Focal mechanisms of small earthquakes are critical for characterizing faults and regional stress. P-wave polarities and S/P amplitude ratios (e.g., in HASH) are widely used to determine focal mechanisms for small earthquakes, but first-motion picking can be difficult for emergent onsets, and S/P ratios are often highly inconsistent because of imperfect velocity models and unknown site responses. Here, I present FocMecDR, a relative focal-mechanism inversion method that uses relative polarities and double ratios of S/P amplitudes. Analogous to double-difference earthquake location, FocMecDR uses a reference event with a known focal mechanism to solve focal mechanisms for nearby earthquakes by minimizing the misfit between observed and theoretical S/P amplitude double ratios while enforcing cross-correlation-based relative polarities. Extensive synthetic tests demonstrate the effectiveness and accuracy of the method. Field verification using an antisimilar earthquake pair and aftershocks from the northwestern end of the 2019 Ridgecrest rupture zone further confirms its reliability. Application to the foreshocks preceding the 2019 Ridgecrest mainshock reveals a highly uniform stress field during nucleation, suggesting a possible framework for evaluating fault-zone stress and rupture readiness before large earthquakes.

Tensor network methods provide a scalable solution to represent high-dimensional data. However, their efficacy is often limited by static, expert-defined structures that fail to adapt to evolving data correlations. We address this limitation by formalizing the tensor network structural rounding problem and introducing the hierarchical structure search algorithm HISS, which automatically identifies near-optimal structures and index reshaping for arbitrary tree networks. To navigate the combinatorial explosion of the structural search space, HISS integrates stochastic sub-network sampling with hierarchical refinement. This approach utilizes entropy-guided index clustering to reduce dimensionality and targeted reshaping to expose latent data correlations. Numerical experiments on analytical functions and real-world physics applications, including thermal radiation transport, neutron diffusion, and computational fluid dynamics, demonstrate that HISS exhibits empirical polynomial scaling with dimensionality relative to the sampling budget, bypassing the scalability barriers in prior work. HISS achieves compression ratios $2.5\times$ to $100\times$ higher than standard fixed formats such as Tensor Trains and Hierarchical Tuckers~(peaking at $1000\times$). Furthermore, HISS discovers structures that generalize effectively: applying a structure optimized for one data instance to a related target data typically maintains compression performance within $10\%$ of the result obtained by performing structure search on that target data. These results highlight HISS as a robust, automated tool for adaptive data representation and high-dimensional simulation compression with tensor network methods.

We present a computational framework for characterizing the molecular self-organization of cocoa butter (Theobroma cacao) during dark chocolate tempering through the lens of Topological Data Analysis (TDA). A physics-inspired particle simulation models N=100 triglyceride molecules across the full temperature range 15--60 degrees C, spanning all six crystalline polymorphs of cocoa butter (Forms I--VI) as well as the melt and superheating regimes. At each temperature tick, we construct a Vietoris-Rips filtration and compute the persistent homology groups H0 (connected components), H1 (independent cycles), and H2 (3D voids). The resulting persistence diagrams are analyzed via persistent entropy E = -sum_i p_i log2(p_i), where p_i = l_i / sum_j l_j and l_i = death_i - birth_i denotes feature lifetime; essential classes are assigned death = m+1 (m = eps_max) following the standard persistent entropy convention (Rucco 2026, arXiv:2602.09058). Our results demonstrate that Form V (the optimal tempering polymorph, 29.5--34 degrees C) is characterized by a distinctive topological signature: a local minimum in the H0 persistent entropy (E0 = 5.74 +/- 0.04 bits), a pronounced depression in the first Betti number beta_1 (1562 +/- 35), and a global minimum in the H2 entropy (E2 = 12.29 +/- 0.25 bits) reflecting coherent inter-bilayer lamellar cavities. Via Theorem 1 and Corollary 1 of Rucco (2026), persistent entropy is proven to separate the ordered and disordered phases by an asymptotically non-vanishing gap whenever a phase transition induces the creation or destruction of topological mass at macroscopic scales -- a condition we verify empirically across all eight cocoa butter regimes. These findings suggest that TDA-based metrics could serve as non-invasive quality indicators for industrial chocolate tempering processes.

The deployment of ultra-dense networks (UDNs), particularly cell-free massive MIMO (CF-mMIMO), is mainly hindered by costly and capacity-limited fronthaul links. This work proposes a two-tiered optimization framework for cost-effective hybrid fronthaul planning, comprising a Near-Optimal Fronthaul Association and Configuration (NOFAC) algorithm in the first tier and an Integer Linear Program (ILP) in the second, integrating fiber optics, millimeter-wave (mmWave), and free-space optics (FSO) technologies. The proposed framework accommodates various functional split (FS) options (7.2x and 8), decentralized processing levels, and network configurations. We introduce the hierarchical scheme (HS) as a resilient, cost-effective fronthaul solution for CF-mMIMO and compare its performance with radio-stripes (RS)-enabled CF-mMIMO, validating both across diverse dense topologies within the open radio access network (O-RAN) architecture. Results show that the proposed framework achieves better cost-efficiency and higher capacity compared to traditional benchmark schemes such as all-fiber fronthaul network. Our key findings reveal fiber dominance in highly decentralized deployments, mmWave suitability in moderately centralized scenarios, and FSO complements both by bridging deployment gaps. Additionally, FS7.2x consistently outperforms FS8, offering greater capacity at lower cost, affirming its role as the preferred O-RAN functional split. Most importantly, our study underscores the importance of hybrid fronthaul effective planning for UDNs in minimizing infrastructural redundancy, and ensuring scalability to meet current and future traffic demands.

Embedding high-dimensional data into resource-limited quantum devices remains a significant challenge for practical quantum machine learning. In multimodal face anti-spoofing, while linear compression methods such as principal component analysis can reduce dimensionality to accommodate limited quantum budgets, such approaches often lose critical high-order cross-modal correlations due to the loss of structural information. To this end, we propose a hybrid Matrix Product State (MPS)-Variational Quantum Circuit (VQC) framework, where the MPS serves as a structured, differentiable pre-quantum compression and fusion module, and the VQC acts as the quantum classifier. Built upon the low-rank structure controlled by the virtual bond dimension and integrated with a configurable nonlinear enhancement mechanism, this MPS module explicitly models long-range cross-modal correlations while compressing multimodal data into a compact representation matching the quantum budget and improving numerical stability under extreme compression. Experiments on the CASIA-SURF benchmark demonstrate that MPS-VQC achieves accuracy comparable to strong classical neural network baselines with fewer than 0.25M parameters, highlighting the parameter efficiency of tensor-network representations for high-dimensional multimodal data under tight resource budgets. Leveraging the intrinsic compatibility between MPS structures and quantum circuit topology, this framework not only provides a viable technological pathway for efficient multimodal anti-spoofing on NISQ devices but also serves as a stepping stone toward fully quantum implementations of such tasks in the future.

In this paper we present a new criterion to determine when the normalized Haar measure on a compact topological group is a Pietsch measure for nonlinear summing mappings. As a consequence, we provide a partial answer to a problem raised by Botelho et al. in \cite{haar botelho} motivated by a question posed to the authors by J. Diestel. It is explicitly shown that this criterion encompasses recent and new results as particular cases.

Little is known about the representations used in qualitative research studies and why. A data-driven literature review was employed to explore the use of media in qualitative research reporting. A study by Verdinelli & Scagnoli (2013) was replicated and extended by conducting a content analysis of papers and figures published across three qualitative methods journals between 2020 and 2022. Figures were categorized by types (e.g., matrix-based, Venn diagrams, flowcharts) and documents were grouped by their epistemological stances (i.e., objectivist, subjectivist, or constructivist) before conducting a correspondence analysis and epistemic network analysis. Our findings suggest that (1) visual media have remained largely absent, (2) figure types have be come more diverse and (3) the use of figure types is likely independent of epistemological stance but provide opportunities for further exploration. These findings provide a foundation for impactful integration of data visualization tools to enhance communicati ve power of findings across disciplines.

We study the Willmore flow for graphs over a bounded domain in $\mathbb{R}^2$ with Dirichlet (clamped) boundary conditions, a still little-studied setting that also serves as a prototype for higher-order flows with fixed boundary data. We develop a low-regularity theory that avoids the classical fourth-order compatibility condition at $t=0$. Combining a reformulation of the graphical equation, which isolates the quasilinear fourth-order principal part from the lower-order terms, with time-weighted parabolic Hölder spaces, we prove short-time existence for initial data in $C^{1+α}(\overlineΩ)$ and, under a smallness assumption, also for Lipschitz data in $C^{0,1}(\overlineΩ)$, even when the initial Willmore energy is not defined. In the Hölder regime, uniqueness is obtained. In the small-data Lipschitz regime, we also prove global existence, uniform gradient bounds, and exponential convergence to a stationary solution of the elliptic Willmore equation with the prescribed boundary data. A key ingredient is an $L^2$-smallness criterion for graphical surfaces with small Willmore energy and small boundary values. The approach is mainly analytic and extends naturally to related higher-order geometric flows and other boundary conditions.

Jets produced in relativistic heavy-ion collisions are modified by their interactions with the quark-gluon plasma (QGP), making jet substructure observables sensitive probes of QGP dynamics. A quantitative description of these modifications requires understanding how the medium affects elementary parton splittings with full dependence on both their energy fraction $z$ and splitting angle $θ$, beyond the widely used soft emitted-gluon approximation. Here, we study medium-induced $1 \to 2$ splittings double-differential in $z$ and $θ$, with full resummation of multiple scatterings, and show that in the large-$N_c$ limit and under the harmonic oscillator (HO) approximation, all path integrals can be evaluated analytically for any splitting channel, providing a computationally efficient semi-analytical result. We also revisit the semi-hard approximation (SHA), extending it to include leading corrections in inverse powers of the partons energies, which we denote the improved semi-hard approximation (ISHA), and assess its validity through a comparison with the large-$N_c$-HO results. Our analysis shows that while the SHA is found to be unreliable across most of phase space, even for high-energy emitters, the ISHA provides a robust approximation for splittings where all partons are sufficiently energetic.

Affective polarization, the emotional divide characterized by in-group love (trust towards fellow partisans) and out-group hate (mistrust towards those with opposite political views), has become prevalent in the current society. Despite its prevalence, the role of social network structure in the dynamics of affective polarization is yet to be well-understood. We provide a mean-field approximation of opinion dynamics under affective polarization on Watts-Strogatz and power-law (scale-free) networks. Our results show that consensus is fragile in social networks with power-law degree distributions, and the smaller average path length of the network (resembling a small-world network) makes achieving the consensus further difficult. Simulations and numerical experiments on real-world networks indicate that the mean-field model is aligned with the actual dynamics. Our findings shed light on how real-world network properties shape the dynamics of affective polarization and why consensus remains elusive in the real-world.

Vortex-induced vibrations (VIV) remain a canonical yet complex manifestation of fluid-structure interactions, where coupled nonlinear dynamics govern the motion of bluff bodies. For several years, we have relied on traditional reduced-order mathematical models derived from empirical and oscillator-based formulations; however, such models often fail to reproduce the quantitative dynamics observed in realistic flow environments. In this study, we explore a data-driven framework that leverages sparse identification of nonlinear dynamics (SINDy) and its weak formulation to uncover the governing equations of a single-degree-of-freedom cylinder undergoing VIV, using both data generated from previously developed reduced-order models and high-fidelity simulation results to assess the interpretation and efficacy of models discovered from a purely data-driven approach, particularly when the underlying dynamics are not fully known. The weak formulation (WSINDy), which replaces numerical differentiation with an integral-based representation, demonstrates marked robustness for aperiodic dynamics in particular. A complementary analysis using proper orthogonal decomposition (POD) is employed to extract the dominant spatio-temporal structures of the flow and to assess whether the temporal evolution of the wake can be represented on a reduced-dimensional manifold. The findings establish that data-driven identification can recover interpretable, quantitatively reliable models of VIV, providing a robust and computationally efficient pathway for modelling fluid-structure interactions directly from data. In particular, WSINDy is shown to be a more robust and interpretable alternative to standard SINDy for discovering VIV equations from aperiodic response dynamics, paving the way for predictive, data-informed design of fluid-structure interaction systems.

In simulation-based engineering, design choices are often obtained following the optimization of complex blackbox models. These models frequently involve mixed-variable domains with quantitative and categorical variables. Unlike quantitative variables, categorical variables lack an inherent structure, which makes them difficult to handle, especially in the presence of constraints. This work proposes a systematic approach to structure and model categorical variables in constrained mixed-variable blackbox optimization. Surrogate models of the objective and constraint functions are used to induce problem-specific categorical distances. From these distances, surrogate-based neighborhoods are constructed using notions of dominance from bi-objective optimization, jointly accounting for information from both the objective and the constraint functions. This study addresses the lack of automatic and constraint-aware categorical neighborhood construction in mixed-variable blackbox optimization. As a proof of concept, these neighborhoods are employed within CatMADS, an extension of the MADS algorithm for categorical variables. The surrogate models are Gaussian processes, and the resulting method is called CatMADS-GP. The method is benchmarked on the Cat-Suite collection of 60 mixed-variable optimization problems and compared against state-of-the-art solvers. Data profiles indicate that CatMADS-GP achieves superior performance for both unconstrained and constrained problems.

This paper presents a novel method for estimating larger Region of Attractions (ROAs) for continuous-time nonlinear systems modeled via the Takagi-Sugeno (TS) framework. While classical approaches rely on a single TS representation derived from the original nonlinear system to compute an ROA using Lyapunov-based analysis, the proposed method enhances this process through a systematic coordinate transformation strategy. Specifically, we construct multiple TS models, each obtained from the original nonlinear system under a distinct linear coordinate transformation. Each transformed system yields a local ROA estimate, and the overall ROA is taken as the union of these individual estimates. This strategy leverages the variability introduced by the transformations to reduce conservatism and expand the certified stable region. Numerical examples demonstrate that this approach consistently provides larger ROAs compared to conventional single-model TS-based techniques, highlighting its effectiveness and potential for improved nonlinear stability analysis.

Lopsided sets were introduced by Jim Lawrence in 1983 when he studied the subsets of $\{-1,+1\}^E$ that encode the intersection pattern of a convex set $K$ with the orthants of ${\mathbb R}^E$. Lopsided sets have been independently rediscovered by several other authors, in particular by Andreas Dress in 1995, who called them \emph{ample} sets. Dress defined ample sets as the set families satisfying equality in a combinatorial inequality, which holds for all set families. In a previous article we characterized ample sets in various combinatorial and graph-theoretical ways. In this paper we study geometric realizations of ample sets as cubihedra (cube complexes), which yields several new characterizations. One such characterization establishes that the cubihedra of ample sets endowed with the intrinsic $\ell_1$-metric are exactly the isometric subspaces of $\ell_1$-spaces (which we call, weakly convex sets). We also view the barycenter maps of faces of cubihedra of ample sets as collections of $\{ \pm 1, 0\}$-sign vectors and, in analogy with the characterization of oriented matroids by the covectors and the cocircuits. Moreover, we characterize the collections of $\{ \pm 1, 0\}$-sign vectors corresponding to barycenter maps of all faces and all maximal faces of an ample set. Furthermore, we show that any ample set $\covectors\subseteq \{ -1,+1\}^E$ is realizable as the intersection pattern of a weakly convex set $K$ with the orthants of ${\mathbb R}^E$. All this testifies that the concept of ample sets is quite natural in the context of cube complexes.

We study nonequilibrium mode selection in dissipative exciton-polariton condensates incoherently pumped through an excitonic reservoir in the presence of pure energy relaxation. For a confined system in which a vortex mode is selected at threshold, we show that energy relaxation qualitatively changes the condensation scenario: as the pump increases, the asymptotic state evolves from a vortex condensate to a rotating mixed state and then to a ground-state condensate. Pure energy relaxation thus destabilizes condensation into excited states and promotes ground-state selection.

The rapid growth of GPU-heavy data centers has significantly increased electricity demand and creating challenges for grid stability. Our paper investigates the extent to which an energy-aware job scheduling algorithm can provide flexibility in GPU-heavy data centers. Compared with the traditional first-in first-out (FIFO) baseline, we show that more efficient job scheduling not only increases profit, but also brings latent power flexibility during peak price period. This flexibility is achieved by moving lower energy jobs, preferentially executing jobs with lower GPU utilization and smaller node requirements, when the electricity price is high. We demonstrate that data centers with lower queue length and higher variance in job characteristics such as job GPU utilization and job size, offer the greatest flexibility potential. Finally we show that data center flexibility is highly price sensitive, a 7% demand reduction is achieved with a small incentive, but unrealistically high prices are required to achieve a 33% reduction.

It is generally accepted that the launching of astrophysical jets requires a large-scale magnetic field threading a central object (black hole or star) and/or its surrounding accretion disc. However, the collimation mechanism far away from the central object has not yet been fully understood. In a previous work we investigated a mechanism in which the jet is self-collimated due to a dominant hoop stress. We ran numerical simulations in which a Jet-Emitting disc (JED) spans the entire lower computational boundary. Those were the first of their kind to showcase the steady recollimation shocks predicted by steady-state analytical studies of jets. However, the huge size of the JED prevented a complete study of the connection between the accelerating and asymptotic electric circuits, as well as the influence of the outer medium. We performed a set of axisymmetric ideal MagnetoHydroDynamics (MHD) non-relativistic jet simulations. In those, only the innermost region of the accretion disc is a jet-launching zone. The jets of finite radial extent in those simulations also produce steady recollimation shocks at large distances from the central object. Standing recollimation shocks are not a bias of self-similarity, but a generic feature of jets emitted from magnetized Keplerian accretion discs. They may produce observable features, such as a standing emission knots, a decrease of the rotation rate or a change in polarisation. We also recover previous results on the influence of external pressure on jet confinement, such as the relation between pressure profile and jet shape, and jet acceleration efficiency.

We present a data-driven approach for constructing generalized collisional kinetic models for inhomogeneous plasmas in one-dimensional physical space and three-dimensional velocity space (1D-3V). The collision operator is directly learned from micro-scale molecular dynamics (MD) and accurately accounts for the unresolved particle interactions over a broad range of plasma conditions. Unlike the standard Landau operator, the present operator takes an anisotropic, non-stationary form that captures the heterogeneous collisional energy transfer arising from the many-body interactions, which is crucial for plasma kinetics beyond the weakly coupled regime. Efficient numerical evaluation is achieved through a low-rank tensor representation with $O(N \log N)$ computational complexity. The constructed kinetic equation strictly preserves conservation laws and physical constraints and therefore, enables us to develop an explicit second-order, energy-conserving scheme that ensures fully discrete conservation of mass and total energy. Numerical results demonstrate that the present model accurately predicts both transport coefficients and several 1D-3V kinetic processes compared with MD simulations across a broad range of densities and temperatures in spatially inhomogeneous settings. This work provides a systematic pathway for bridging micro-scale MD and inhomogeneous plasma kinetic descriptions where empirical models show limitation.

We consider a Dirichlet waveguide in $\mathbb{R}^n$ ($n = 2,3$) with an attached cavity. We show that if the cavity admits a small gap, then the original embedded eigenvalues turn into resonances. The main question we address is how the size of the gap affects the resonant properties, in particular the imaginary part of the resonant pole. For example, in the case of a two dimensional waveguide with a gap of size $\varepsilon$, we show that the leading order term of the resonance behaves as $\mathcal O (\varepsilon^2)$. In the three-dimensional case, if the aperture is defined by a rectangular opening with volume proportional to $\varepsilon^2$, the resonant component behaves as $\mathcal{O}(\varepsilon^4)$. This shows that, in the analyzed class of models, the characteristic time scale associated with the resonances is generically of order $\mathcal{O}((\mathrm{vol}_\varepsilon)^{-2})$, where $\mathrm{vol}_\varepsilon$ denotes the volume of the aperture inducing the resonance.

Following the HL-LHC era, proposed lepton colliders highlight the need to study various important Higgs boson production mechanisms to precisely probe the Standard Model Higgs sector. We propose a novel mechanism $e^\pm γ\rightarrow {\bar ν}_e (ν_e) H W^\pm$, which can be useful to study Higgs boson properties. This channel is relatively free from the background and can be used to measure the Higgs boson properties, in particular $WWH$ coupling. We examine the viability of this production mechanism. We show that the process can be observed at the planned FCC-ee with the center-of-mass energy of 365 GeV. At the center-of-mass energy of 500 GeV, the process can be observed within a few months of the operation. We use an in-house Monte Carlo event generator that simultaneously incorporates the photon distribution and electron/positron distribution. Our work is also a step towards realistic simulations of lepton- and photon-initiated processes at lepton colliders.

In recent years, the contribution of renewable energy resources to the electrical grid has increased drastically; the most common of these are photovoltaic solar panels and wind turbines. These resources rely on inverters to interface with the grid, which do not inherently exhibit the same fault characteristics as synchronous generators. Consistently, they can strain grid reliability and security, cause increased number of blackouts, and, in some cases, allow relatively minor faults to turn into cascading failures. Solar and wind energy provide benefits and can support grid stability; however, several challenges and gaps in understanding must be explored and addressed before this can be realized. This paper provides a comprehensive literature review of grid codes, modeling techniques, and tools, as well as current methods for responding to various faults. It also presents an overview of the industry's state as it relates to grid fault response in the presence of inverter-based resources.

Accurate spatio-temporal and spatio-spectral metrology is critical to the characterization and use of ultra-short, high-power lasers. The emergence of few cycle pulses, with bandwidths of tens or hundreds of nanometers, poses a significant challenge to existing metrology techniques. This is due both to large discrepancies in the sensitivities of the measurements at different wavelengths and to variation in the spectral intensity at those wavelengths. In this paper, the authors propose spectral filtering and stitching of the measurements as a robust, simple solution that enhances the dynamic range of the measurements, allowing accurate few-cycle pulse reconstruction. This enhancement is demonstrated using INSIGHT -- the most commonly used spatio-spectral measurement device -- as well as using IMPALA and spatially resolved Fourier transform spectrometry.

Parliaments dominated by two political blocs often face legislative inefficiencies as polarization increases. A central institutional question concerns how majority and supermajority rules interact with parliamentary composition to balance governability and minority protection. This article examines how the inclusion of independent legislators, introduced through sortition, affects collective decision-making under different majority and supermajority quorum requirements. Using an agent-based model that combines analytical threshold derivations with numerical simulations, we identify four critical thresholds that partition the parameter space into distinct coalition regimes. These regimes range from majority-party dominance to minority veto and, at high levels of diversity, to structural fragmentation in which coordination becomes increasingly difficult. Parliamentary efficiency emerges from non-linear interactions among quorum thresholds, party-size distribution, and the proportion of independents. Under simple majority, the system exhibits an interior efficiency maximum associated with a transition from unilateral control to pivot-based coalition formation. Under strong supermajorities, however, the same increase in diversity may induce minority veto dynamics or coordination breakdown, significantly reducing legislative performance. These results show that institutional performance is not an intrinsic property of a particular decision rule but an emergent outcome of the interaction between approval thresholds, parliamentary composition, and the coalition structures they generate.

Let $Ω\subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is proportional to the length $\mathcal{H}^1(\ell \cap Ω)$ as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy $\lesssim L^{1/3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L \rightarrow \infty$.

We study the physical mechanisms underlying the production of orbitally-modulated PeV photons from Cyg X-3, recently discovered by the LHAASO collaboration. Our key findings are as follows. Helium nuclei are accelerated in a compact and strongly magnetized region within the jet, but they then quickly advect downstream to regions with a weaker field, allowing them to diffuse out of the jet, where they produce pions in hadronic collisions with both the stellar photons and the stellar wind of the Wolf-Rayet donor. The optical depths across the binary are $\lesssim$1 for both types of interactions, implying that their rates are proportional to the column densities along the particle paths. Given the low viewing angle of Cyg X-3 ($i\approx26^\circ$--$28^\circ$), most of the observed photons are produced by the relativistic hadrons accelerated in the counterjet (for which the column densities toward the observer are much longer than for the jet). This also explains the peak of the phase-folded PeV photon flux to be on the opposite side of the superior conjunction than that for the (also orbitally-modulated) GeV photons, which are produced by collisions of relativistic electrons with stellar photons in the optically thick regime. This then implies that the GeV emission is produced in the approaching jet.

In this work, we derive reduced interface models for hydroelastic water waves coupled to a nonlinear viscoelastic plate. In a weakly nonlinear small-steepness regime we obtain bidirectional nonlocal evolution equations capturing the interface dynamics up to quadratic order, and we also derive two unidirectional models describing one-way propagation while retaining the leading dispersive and dissipative effects induced by the plate. Remarkably, one of the bidirectional model has a doubly nonlinear structure in the sense that there there is a nonlinear elliptic operator acting on the acceleration of the interface. We prove local well-posedness for the bidirectional model for small data via a two-parameter regularization and nested fixed points. For the unidirectional models, we obtain local well-posedness for arbitrary data and global well-posedness for small data.

The goal in the stochastic vertex cover problem is to obtain an approximately minimum vertex cover for a graph $G^\star$ that is realized by sampling each edge independently with some probability $p\in (0, 1]$ in a base graph $G = (V, E)$. The algorithm is given the base graph $G$ and the probability $p$ as inputs, but its only access to the realized graph $G^\star$ is through queries on individual edges in $G$ that reveal the existence (or not) of the queried edge in $G^\star$. In this paper, we resolve the central open question for this problem: to find a $(1+\varepsilon)$-approximate vertex cover using only $O_\varepsilon(n/p)$ edge queries. Prior to our work, there were two incomparable state-of-the-art results for this problem: a $(3/2+\varepsilon)$-approximation using $O_\varepsilon(n/p)$ queries (Derakhshan, Durvasula, and Haghtalab, 2023) and a $(1+\varepsilon)$-approximation using $O_\varepsilon((n/p)\cdot \mathrm{RS}(n))$ queries (Derakhshan, Saneian, and Xun, 2025), where $\mathrm{RS}(n)$ is known to be at least $2^{Ω\left(\frac{\log n}{\log \log n}\right)}$ and could be as large as $\frac{n}{2^{Θ(\log^* n)}}$. Our improved upper bound of $O_{\varepsilon}(n/p)$ matches the known lower bound of $Ω(n/p)$ for any constant-factor approximation algorithm for this problem (Behnezhad, Blum, and Derakhshan, 2022). A key tool in our result is a new concentration bound for the size of minimum vertex cover on random graphs, which might be of independent interest.

This study investigates the dynamics of passively morphing foils under accelerated plunging, establishing mechanistic links between transient kinematics, structural compliance, and aerodynamic performance. Two-way coupled simulations are performed for three wing geometries: a symmetric NACA0012 foil and two bio-inspired geometries based on falcon and owl wing sections, across non-dimensional bending rigidity values, chordwise flexible segment extents from the trailing-edge (25%, 50%, and 75%), and transition speed parameters. The present findings reveal that flexible trailing-edge configurations exhibit improved aerodynamic performance relative to stiffer foils, and the aerodynamic benefit of trailing-edge compliance is strongly influenced by wing geometry. A geometry-specific optimal bending stiffness exists beyond which additional flexibility degrades performance. The extent of the chordwise flexible segment critically governs the aeroelastic response. Whilst a 25% flexible segment produces behaviour indistinguishable from a rigid wing, extending flexibility to 75% of the chord induces highly unsteady lift fluctuations, particularly for the NACA0012 foil, for which the root-mean-square lift coefficient increases sharply. The bio-inspired foils, in contrast, exhibit a moderate reduction in root-mean-square lift coefficient for the 50% and 75% cases, reflecting the stabilising influence of their cambered geometry. Increasing the transition speed parameter monotonically amplifies trailing-edge deflection, strengthens the leading- and trailing-edge vortices, and intensifies the coupling between structural deformation and instantaneous lift. These findings provide new physical insight into bio-inspired propulsion and manoeuvring strategies, with implications for the design of passively adaptive lifting surfaces in unsteady environments.

In this note, we study the classification of four-dimensional complete gradient steady and expanding Ricci solitons. Specifically, under the asymptotically cylindrical (respectively, asymptotically conical) assumption, we classify gradient steady (respectively, expanding) Ricci solitons with half-harmonic Weyl curvature. In addition, we obtain a partial classification of four-dimensional gradient expanding Ricci solitons with half-nonnegative isotropic curvature.

We consider on Riemannian manifolds the Leibenson equation $\partial _{t}u=Δ_{p}u^{q}$ that is also known as a doubly nonlinear evolution equation. We prove sharp upper estimates of weak subsolutions to this equation on Riemannian manifolds with non-negative Ricci curvature in the whole range of $p>1$ and $q>0$ satisfying $q(p-1)<1$. In this way, we improve the result of \cite{Grigoryan2024a} and prove Conjecture 1.2 from \cite{Grigoryan2024a}.