The 2-μm waveband, emerging as a highly promising candidate for optical communication, offers an extended wavelength window for high-speed optical transmission. Despite its potential, the development of integrated electro-optic (E/O) modulators operating at this wavelength range has been limited. Such E/O modulators are crucial for high-speed optical communication systems at the 2-μm waveband. In this work, we propose and experimentally demonstrate high-performance E/O absorption modulators based on a graphene-silicon slot waveguide. Our approach enables wideband, high-speed, efficient, robust and compact modulators at both 2-μm and 1.5-μm wavebands. This work represents a significant advancement towards the realization of high-speed integrated E/O modulators for optical communication systems operating at the 2-μm wavelength range.

In the dynamic set cover problem, the input is a dynamic universe of elements and a fixed collection of sets. As elements are inserted or deleted, the goal is to efficiently maintain an approximate minimum set cover. While the past decade has seen significant theoretical breakthroughs for this problem, a notable gap remains between theoretical design and practical performance, as no comprehensive experimental study currently exists to validate these results. In this paper, we bridge this gap by implementing and evaluating four greedy-based dynamic algorithms across a diverse range of real-world instances. We derive our implementations from state-of-the-art frameworks (such as GKKP, STOC 2017; SU, STOC 2023; SUZ, FOCS 2024), which we simplify by identifying and modifying intricate subroutines that optimize asymptotic bounds but hinder practical performance. We evaluate these algorithms based on solution quality (set cover size) and efficiency, which comprises update time (the time required to update the solution following each insertion or deletion) and recourse (the number of changes made to the solution per update). Each algorithm uses a parameter $β$ to balance quality against efficiency; we investigate the influence of this tradeoff parameter on each algorithm and then perform a comparative analysis to evaluate the algorithms against each other. Our results provide the first practical insights into which algorithmic strategies provide the most value in realistic scenarios.

This paper presents a systematic observer design methodology for a class of port-Hamiltonian (pH) systems with state-dependent input matrices. Such systems can model a wide range of electromechanical systems, including magnetic levitation systems, MEMS devices, and electro-active polymer actuators such as DEA actuators, HASEL actuators, etc. In these applications, state-dependent input matrices naturally arise when the system is modeled under quasi-static electrical assumptions. An LPV polytopic embedding framework, together with LMI-based synthesis conditions, is proposed. The nonlinear error dynamics are represented as a convex combination of linear vertex systems using an integral mean value representation, which enables systematic computation of the observer gains that ensures exponential convergence. Both constant-gain and gain-scheduled observers are derived. Numerical results demonstrate the effectiveness of the proposed observer, with the gain-scheduled design achieving a significant increase in the maximum certifiable decay rate compared with constant-gain approaches, thereby reducing conservatism.

The propagation of the transverse electric (TE) and transverse magnetic (TM) waves in an effectively two-dimensional (2D) isotropic medium is described by Bergmann's equation of acoustics. We develop a dynamical formulation of the stationary scattering of these waves and explore its application in the study of the low-frequency behavior of the scattering data. Specifically, we introduce a suitable notion of fundamental transfer matrix for TE and TM waves in 2D. This is an integral operator $\widehat{\mathbf{M}}$ that carries the information about the scattering properties of the medium and admits a Dyson series expansion involving a non-Hermitian Hamiltonian operator. For situations where the inhomogeneities of the medium are confined to a layer of thickness $\ell$, we use the Dyson series for $\widehat{\mathbf{M}}$ to construct the series expansion of the scattering amplitude in powers of $k\ell$, where $k$ is the incident wavenumber. We derive analytic expressions for the leading- and next-to-leading-order terms of this series, verify the effectiveness of their application to a class of exactly solvable models, and use them to study low-frequency invisibility. In particular, we develop a low-frequency cloaking scheme which is applicable for both TE and TM waves. Our results have immediate applications in the study of low-frequency scattering of acoustic waves in a 2D fluid as these waves are also described by Bergmann's equation.

The absorption of light by materials is one of the most fundamental processes in optics and condensed-matter physics. Here we investigate whether laser light is absorbed by a crystalline material as an electromagnetic wave or as localized photon energies. We excite the first-order phase transition of vanadium dioxide with laser pulses of sufficient frequency to overcome the band gap but with insufficient pulse energy to overcome the latent heat. According to Maxwell's equations and Bloch theory, no transition should occur, because nowhere in the material is enough energy. Nevertheless, we observe with ultrafast electron diffraction for short times a disordered crystal geometry with nanometer-sized spots of switched material. Their amount matches approximately to the number of photons in the absorbed laser wave. Two optical experiments confirm this phenomenon, and simulations of single absorbed photons reproduce all measurements results. Although laser light and Bloch electrons are extended quantum objects, the energy of the individual photons is localized into nanometer dimensions, enabling local consequences at substantially higher energy than average.

Resource sharing in multi-tenant cloud environments enables cost efficiency but introduces the Noisy Neighbor problem, i.e., co-located workloads that unpredictably degrade each other's performance. Despite extensive research on detecting such effects, there are no explainable methodologies for quantifying the severity of impact and establishing causal relationships among tenants. We propose an analytical that combines controlled experimentation with multi-stage causal inference and validates it across 10 independent rounds in a Kubernetes testbed. Our methodology not only quantifies severe performance degradations (e.g., up to 67\% in I/O-bound workloads under combined stress) but also statistically establishes causality through Granger causality analysis, revealing a 75\% increase in causal links when the noisy neighbor activates. Furthermore, we identify unique "degradation signatures" for each resource contention vector (i.e., CPU, memory, disk, network), enabling diagnostic capabilities that go beyond anomaly detection. This work transforms the Noisy Neighbor from an elusive problem into a quantifiable, diagnosable phenomenon, providing cloud operators with actionable insights for SLA management and smart resource allocation.

Multi-agent LLM applications organize execution in synchronized rounds where a central scheduler gathers outputs from all agents and redistributes the combined context. This All-Gather communication pattern creates massive KV Cache redundancy, because every agent's prompt contains the same shared output blocks, yet existing reuse methods fail to exploit it efficiently. We present TokenDance, a system that scales the number of concurrent agents by exploiting the All-Gather pattern for collective KV Cache sharing. TokenDance's KV Collector performs KV Cache reuse over the full round in one collective step, so the cost of reusing a shared block is paid once regardless of agent count. Its Diff-Aware Storage encodes sibling caches as block-sparse diffs against a single master copy, achieving 11-17x compression on representative workloads. Evaluation on GenerativeAgents and AgentSociety shows that TokenDance supports up to 2.7x more concurrent agents than vLLM with prefix caching under SLO requirement, reduces per-agent KV Cache storage by up to 17.5x, and achieves up to 1.9x prefill speedup over per-request position-independent caching.

Within this research, two combinatorial bijections using Young diagrams were studied. The first is a special case of a bijective correspondence between two classes of combinatorial objects. Its proof, based on Young diagrams, establishes equinumerosity and provides an explicit constructive mapping. The second is a generalization to any natural d, preserving bijectivity. It shows the combinatorial structure remains stable under changes in the parameter, with Young diagrams serving as a universal language. A notable and non-obvious aspect of this generalization is the symmetry revealed in the construction. Intuitively, it was not evident that one could consider not only the natural order of residues but also any permutation of them.

This paper presents a methodology for Practically Safe Extremum Seeking (PSfES), designed to optimize unknown objective functions while strictly enforcing safety constraints via a Logarithmic Barrier Function (LBF). Unlike traditional safety-filtered approaches that may induce chattering, the proposed method augments the cost function with an LBF, creating a repulsive potential that penalizes proximity to the safety boundary. We employ averaging theory to analyze the closed-loop dynamics. A key contribution of this work is the rigorous proof of practical safety for the original system. We establish that the system trajectories remain confined within a safety margin, ensuring forward invariance of the safe set for a sufficiently fast dither signal. Furthermore, our stability analysis shows that the model-free ESC achieves local practical convergence to the modified minimizer strictly within the safe set, through the sequential tuning of small parameters. The theoretical results are validated through numerical simulations.

Power grids must modernize to meet climate goals while maintaining reliable and stable operating conditions. Yet progress is hindered by a limited understanding of the stochastic processes underlying grid frequency and phase-angle fluctuations, which are induced by the growing penetration of renewable generation, consumer demand fluctuations, and market trading. This issue is particularly acute in Africa, where grids often face weak investment. Here, we present results from a newly collected, large-scale, high-resolution dataset of grid frequency and phase angles for the United Kingdom and South Africa, comprising close to one billion data points. Using superstatistical modeling, we treat market-driven power fluctuations as a slowly varying parameter driving grid dynamics and incorporate nonlinear frequency control. As a result, we derive an analytical model that reproduces multimodal frequency distributions previously obtained from numerical simulations, as well as heavy-tailed fluctuations and double-exponential frequency autocorrelation decays, all in excellent agreement with experimental measurements. Beyond frequency, we also address the so far largely overlooked problem of characterizing spatial phase-angle fluctuations. By comparing our predictions with measurement data, we demonstrate that a low-dimensional effective grid model accurately fits South African data despite the grid's complexity. We also highlight significant differences between the grids of South Africa and the United Kingdom. Our results clarify how energy markets and control policies shape grid dynamics across countries with contrasting infrastructure maturity.

We study the persistent homology of the offset filtration generated by the range of a planar Brownian motion with constant nonzero drift. The members of this filtration are the Wiener sausages of increasing radius, and the degree-one persistence diagram records the birth and death of holes in the thickened trace as the radius varies. Our first result is a sampling theorem: for any continuous path in R d observed on a time grid $π$n the bottleneck distance between the persistence diagram of the continuous offset filtration and that of the sampled point cloud is bounded by the pathwise modulus of continuity $ω$X (|$π$n|). For Brownian motion this yields the almost-sure rate O |$π$n| log(1/|$π$n|) . Our second and main result is a law of large numbers for the drifted planar case. For every bounded Borel weight $ψ$ supported on a compact radius window [r0, r1] with r0 > 0, the smoothed persistence functional $Φ$ $ψ$ (T ), where $β$ T 1 (r) counts the holes in the radius-r sausage at time T , satisfies $Φ$ $ψ$ (T )/T $\rightarrow$ $ρ$ $ψ$ almost surely and in L 1 for a deterministic constant $ρ$ $ψ$ . This yields a finite positive intensity measure on the radius axis that governs the linear growth of topological complexity. The proof introduces a regeneration scheme along the drift direction: projecting the planar path onto the drift axis produces a one-dimensional Brownian motion with positive drift, whose ladder hits and bounded-backtracking events generate i.i.d. path blocks. The non-additivity of topology under concatenation is controlled by a Boundary Lemma, which combines a deterministic Mayer-Vietoris estimate with a geometric bound relating integrated Betti numbers to sausage area via the coarea formula. A Betti-curve representation converts the two-parameter persistence problem into a one-parameter family of fixed-radius hole counts, making the regeneration argument possible.

We study exit times from time-dependent domains under joint perturbations of the trajectory and the domain. Representing a moving domain by a continuous barrier $Φ$ on space-time, we reduce the exit problem to a one-dimensional first-passage problem for the scalarised path $y(t) := Φ(t,x(t))$. Our first main result is a deterministic continuity theorem: the exit-time functional is continuous, under local Skorokhod $J_1$ convergence of the path and local uniform convergence of the barrier, at every configuration satisfying an explicit non-tangency condition (NT). We show that NT is sharp in the sense that it characterises the continuity set of the functional. As a direct consequence, weak convergence of exit times follows from joint weak convergence of paths and barriers whenever the limiting pair satisfies NT almost surely; no independence or structural restrictions between trajectory and domain are required. Our second main result is a functional limit theorem: the exit-time profile $u\mapstoτ(u)$, viewed as a càdlàg function of the barrier level, converges in the Skorokhod $M_1$ topology under the same hypotheses, with a concrete example showing that $J_1$ convergence can fail. Concrete verification routes for NT are provided, including a non-characteristic/Itô criterion for diffusions, and the full framework is illustrated through a worked Donsker-type example.

We derive a concrete closed string dual to any interacting Hermitian one-matrix model, away from the double-scaling limit. Matrix and string correlators manifestly agree, to all orders in the genus expansion and all orders in the 't Hooft coupling(s). The worldsheet theory consists of a supersymmetric B-twisted Landau-Ginzburg model coupled to 2d topological gravity. We provide a precise dictionary between traces of the matrix and vertex operators on the worldsheet. Matrix model correlators are explicitly mapped to computable integrals over the moduli space of Riemann surfaces. We perform several direct cross-checks on both sides of the duality. This work furnishes a detailed instantiation of gauge/string duality, in the standard 't Hooft regime, and hopefully a useful worldsheet toy model for the AdS/CFT correspondence, away from the free field limit.

Let t be the first-passage time of a continuous barrier by a c{à}dl{à}g adapted process. We show that t admits a canonical fourfold pathwise decomposition into continuous contact, contact from the left followed by an upward jump, exact hit by jump, and strict overshoot by jump from below. This refinement is more informative than the classical contact-versus-overshoot dichotomy for random-time purposes, because it separates modes with different predictability properties. In particular, the left-contact component always defines an accessible stopping time and becomes predictable under a no-premature-left-contact condition, which we prove to be both sufficient and necessary for the canonical running-supremum announcing sequence to work. On the gap side, under a structural exclusion of predictable gap-crossings, the corresponding restricted time is totally inaccessible. In the semimartingale setting, we obtain a sharp compensator criterion for the predictable-side condition, explicit compensator formulas for the jump-driven crossing modes, and a decomposition of the compensator of the default indicator into its predictable jump part and continuous part. As an application, for a mean-reverting affine jump-diffusion with upward exponential jumps, we derive the boundary-value problem governing the overshoot mode, prove that the differentiated third-order ODE is equivalent to the original problem only when a boundary compatibility condition is retained, and establish verification and uniqueness for the discounted problem. This yields an explicit Green-Volterra representation, a first-order small-q expansion expansion, and, in the undiscounted case, closed formulas for the overshoot and creeping probabilities

We introduce a new class of swarm-based inertial methods (SBIMs) for global minimization, formulated as coupled dissipative inertial dynamical systems derived from the generalized Onsager principle. The proposed framework identifies the friction operator and the scaling of the potential energy, namely the objective function to be minimized, as the key ingredients governing relaxation dynamics over the energy landscape. Within this framework, we propose a new underdamped inertial dynamics whose damping mechanisms incorporate both gradient and Hessian information, allowing the system to adjust damping or acceleration according to the agent trajectories and the curvature of the landscape. Under suitable conditions, we prove that the underdamped system satisfies an energy dissipation law, from which we establish an upper bound on the asymptotic decay rate of the gap between the objective function and its global minimum, given by $O(1/δ(t))$ (defined in §3). We further construct structure-preserving discretizations that retain both discrete energy dissipation and the convergence rate estimate, $O(1/δ_k)$ (defined in \S3). In addition, we present several other efficient numerical algorithms for the dynamical system. Numerical experiments for all proposed algorithms validate the theory on convex test problems and demonstrate convergence rates in function values that are substantially faster than the theoretical guarantees ($O(1/δ_k)$). On nonconvex benchmark problems, the proposed methods achieve high success rates in reaching the global minimum, and exhibit more stable energy decay than swarm-based gradient descent and Nesterov methods. Overall, this work provides a systematic framework for the construction and analysis of SBIMs from an energy-dissipative perspective.

Existing cyberattack detection methods for smart grids such as Artificial Neural Networks (ANNs) and Deep Reinforcement Learning (DRL) often suffer from limited adaptability, delayed response, and inadequate coordination in distributed energy systems. These techniques may struggle to detect stealthy or coordinated attacks, especially under communication delays or system uncertainties. This paper proposes a novel Snitch Digital Twin (Snitch-DT) architecture for cyber-physical anomaly detection in grid-connected wind farms using Smart Voltage Source Converters (VSCs). Each wind generator is equipped with a local Snitch-DT that compares real-time operational data with high-fidelity digital models and generates trust scores for measured signals. These trust scores are coordinated across nodes to detect distributed or stealthy cyberattacks. The performance of the Snitch-DT system is benchmarked against previously published Artificial Neural Network (ANN) and Deep Reinforcement Learning (DRL)-based detection frameworks. Simulation results using an IEEE 39-bus wind-integrated test system demonstrate improved attack detection accuracy, faster response time, and higher robustness under various cyberattack scenarios.

Nested Monte Carlo is widely used for risk estimation, but its efficiency is limited by the discontinuity of the indicator function and high computational cost. This paper proposes a nested Multilevel Monte Carlo (MLMC) method combined with preintegration for efficient risk estimation. We first use preintegration to integrate out one outer random variable, which effectively handles the discontinuity of the indicator function, then we construct the MLMC estimator with preintegration to reduce the computational cost. Our theoretical analysis proves that the strong convergence rate of the MLMC combined with preintegration reaches -1, compared with -1/2 for the standard MLMC. Consequently, we obtain a nearly optimal computational complexity. Besides, our method can also handle the high-kurtosis phenomenon caused by indicator functions. Numerical experiments verify that the smoothed MLMC with preintegration outperforms the standard MLMC and the optimal computational cost can be attained. Combining our method with quasi-Monte Carlo further improves its performance in high dimensions. Keywords: Nested simulation, Multilevel Monte Carlo, Risk estimation, Preintegration

We prove that random $\mathbb{Z}$-homomorphisms on weakly expanding bipartite graphs exhibit a strong "flatness" phenomenon. Extending prior work of Peled, Samotij, and Yehudayoff for expanders, we first show that on any bipartite $(n, d, λ)$-graph with $λ\leq (1-δ)d$, a uniformly chosen $\mathbb{Z}$-homomorphism has a range at most $O(\log \log n)$ with high probability, which is tight up to a constant factor. This provides an affirmative answer to their question in the spectral setting. As a concrete application, we prove that a random $\mathbb{Z}$-homomorphism on the middle layers of the Hamming cube takes at most $5$ values with high probability. This shows that the $O(1)$-flatness for the full Hamming cube, proved by Kahn and Galvin, persists even when the rigid structural properties are relaxed.

A novel design is presented for a permanent magnet array to address specific challenges with scalable trapped-ion quantum computing systems. Design and optimization of this magnet geometry is motivated by concepts for large-scale Quantum Charge-Coupled Device (QCCD) architectures. This proposal is relevant to magnetic field gradient schemes for laser-free entanglement using long-wavelength radiation, and individual addressing based on spatially dependent, magnetic field sensitive qubits. This configuration generates a localized, asymmetric magnetic field, yielding a region for ion transport into and out of a strong magnetic field gradient, while minimizing the absolute field experienced by the ion. This is a distinct improvement for scalability over dipolar magnet geometries where a strong magnetic field surrounds a magnetic field nil in three dimensions, which is problematic for ion transport applications. The design also relaxes the alignment constraints for experimental setup by allowing greater tolerance to misalignment in two dimensions. Additionally, the potential to scale a permanent magnet scheme in QCCD systems circumvents engineering challenges associated with using large electrical currents to generate the field gradient. Finally, a conceptual discussion is given for incorporating the design into a scalable QCCD type architecture.

High-order extensions of the Hopfield model are known to exhibit dramatically enhanced storage capacity at equilibrium, while their dynamical retrieval properties remain less well understood. In our previous work, we carried out a dynamical mean-field theory (DMFT) analysis of the Krotov--Hopfield-type dense associative memory and found that the transition between successful and failed retrieval is accompanied by pronounced slow dynamics. As a consequence, the effective basin of attraction observed in numerical simulations extends well beyond that predicted by equilibrium statistical mechanics. A natural hypothesis is that this discrepancy originates from diagonal (self-interaction) contributions in the Krotov--Hopfield model, which generate a large number of lower-order interaction terms and may induce glassy relaxation near the retrieval boundary. To test this hypothesis, we analyze an alternative high-order associative memory model, namely the Abbott--Arian-type $p$-body Hopfield model, in which such diagonal contributions are absent by construction. Using dynamical mean-field theory, we derive an effective single-site process together with closed macroscopic equations governing the retrieval dynamics. Our analysis reveals that both slow dynamics and a substantial enlargement of the apparent basin of attraction persist even in this model. These results indicate that the dynamical slowdown near the retrieval boundary cannot be attributed primarily to diagonal self-interaction effects, but instead originates from intrinsic properties of high-order interactions.

This work presents a space-time isogeometric analysis of biharmonic wave problem, in contrast to the more common application of space-time methods to second order wave equations. We first establish the unique solvability of the continuous space-time variational formulation. In order to obtain $H^2$- conforming discretization of the biharmonic wave equation, we consider globally smooth B-spline functions having continuity higher than $C^0$. We prove that the resulting space-time discrete formulation is stable under a Courant-Friedrichs-Lewy (CFL) condition. Furthermore, we propose a stabilized formulation, achieved by adding a non-consistent penalty term, which yields unconditional stability. Exploiting the tensor product structure, an efficient direct solver is also provided for solving the linear system arising from the discrete formulation. A few numerical experiments are presented to demonstrate the stability and convergence properties of the proposed scheme as well as the efficiency of the proposed solver.

Let $\overlineμ_Λ(t):=\sum\limits_{m\geq1}μ_Λ(m)t^m$ be the \emph{$μ$-series} of a finite-dimensional tame algebra $Λ$ over an algebraically closed field, where $μ_Λ(m)$ denotes the minimal number of one-parameter families of $Λ$-modules with total dimension $m$. When $Λ$ is a string algebra with $\mathrm{Ba}(Λ)$ as its set of bands up to cyclic permutation, define the \emph{zeta function} $ζ_Λ(t):=\prod\limits_{\mathfrak b\in\mathrm{Ba}(Λ)}(1-t^{|\mathfrak b|})^{-1}$, where $|\mathfrak b|$ is the length of $\mathfrak b$. We prove an analogue of the prime number theorem for string algebras and use it to conclude that non-domestic string algebras are of exponential growth. Finally, we show that a string algebra is domestic if and only if its $μ$-series is rational.

We propose a new scintillator-based tracker concept based on a monolithic plastic scintillator plate with embedded scatterers and wavelength-shifting fiber readout. The embedded scatterers localize scintillation light so that channels closer to the charged-particle crossing point collect more light. The particle crossing position is reconstructed from the channel-to-channel light yield distribution with a position resolution well below the readout pitch. We performed a positron beam test with prototypes to validate the reconstruction principle and to evaluate the detection efficiency and position resolution. The beam test validated the position reconstruction principle, and demonstrated a near-100% detection efficiency and a position resolution of 1.47 mm for normal incidence and 1.85 mm for an incidence angle of 45°, with the 10-mm readout pitch. In this paper, we describe the detector concept, the reconstruction method, and the results of the beam test.

We propose a family of high-order local discontinuous Galerkin (LDG) methods, built on a parametric representation and coupled with a semi-implicit backward Euler time discretization, for isotropic and anisotropic curve-shortening flows. The spatial LDG formulation introduces auxiliary variables and carefully designed numerical fluxes which inherit the underlying variational structure. We prove the unconditional energy dissipation for the semi-discrete scheme, and establish the well-posedness for the fully discrete scheme under mild assumptions. For $P^k$ approximations, the LDG method achieves high-order spatial convergence; extensive numerical experiments confirm optimal $(k+1)$-order accuracy when the surface energy is isotropic or weakly anisotropic. Compared to classical parametric finite element methods (PFEM), the proposed LDG schemes do not need to rely on good mesh distributions or auxiliary symmetrized surface energy matrices for strongly anisotropic surface energy cases, and remain numerically stable on severely degraded meshes that typically cause PFEMs failure. This intrinsic stability enables effective capture of complex geometric evolution and sharp corner singularities produced by strong anisotropy. The approach thus provides a flexible and reliable framework for the numerical simulation of a broader class of geometric flows.

We investigate the co-evolution of the stellar and dark matter mass of satellite galaxies using the COLIBRE cosmological hydrodynamical simulations with subhaloes resolved by the history-based HBT-HERONS subhalo finder. We identify a universal tidal track connecting stellar mass loss to subhalo mass loss characterized by two distinct phases, which can be well described by the two-parameter model. The initial phase consists primarily of dark matter stripping, whereas stellar stripping becomes significant only after the subhalo bound mass fraction drops below a critical value ($\sim 0.057$). We find a bimodal mass loss rate distribution of subhaloes. In satellites with modest mass loss rates, the stellar mass is largely frozen. By contrast, the galaxy quickly becomes unresolved, along with the dark matter component for the extreme-mass-loss population, naturally explaining the lack of ``orphan'' galaxies in previous hydrodynamical simulations. Our model also predicts the formation condition for dark-matter-deficient galaxies (DMDGs), whose abundance peaks at $m_{*}\sim 10^{9.5}\,\rm{M}_{\odot}$. The abundance of DMDGs can be very sensitive to numerical effects, with COLIBRE resolving a much larger DMDG population than previous hydrodynamical simulations. We also estimate the influence of artificial disruption on the satellite stellar mass function, which can amount to 20 (50) per cent at $m_* \sim 10^{9} (10^{8}) \, \rm M_\odot$, given a baryonic mass resolution of $\sim 10^{6}\,\rm{M}_{\odot}$.

Decisions to pursue higher education are not fully explained by economic incentives, with social influence and peer effects playing a crucial, yet dynamically understudied, role. This paper develops a theoretical non-linear dynamics model analysing the interplay between economic returns and social pressure. We model a heterogeneous population of "Followers" who exhibit imitative behaviour, and "Positional Agents" who display counter-adaptive behaviour. Agents' preferences for education evolve endogenously, reacting to both aggregate enrolment and an endogenous wage premium that declines with the supply of educated workers. The aggregate dynamics are governed by a one-dimensional non-linear map. By assuming fixed population structure. we show that the social conflict between pro-cyclical imitative forces and counter-cyclical positional forces can destabilize the steady state, generating a period-doubling route to chaos. These complex, endogenous fluctuations in enrolment emerge only for intermediate, heterogeneous population mixes, while homogeneous populations remain stable. We argue that this instability represents a significant coordination failure, scrambling economic signals and hindering rational long-term planning for both students and institutions, making it a key policy concern. Finally, we also extend the result to the case where the population structure is endogenous.

Let $R$ be a commutative ring with identity and let $Z^{\ast}(R)$ denote the set of nonzero zero-divisors of $R$. The \emph{zero-divisor graph} $ \varGamma(R)$ is the simple graph with vertex set $V( \varGamma(R))=Z^{\ast}(R)$, where two distinct vertices$x,y\in Z^{\ast}(R)$ are adjacent if and only if $xy=0$ in $R$. In this paper we investigate the zero-divisor graph of the truncated polynomial ring $R=\mathbb{Z}_{p}[x]/\langle x^{c}\rangle,$ for $c\in\mathbb{N}.$ We determine the spectrum of the $A_α$-matrix associated with $ \varGamma(R)$, and, as special cases, explicitly obtain both the adjacency spectrum and the signless Laplacian spectrum of $ \varGamma(R)$. Furthermore, we prove that the Laplacian eigenvalues, as well as the distance eigenvalues, of these graphs are all integers.

In the paper we study expansiveness along distinguished subsets in the case of a continuous action of the discrete Heisenberg group on a compact metric space $(\mathbb X,ρ)$. Transferring the ideas proposed by Boyle and Lind for continuous actions of $\mathbb{Z}^D$, we embed the acting group in the (continuous) $(2D+1)$-dimensional Heisenberg group $\mathcal H$ and define expansive subsets of $\mathcal H$. We focus on the expansiveness of vertical subgroups of the Heisenberg group. In particular, we show that, if only the space $\mathbb X$ is infinite, the center of $\mathcal H$ cannot be expansive, and that there always exists at least one nonexpansive $2D$-dimensional vertical subgroup.

We develop a triangular formulation of the hierarchical Poincaré-Steklov (HPS) method for elliptic partial differential equations on surfaces, allowing high-order discretizations on unstructured meshes and complex geometries. Classical HPS formulations rely on high-order quadrilateral meshes and tensor-product spectral discretizations, which enable efficient algorithms but restrict applicability to structured geometries. To overcome this restriction, we introduce a triangle-based hierarchical Poincaré-Steklov scheme (THPS) built on orthogonal Dubiner polynomial bases. As in the classical HPS framework, local solution operators and Dirichlet-to-Neumann maps are constructed and merged hierarchically, yielding a fast direct solver with $O(N \log N)$ complexity for repeated solves on meshes with $N$ elements. The reuse of precomputed operators makes the method particularly effective for implicit time-stepping of surface PDEs. Numerical experiments demonstrate that the proposed method retains spectral accuracy and achieves high-order convergence for a range of static and time-dependent test problems.

We study the functoriality of the local theta correspondence for classical $p$-adic groups. This is realized via the adaptation of the Adams conjecture to ABV-packets. We provide evidence for the conjecture, especially in the case of general linear groups.