Dirichlet or Neumann boundary conditions, as well as the linearized stochastic Cahn-Hilliard equation with Neumann boundary conditions. We prove that the supremum of the solution admits a density with respect to Lebesgue measure. Our approach is based on Malliavin calculus, and in particular on the version of the Bouleau-Hirsch criterion for suprema developed by Nualart and Vives. One of the main difficulties lies in the analysis of the argmax set of the solution and in showing that the Malliavin derivative is almost surely nondegenerate on this set. As a byproduct of our arguments, we also establish Hölder continuity properties for the Malliavin derivative of the solution as an $L^2-$valued process in the regimes considered in this work.

Topological methods have emerged as valuable tools for analyzing the structural properties of directed graphs, particularly connectome data in computational neuroscience. This paper investigates the construction of topological spaces from combinatorial data of directed graphs using the polyhedral product functor, with particular emphasis on understanding their homotopy type, which is also of independent interest in topology and combinatorics. Specifically, we compute the homotopy type of the moment-angle complex over the face poset of the complex of injective words. This reveals a tight connection between homotopy and combinatorics: its homotopy type is determined by the $h$-vector of complexes of injective words. We also construct an associated homotopy fibration of polyhedral products associated to ordered simplicial complexes, which in a way generalizes the analogous homotopy fibration for polyhedral products over abstract simplicial complexes.

We delve into the estimation of the functional coefficients and inference for varying coefficient model. Applying Laguerre series, we develop an estimator for the vector of functional coefficients that attains asymptotically optimal convergence rates in the minimax sense. These rates are derived for functional coefficients that belong to Laguerre-Sobolev space. The method is based on approximating the functional coefficients using truncated Laguerre series and choosing empirical Laguerre coefficients that minimize the least squares criterion. In addition, we establish the asymptotic normality of the estimator for the functional coefficients, construct their confidence intervals, and establish point-wise hypothesis tests about their true values. A simulations study is carried out to examine the finite-sample properties of the proposed methodology. A real data set is considered as well, and results based on the proposed methodology are compared to those based on selected existing approaches.

In this note, we summarize known results and open questions on the existence of isometric embeddings between different Schatten classes as well as obtain a new non-embeddability result using a novel method. We also provide a brief overview of the relevant methods.

Here we develop a formalism for reversible ionic aggregation kinetics in an example concentrated electrolyte, building on previous equilibrium work of McEldrew and co-workers, and thermoreversible polymers and patchy particle systems. This is achieved through solving a macroscopic rate equation of open/occupied association sites, shown to be a solution of the reversible Smoluchowski aggregation equation, which predicts how ionic associations in electrolytes change subject to a step-change in conditions. We test the derived equations against atomistic molecular dynamics simulations of a salt-in-ionic liquid, where good qualitative agreement is obtained, but quantitative differences are found. This highlights the multiple time scales that exist in concentrated electrolytes, with a fast timescale preceding a longer timescale. We hope this formalism opens new avenues in understanding the dynamics and non-equilibrium behaviour in electrolytes. For example, the formalism can be developed further to investigate the non-Newtonian behaviour of concentrated electrolytes, double layer charging, and the slow dynamics of these electrolytes in confinement.

Failures in clusters running large-scale AI workloads can result in decreased utilization. Because the cost of a failure in such AI workloads is high (as it requires restarting the entire job from a previous checkpoint), there are many mechanisms in place to ensure that the failures are mitigated, and the impact of a failure is minimized. However, these mechanisms have many knobs and parameters, all of which must be carefully tuned based on the system and cluster's characteristics. We built AIReSim, a discrete event simulator to evaluate the different design choices during the failure, recovery, scheduling and repair processes for a cluster running a large-scale AI workload. AIReSim allows the system designer to systematically evaluate the effects of the different knobs and parameters on the overall end-to-end reliability of the system. Further, AIReSim can be used to identify which knobs or parameters are important in order to prioritize the investment of effort in improving the system. AIReSim also allows tuning of the knobs for achieving different tradeoffs in the system, as well as to consider various ``what-if'' scenarios. We present a case study of applying AIReSim for capacity planning for large-scale clusters running AI workloads.