We characterize the boundedness and compactness of dyadic paraproducts on local dyadic fractional Sobolev spaces. Our conditions are stated in terms of new dyadic fractional BMO and CMO conditions involving the dyadic fractional Sobolev capacity, and our proofs use a new dyadic fractional version of the Carleson embedding theorem.

The Traveling Salesman Problem (TSP) is one of the classic and hard problems in combinatorial optimization. We develop a new heuristic that uses a connection between Minimum Cost Flow Problems and the TSP to improve on a given suboptimal tour, such as a local optimum found using a classic heuristic. Minimum Cost Flow Problems can be solved efficiently through linear programming or combinatorial algorithms based on cycle canceling. We investigate the potential of flow-canceling in the context of the TSP. Through a restriction of the search space to cycles and circulations that alternate between arcs in- and outside of the tour, practical results exhibit that only a low number of subtours is created, and a lightweight patching step suffices for a high success rate and gap closure towards an optimum.

We present a theoretical analysis of frictional transitions along the Stribeck curve for rough elastic contacts lubricated by a Newtonian fluid. Building on the mean-field framework of Persson and Scaraggi (J. Phys.: Condens. Matter 21 (2009) 185002), we formulate a minimal elastohydrodynamic model that couples contact mechanics and lubrication through a homogenized pressure decomposition. Dimensional analysis reveals three independent dimensionless parameters governing the frictional response, which correspond to a dimensionless speed, normal load, and surface roughness. Using asymptotic expansions, we first characterize the boundary and hydrodynamic lubrication regimes, which arise naturally as the quasistatic and smooth-surface limits of the model. In both limits, the contact morphology converges toward Hertzian contact in the regime of large elastic deformation, with boundary layers regularizing the separation profile at the edge of the contact zone. We then analyze the mixed lubrication regime and derive asymptotic expressions for the friction coefficient in the low- and high-speed limits. At high speeds, friction decomposes into a viscous contribution and a residual contact term, leading to a roughness- and load-dependent criterion for the transition to hydrodynamic lubrication that departs from constant-Λ ratio theories. At low speeds, friction reduction results from the progressive redistribution of the applied load between asperity contact and hydrodynamic pressure, yielding a characteristic transition speed from boundary to mixed lubrication. These results are summarized in a phase diagram that generalizes the classical Stribeck curve to a multidimensional parameter space.

We investigate static, spherically symmetric halo configurations within Unified Dark Matter (UDM) scalar-field models, developing a systematic mapping between standard cold dark matter (CDM) density profiles and their UDM counterparts. Exploiting the equivalence-class structure of UDM models, we show that, in principle, different Lagrangian realisations can share the same weak-field rotation curve while exhibiting distinct field properties. We reconstruct the effective energy density, radial and tangential pressures from a phenomenological circular velocity profile, ensuring the absence of ghosts and instabilities and the preservation of the Null Energy Condition (NEC). Applying our procedure to several commonly used CDM halo profiles -- including Persic, Salucci \& Stel, NFW, and Burkert models -- we demonstrate that their phenomenological success can be retained within a relativistic UDM framework, reproducing the observed flatness of rotation curves without introducing separate dark matter and dark energy components.

Large-eddy simulations (LES) with an appropriate subgrid-scale (SGS) model provide a powerful tool for investigating real-world turbulence. The Smagorinsky model, one of the simplest and most used SGS models, often shows an over-dissipative behavior even when using dynamic procedures to adjust the model coefficient. By incorporating the structural or geometrical information of turbulence provided by helicity (velocity-vorticity correlations), the helicity SGS model is expected to alleviate these issues in the standard Smagorinsky framework, in which only information of turbulence intensity is considered through the turbulent energy. The validity of helicity SGS models is investigated here with the aid of direct numerical simulations (DNSs). Using configurations with and without net rotation, and with large-scale helicity gradients sustained by a mechanical forcing, we show that to better model SGS turbulence, SGS helicity effects should be incorporated into the model together with the Smagorinsky-like eddy viscosity.

Background: Right anomalous aortic origin of coronary arteries (R-AAOCA) involves fixed compression, assessable with adenosine-derived fractional flow reserve (FFRAdnosine), and additional stress-induced dynamic compression captured by dobutamine-derived FFR (FFRDobutamine). We hypothesized that coronary CT angiography (CCTA)-derived fluid dynamics-informed parameters outperform conventional metrics in predicting both FFR types. Methods: We retrospectively analyzed CCTA data from R-AAOCA patients who underwent invasive FFRAdnosine and FFRDobutamine assessment. Parameters were categorized as: (1) conventional metrics (cross-sectional area, perimeter, minor/major axis, intramural lumen area [ILA], effective diameter, area and diameter stenosis ratios) and (2) fluid dynamics-informed metrics (hydraulic diameter, elliptic ratio, circularity, hydraulic diameter stenosis ratio, resistance index [RI], ostial angulation penalty [OAP], and comprehensive stenosis score [CSS]). Hemodynamic relevance was defined as FFR$\leq$0.80. Results: 81 patients were included. FFRAdnosine$\leq$0.80 occurred in 5 (6.2%) and FFRDobutamine$\leq$0.80 in 16 (19.8%) patients. For FFRAdnosine, top discriminators were RI (AUC=0.97), OAP (AUC=0.97), ostial area (AUC=0.96), and CSS (AUC=0.95). For FFRDobutamine, ostial minor diameter led (AUC=0.85), followed by RI (AUC=0.83) and ILA minor diameter (AUC=0.81). RI explained 45% and 43% of FFRAdnosine and FFRDobutamine variance, respectively. At optimal thresholds, RI achieved 100% sensitivity and 95% specificity for FFRAdnosine; ostial minor diameter achieved 100% sensitivity and 57% specificity for FFRDobutamine. Conclusions: In R-AAOCA, CCTA-derived fluid dynamics-informed metrics provide excellent and superior performance compared with conventional geometric parameters in predicting hemodynamic relevance of fixed compression.

Incomplete pairwise comparison matrices are increasingly employed to save resources and reduce cognitive load by collecting only a subset of all possible pairwise comparisons. We present their graph representation and some completion algorithms, including the incomplete eigenvector and incomplete logarithmic least squares methods, as well as a lexicographical minimisation of triad inconsistencies. The issue of ordinal violations is discussed for matrices generated by directed acyclic graphs and the best--worst method. We also show a reasonable approach to generalise the inconsistency threshold based on the dominant eigenvalue to the incomplete case, and state recent results on the optimal order of obtaining pairwise comparisons. The benefits of using incomplete pairwise comparisons are highlighted by several applications.

Rotation and magnetic fields in the cores of evolved massive stars in their final phase are thought to play an important role in the subsequent supernova explosion and the formation of a compact object, especially in hyperenergetic explosions. However, the interplay between rotation, magnetic fields, and convection up to the final collapse is a nonlinear, multidimensional effect that is difficult to capture with standard one-dimensional (1D) stellar evolution models. We quantify the magnetic angular momentum (AM) transport in the convective oxygen burning shell in a three-dimensional (3D) rotating core-collapse progenitor model. We find that the radial direction of magnetic AM transport is directly related to the Rossby number of the convective oxygen shell. We also analyze the magnetic energy, which sets the amplitude of the magnetic AM flux. The magnetic energy is determined both by rotation and the nuclear energy generation rate analogously to low-mass stars like the Sun. Based on these results, we construct a 1D model of magnetic AM transport in the convection zone for the first time in terms of properties of a given stellar evolution model. This model successfully reproduces the AM transport in the 3D model when the convective dynamo is in a quasi-steady state. Notably, our model for radial AM transport is the first to account for inward AM flux. This may result in interesting differences compared to the conventional treatment of magnetic AM transport in stellar evolution models, which assume AM is transported outward by a purely diffusive process.

Transient absorption spectroscopy is routinely used to study the electron dynamics in plasmonic gold nanoparticles. Typically, the transient absorption bleach is analyzed as measure for the electron temperature. However, the implicitly assumed linear dependence between bleach intensity and temperature has not been systematically studied. Similarly, the influence of lattice heating also lacks a detailed analysis. Here, we solve momentum-resolved metal Boltzmann-Bloch equations for a semi-analytic access to the temperature-dependent gold nanoparticle absorption. We confirm the theory with steady state and transient absorption experiments, define regions of linear correlation between transient absorption bleach intensity and electron temperature and reveal a strong impact of the lattice temperature on the TA bleach intensity.

This study examines the impact of residential energy retrofits on household energy consumption in France using smart meter data from nearly 2,500 Hello Watt users, using a two-period difference-in-differences design. The dataset combines daily electricity and gas consumption collected through smart meters, hourly temperatures from Météo France, and user-declared home and retrofit information. As a control, we use a group composed of homes of Hello Watt users that are similar to the treated homes, but did not undergo any renovations. The average treatment effect on the treated is estimated with the estimator of Sant'Anna & Zhao (2020). Estimates are reported by energy source (electricity vs. gas) and by retrofit type. The retrofit measures considered are limited to single interventions: wall insulation, attic insulation, floor insulation, installation of an air-to-air heat pump, or installation of an air-to-water heat pump. A comprehensive retrofit is defined separately as the simultaneous implementation of at least two of these measures. Our results show that insulation works cause a significant decrease in both electricity and gas consumption (3% to 13% and 5% to 16% respectively, depending on the retrofit type). We also estimate the reduction on the heating consumption only (7% to 27% for electrical heating and 7% to 19% for gas heating). We also study retrofits that consist in replacing a gas boiler with an air-to-water heat pump, resulting in a cut of 85% in carbon emissions.

Quantum electrodynamics (QED) becomes nonlinear when the magnetic field strength surpasses the critical Schwinger limit $B_Q \approx 4.41\cdot 10^{13}$ G. This limit is surpassed, for example, in the magnetospheres of a specific class of neutron stars known as magnetars, which has important consequences for magnetospheric plasma dynamics due to modifications in scattering cross sections. Using a formalism previously applied to the study of magnetic catalysis, I calculate the cross sections of all tree-level 1-to-2, 2-to-1, and 2-to-2 particle QED scattering processes that do not include a photon propagator. The calculations are done in a strong background magnetic field and the results are implemented into an open-source Python package. This article focuses on presenting the formalism and computational techniques required for the calculations, while the impact of the results on, e.g., magnetospheric plasma dynamics is discussed in a companion letter (Kiuru et al. 2026).

We study diagnostics of thermalization in quantum many-body systems with global SU(2) symmetry, where the standard eigenstate thermalization hypothesis (ETH) is generalized to its non-Abelian form. As an eigenstate-level probe, we introduce a symmetry-resolved trace distance constructed from the block structure of the reduced density matrix. This block structure separates spin-sector probabilities from configurational fluctuations within each sector, naturally leading to a decomposition into a probability trace distance and a configurational trace distance. The microcanonical average of the former is bounded by fluctuations of the corresponding spin-sector probabilities within a microcanonical energy window, whereas the latter captures finer intra-sector fluctuations. In non-Abelian thermalizing systems, these spin-sector-probability fluctuations are constrained by the non-Abelian ETH and therefore become exponentially suppressed with system size. Numerical studies of the one-dimensional \(J_1\)--\(J_2\) Heisenberg chain are consistent with this picture and suggest that, in the thermal regime, the trace distance is asymptotically dominated by the configurational trace distance.

Directed Acylic Graphs with a single entry vertex and a single exit vertex (st-DAGs) have many applications. For instance, they are frequently used for modelling flow problems or precedence conditions among tasks, work packages, etc.. This paper presents an algorithm for finding special types of subgraphs in such st-DAGs, called clusters. Knowing the clusters of a given st-DAG is very useful during DAG analysis. Clusters are characterized by a kind of synchronizing behaviour at their entry border and at their exit border. In this context, we introduce the notion of syncpoint, a type of synchronisation point within a DAG, and for a given st-DAG we construct a second DAG, called MSP-DAG, whose edges are given by the precedence relation among maximum size syncpoints (MSPs). Our new cluster finding algorithm searches for clusters between potential pairs of enclosing MSPs. The efficiency of the algorithm stems from the fact that it works on the MSP-DAG, which is usually much smaller than the original st-DAG. The paper includes a thorough complexity analysis of the algorithm's runtime, which turns out to be quadratic in the number of DAG vertices and exponential in the number of MSP-DAG vertices. There is also a section reporting on experiments with randomly generated DAGs, which shows the practical applicability of the algorithm and confirm our theoretical findings.

There is growing interest in anticipating critical transitions in natural systems, often pursued through statistical detection of early warning signals associated with dynamical bifurcations. In stochastic dynamical systems, such signals commonly rely on manifestations of critical slowing down. However, we still need additional development for the underlying theory for critical transitions in non-autonomous systems. This extension is relevant for natural systems, whose behaviour often emerges from seasonal periodic forcing. In this study, we systematically investigate the feasibility of anticipating the termination of oscillatory behavior in a bistable system with slow periodic forcing. In this setting, existing approaches of estimating linear characteristics of the return map fail in practical scenarios due to the unfavourable time-scale separation. Instead, we propose two statistical indicators for the anticipation of critical transitions in the periodic behaviour: (i) conventional early warning indicators, such as increasing variance and autocorrelation, evaluated across system cycles, and (ii) indicators derived from the phase of the seasonal forcing. By statistically comparing their predictive performance, we find that phase-based indicators provide the strongest early warning capability. Our results offer guidance for the detection of critical transitions in periodically forced systems and, more broadly, systematically extend early-warning signs towards non-autonomous dynamical systems.

We present the first radiation-hydrodynamical simulations of the formation of a born-again planetary nebula (PN) triggered by a late thermal pulse (LTP). The 2D radiation-hydrodynamic simulations, performed with the {\sc pluto} code, have been consistently coupled to stellar evolution calculations using the Modules for Experiments in Stellar Astrophysics ({\sc mesa}) code. Very particularly the stellar evolution model uses (i) updated opacity tables for H-deficient, C-rich mixtures during the LTP, and (ii) a mass-loss prescription tailored for H-deficient [Wolf-Rayet]([WR])-type winds during the post-LTP phase. Our stellar model reproduces the nearly complete depletion of H expected after an LTP event, while matching the observed abundances and spectral types of iconic [WR]-type central stars of PNe. The simulations show for the first time that the H-deficient LTP ejecta forms a transient double-shell structure which, after $\sim$1000 yr, becomes fully mixed with the H-rich PN. The ejecta mass ($\sim3.4\times10^{-4}$~M$_\odot$) is too small to leave a lasting imprint on the nebular abundances, predicting H-rich PNe around [WR] central stars. The injection of LTP material into the hot bubble drives turbulence, clump formation, and enhanced mixing, providing an explanation to the larger expansion velocities and larger turbulent nebular structures of PNe with [WR] central stars compared to those with H-rich central stars. These results provide robust support for the born-again scenario as the origin of H-deficient [WR] central stars within H-rich PNe.

This paper is concerned with the local well-posedness, wave breaking, blow-up rate for a Camassa-Holm type equation with time-dependent weak dissipation. Firstly, we obtain the local well-posedness of solutions by using Kato's theory. Secondly, by using energy estimates, characteristic methods, and comparison principles, we derive two blowup criteria involving both pointwise gradient conditions and mixed amplitude-gradient conditions, and prove the blowup rate is universally $-2$. Our results extend wave breaking analysis to physically relevant variable dissipation regimes.

Lightcone observations are the natural data format of galaxy surveys, but their evolving geometry breaks the translational symmetry assumed by standard convolutional neural networks (CNNs). In particular, applying CNNs to 3D gridded lightcone data implicitly treats the line-of-sight direction as translationally invariant, despite encoding cosmic time evolution. We propose a simple alternative (CNN+2D) that divides the lightcone into redshift slices, projects each onto a HEALPix sphere, and analyzes them with a 2D CNN. Using \texttt{AbacusSummit} halo lightcone mocks ($0.3<z<0.8$, $40^\circ\times40^\circ$), we compare this approach with fully connected networks (FC) applied to different summary statistics, including spherical harmonic coefficients ($a_{\ell m}$), wavelet scattering transform (WST) coefficients, and the angular two-point correlation function (2PCF), along with standard 2PCF likelihood and Fisher forecasts. We find that multiple statistics beyond CNNs can achieve competitive performance: FC networks combined with $a_{\ell m}$ and especially WST significantly outperform 2PCF-based methods, with FC+WST yielding the best overall parameter constraints across cosmologies. Meanwhile, for a fiducial cosmology with multiple realizations, the CNN+2D approach achieves the smallest statistical uncertainties. These results demonstrate that both learned features and carefully constructed summary statistics can effectively extract cosmological information from lightcone data, providing flexible and robust analysis strategies for upcoming surveys such as DESI, Euclid, and CSST.

Fluorescence imaging is an essential diagnostic tool in many fields, but diffraction-limited optical imaging at depth is limited by scattering. Here, we present a method based on multiple random illuminations, combined with a computational framework that retrieves high-resolution images by aligning local speckle replicas and computing their pixel-wise variance. We demonstrate its versatility in two regimes: linear wide-field one-photon (1P) fluorescence imaging and nonlinear two-photon (2P) fluorescence imaging where the object is excited by a scanned speckle field and detected with a single-pixel detector. This approach outperforms standard autocorrelation techniques in terms of resolution and convergence.

Designing mobile and interactive technologies requires understanding how users sample dynamic environments to acquire information and make decisions under time pressure. However, existing computational user models either rely on hand-crafted task representations or are limited to static or non-interactive visual inputs, restricting their applicability to realistic, pixel-based environments. We present CR-Eyes, a computationally rational model that simulates visual sampling and gameplay behavior in Atari games. Trained via reinforcement learning, CR-Eyes operates under perceptual and cognitive constraints and jointly learns where to look and how to act in a time-sensitive setting. By explicitly closing the perception-action loop, the model treats eye movements as goal-directed actions rather than as isolated saliency predictions. Our evaluation shows strong alignment with human data in task performance and aggregate saliency patterns, while also revealing systematic differences in scanpaths. CR-Eyes is a step toward scalable, theory-grounded user models that support design and evaluation of interactive systems.

We compute non-eikonal corrections to dijet production in deep inelastic scattering off a nucleus. Such corrections are expected to be quantitatively important at the energies of the future Electron Ion Collider. We focus on those corrections stemming solely from the finite longitudinal size of the nucleus. For both longitudinally and transversely polarized photons, we provide general, all-order expressions in terms of two-dimensional path integrals. To proceed further, we use the harmonic oscillator approximation for the target averages of Wilson lines. We then expand the general expressions order by order beyond the shockwave limit which provides the eikonal results, up to next-to-next-to-eikonal accuracy. We observe that next-to-eikonal corrections to this observable vanish for the mentioned approximation for target averages, as previously found for single gluon production in proton-nucleus collisions. Finally, we calculate the back-to-back of correlation limit of our expressions.

Political discourse attributes the pressure on European welfare systems to foreign nationals. Yet projections of service demand rarely disaggregate service demand by citizenship status. We develop a structural demographic model and project healthcare, education, and housing demand in Austria through 2050, disaggregated by citizenship status and regions across migration scenarios. We find that migration, ageing, and fertility shape each sector differently. In healthcare, the ageing of Austrian nationals contributes 4.7 times more to demand growth than immigration, with the most acute pressures in rural, low-migration regions. In housing, migration accounts for the entire net growth in demand, concentrated in metropolitan hubs. In education, aggregate demand contracts regardless of migration assumptions, whereas future needs are driven more by the births of foreigners in Austria than by new arrivals. Foreign nationals consume services in proportion to their demographic weight, with deviations explained by age structure rather than over-utilisation. These results show that the drivers of service demand are sector-specific: migration restrictions could ease housing pressure, but would not address ageing-driven healthcare demand and may accelerate contraction in the education system.

Artificial intelligence (AI) is the name popularly given to a broad spectrum of computer tools designed to perform increasingly complex cognitive tasks, including many that used to solely be the province of humans. As these tools become exponentially sophisticated and pervasive, the justifications for their rapid development and integration into society are frequently called into question, particularly as they consume finite resources and pose existential risks to the livelihoods of those skilled individuals they appear to replace. In this paper, we consider the rapidly evolving impact of AI to the traditional questions of philosophy with an emphasis on its application in mathematics and on the broader real-world outcomes of its more general use. We assert that artificial intelligence is a natural evolution of human tools developed throughout history to facilitate the creation, organization, and dissemination of ideas, and argue that it is paramount that the development and application of AI remain fundamentally human-centered. With an eye toward innovating solutions to meet human needs, enhancing the human quality of life and expanding the capacity for human thought and understanding, we propose a pathway to integrating AI into our most challenging and intellectually rigorous fields to the benefit of all humankind.

Children's agency plays a critical role in shaping children's autonomy, participation, and well-being in their interactions with digital systems, particularly in emerging child-AI contexts. However, how designers currently understand and reason about children's agency in practice remains underexplored. In this paper, we examine designers's engagement with children's agency through a participatory workshop in which we introduce a design-for-agency framework that supports designers externalising the consideration of agency in their design contexts. We find that while participants are committed to implementing ethical AI systems for children, they often struggle to understand why agency matters and how it can be operationalised in practice. Our agency design framework provided designers with a structured way to translate implicit, experience-based judgments into explicit articulation of agency trade-offs while acknowledging the associated design complexity. We conclude by offering initial insights into supporting designers' reasoning about children's agency and outlining directions for future research.

Generative AI (GenAI) is increasingly used as a knowledge partner in higher education, raising the need for instructional designs that emphasize AI literacy practices such as evaluating output credibility and maintaining human accountability. Existing AI literacy frameworks focus more on what learners should do than on how these practices are enacted in routine student-GenAI collaboration. We address this gap by framing student-GenAI interaction as a transactive memory partnership, where credibility regulates reliance and verification. To make this process visible during coursework, we used a weaker large language model (LLM): small enough to run on most students' computers during class, helpful enough to support learning, but not so capable that it removes the need for verification. In an undergraduate STEM course, students were randomly assigned to one of three conditions across repeated activities: reflection-first (think first, then consult AI), verification-required (use AI, then evaluate the output), or control (unrestricted use). Students completed a transactive memory survey at three time points (N = 42). Weighted credibility diverged by condition over time. ANCOVA controlling for baseline credibility showed a condition effect at mid-semester, F(2, 38) = 4.02, p = .026, partial eta squared = .175, and a stronger effect at post-intervention, F(2, 38) = 5.48, p = .008, partial eta squared = .224; adjusted means were lowest in reflection-first, intermediate in verification-required, and highest in control. Parallel analyses of specialization and coordination were not significant. These findings suggest that workflow sequencing, deliberate use of weaker LLMs, and accountability cues embedded in assignment instructions can recalibrate students' credibility judgments in GenAI use, with reflection-first producing the strongest downward shift in reliance.

Binary black hole (BBH) mergers detected via gravitational waves are addressing key open questions in astrophysics, cosmology, and fundamental physics. Our scientific conclusions rely on extracting accurate source parameters, for which we require accurate signal modelling. It is well known that current BBH waveform models need to be improved for high-mass-ratio, strongly precessing systems, and in this paper we provide a concrete illustration of this issue, showing that the degradation in model performance is substantially more severe than might have been anticipated. We present numerical relativity (NR) simulations of precessing BBH systems with a mass ratio of 18 and a dimensionless spin of 0.8 on the larger black hole (with the smaller black hole non-spinning), covering five values of spin misalignment. We assess the accuracy of state-of-the-art waveform models in this region of parameter space by computing the standard mismatch between the models and the NR waveforms. We find that all current waveform models often exhibit significant mismatches ($\gtrsim$0.1), indicating poor performance in this regime. We also perform limited parameter estimation using a subset of state-of-the-art waveform models, injecting these NR simulations as signals into the three-detector LIGO-Virgo network. In some cases we find errors in mass measurements of over 100%, dramatically illustrating that substantial improvements are required in existing waveform models. The numerical simulations presented here will be valuable for calibrating future BBH waveform models in this region of parameter space.

A new form of human trafficking has emerged across Chinese borders, where individuals are lured to Southeast Asia with fraudulent job offers and then coerced into operating online scams. Despite its massive economic and human toll, this scam-driven trafficking remains underexplored in academic research. Through qualitative analysis of 158 RedNote posts, we examined how Chinese online communities respond to this threat. Our findings reveal that perpetrators exploit cultural ties to recruit victims for cybercriminal roles within self-sustaining compounds, using sophisticated manipulation tactics. Survivors face serious reintegration barriers, including family rejection, as the cultural values that enable trafficking also hinder their recovery. While communities present protective strategies, efforts are complicated by doubts about the reliability of support and cross-border coordination. We discuss key implications for prevention, platform governance, and international cooperation against scam-driven trafficking. Warning: This paper contains descriptions of physical, psychological, and sexual abuse.

It is well known that Hermitian and non-Hermitian models exhibit distinct physics and require different theoretical tools. In this work, we propose a unified generating-function framework for both classes with generic boundary conditions and local impurities. Within this framework, any finite lattice model can be mapped to a generating function of the form G(z)=P(z)/Q(z), where Q(z) and P(z) denote the bulk recurrence relation and boundary terms or impurities, respectively. The problem of solving for eigenstates reduces to a simple criterion based on the cancellation of zeros of Q(z) and P(z). Applying this method to the Hatano-Nelson (HN) model, we show how boundary conditions and impurities determine the location of the zeros, thereby demonstrating the boundary sensitivity of non-Hermitian systems. We further investigate topological edge states in the non-Hermitian Su-Schrieffer-Heeger (SSH) model and identify its topological phase transition. Inspired by generating-function techniques widely used in discrete mathematics, particularly in the study of the Fibonacci sequence, our results establish a direct connection between non-Hermitian physics and recurrence relations, providing a new perspective for analyzing non-Hermitian systems and exploring their connections with discrete mathematical structures.

This paper was originally written by the authors circa 2005 but was never submitted for publication. The present version corrects minor errors, adds references to work published since the original draft, and includes a section discussing further research directions. The core framework -- the property function $ψ_G$, the density function $φ^ψ_μ$, and all threshold results -- is unchanged from the original.

In this paper, we develop a general rough volatility model for commodities that provides an automatic calibration of the initial term structure of the futures prices and an appropriate treatment of the Samuelson effect. After the theoretical analysis of this general model, we focus on the rBergomi and rHeston models and their calibration to market data of vanilla futures options on WTI Crude Oil. Finally, numerical results illustrate the performance of the proposed rough volatility models for commodities pricing.

Algebraic multigrid (AMG) methods derive their optimal efficiency from the interplay between a relaxation process and a corresponding coarse grid correction. In many standard formulations, relaxation and coarse-graining are analyzed and treated as largely separate of one another. Here we propose an alternative theoretical approach centered entirely on the relaxation process, which exposes its fundamental role in the coarse-graining of the fine-scale problem. By treating the relaxation of the error as a dynamical system and applying a dimensional-reduction procedure analogous to the Mori-Zwanzig-Nakajima formalism, we derive exact expressions for the coarse-level equations and the interpolation operations, as well as a natural way of computing complementary transfer operators. We illustrate the unifying nature of this framework by recovering several well-known results for general non-symmetric systems, including ideal and optimal restriction and interpolation, as well as the limiting case of exact elimination. We further emphasize the pivotal importance of compatible-relaxation and identify dynamical corrections that naturally arise in our theory, which have the potential to enhance the convergence, robustness, and adaptivity of future algebraic multigrid methods.