We prove that for each fixed $m \ge 2$, there are only finitely many disjoint covering systems with minimum modulus at least $3$ in which precisely one modulus is repeated, namely the largest modulus, and it occurs exactly $m$ times.

Flat bands in driven-dissipative systems offer a route to engineer strongly localized, long-lived excitations, yet their selective population via incoherent pumping remains an open challenge. We study a one-dimensional chain of coupled lasing dimers arranged in a cross-stitch geometry and show that the synchronization regime of the individual dimers, controllable through pump intensity or inter-resonator distance, determines the character of the flat band hosted by the chain. In the in-phase (ferromagnetic) regime, the flat band appears as a subdominant, damped mode in the linear excitation spectrum. In the antiphase (antiferromagnetic) regime, by contrast, the dimers decouple and the flat band becomes the dominant, neutrally stable mode: it corresponds to an infinite family of Goldstone modes arising from the independent phase rotations of non-interacting dimers, and its compact localized states are directly observable in the noise response spectrum. Switching between these two regimes via pump control constitutes a pump-induced phase transition of the lasing lattice. Our results establish synchronization engineering as a practical mechanism for selective flat-band population in driven-dissipative optical systems, and open new avenues for studying flat-band physics, including nonlinear effects, Fano resonances, and excitation coherence in experimentally accessible laser and polariton platforms.

Computability logic (CoL) provides a semantic foundation in which formulas represent interactive computational problems and validity corresponds to uniform algorithmic solvability. Building on this foundation, clarithmetics -- CoL-based axiomatic number theories -- combine the full arithmetical strength of Peano arithmetic (PA) with explicit control over computational resources. In contrast to traditional bounded arithmetic and related complexity-oriented systems, they strengthen rather than weaken PA. This paper, after briefly surveying the relevant fragment of CoL, presents the systems CLA4-CLA7 and CLA11 of clarithmetic, and outlines their soundness and completeness with respect to natural classes of time, space, and so called amplitude complexities. We argue that, by weakening PA, traditional complexity-oriented systems of arithmetic throw out the baby with the bathwater, discarding large amounts of innocent and useful arithmetical information and losing intensional flexibility essential for natural specification and program extraction. Clarithmetics avoid this loss while supporting direct extraction of optimal or near-optimal algorithms from proofs and providing strong intensional completeness properties absent from bounded arithmetic and related systems. A central message of the paper is that the argued advantages of clarithmetics deserve either acknowledgment or serious refutation from the weak-arithmetics community. To date, neither has occurred.

In this paper we consider a one-dimensional reaction-diffusion model with piecewise continuous reaction term that describes propagation of autoignition fronts in reactive co-flow jets in a certain parametric regime. The model is reduced to a free boundary problem with two interfaces. It is shown that this problem admits permanent traveling front solution which is unique up to translations. The result is obtained using dynamical system approach employing Stable Manifold Theorem and the Melnikov integral as the main tools.

Mathematics curriculums at most universities tend to perpetuate a belief that higher mathematics is historically and culturally European. First Nations and minority students may not see their identities and cultures reflected in the discipline, yet university mathematics educators are keen to diversify and broaden the appeal of their courses. This article presents an investigation on the mathematics of smoke telegraphy, as a contribution to inlaying cross-cultural mathematical heritage in the curriculum. Across Indigenous societies of Australia the technology and practice of smoke telegraphy was developed to a sophisticated level over millennia to fill a need for long-distance communications. Through an original bibliographic and archival analysis, we show that smoke signalling and telegraphy used empirical mathematics of symmetries, frequency coding, and understanding of fluid dynamics. We juxtapose this applied mathematical knowledge, within context, against the timeline of Western understanding and development of these strands of mathematics.

Let $K$ be a $p$-adic field and $T$ a lattice in a semistable representation of $\mathrm{Gal}(\overline{K}/K)$ with Hodge-Tate weights in $[0, r]$. Assuming $0\leq r<p-1$, we prove that for a semistable extension of $\mathbb{Z}_p$ by $T$, the corresponding sequence of strongly divisible modules is exact. The same statement is proved for Breuil-Kisin modules for all $r\geq 0$. In the crystalline case, we deduce that the integral Bloch-Kato Selmer group $H^1_f(K, T)$ is computed by $\mathrm{Ext}^1$ in the category of crystalline strongly divisible modules. Using further exactness results, we define a tensor product of strongly divisible modules, which commutes with the functors to Galois representations. As an application, we show that for abelian varieties $A_1, A_2$ over $K$ with good reduction, the cup product map $δ_1\cupδ_2:A_1(K)\otimes A_2(K)\rightarrow H^2(K, T_p(A_1)\otimes T_p(A_2))$ induced by the Kummer sequences of $A_1, A_2$ factors through an $\mathrm{Ext}^2$ group of strongly divisible modules.

Large Language Models (LLMs) generate responses based on user prompts. Often, these prompts may contain highly sensitive information, including personally identifiable information (PII), which could be exposed to third parties hosting these models. In this work, we propose a new method to sanitize user prompts. Our mechanism uses the randomized response mechanism of differential privacy to randomly and independently perturb each character in a word. The perturbed text is then sent to a remote LLM, which first performs a prompt restoration and subsequently performs the intended downstream task. The idea is that the restoration will be able to reconstruct non-sensitive words even when they are perturbed due to cues from the context, as well as the fact that these words are often very common. On the other hand, perturbation would make reconstruction of sensitive words difficult because they are rare. We experimentally validate our method on two datasets, i2b2/UTHealth and Enron, using two LLMs: Llama-3.1 8B Instruct and GPT-4o mini. We also compare our approach with a word-level differentially private mechanism, and with a rule-based PII redaction baseline, using a unified privacy-utility evaluation. Our results show that sensitive PII tagged in these datasets are reconstructed at a rate close to the theoretical rate of reconstructing completely random words, whereas non-sensitive words are reconstructed at a much higher rate. Our method has the advantage that it can be applied without explicitly identifying sensitive pieces of information in the prompt, while showing a good privacy-utility tradeoff for downstream tasks.

In this paper, we construct novel first- and second-order decoupled schemes for the Navier-Stokes equations based on the penalty method and the sequential regularization method (SRM), respectively. These schemes do not require the boundary condition on the pressure and thus preserve the original velocity boundary conditions. By using the idea of the scalar auxiliary variable (SAV), the nonlinear terms of these schemes are treated explicitly, which improves computational efficiency while maintaining stability. It is important to note that we carefully reformulated the Navier-Stokes system to ensure convergence of the proposed scheme without any restriction on the time step. For the Penalty-SAV (P-SAV) schemes, at each time step it is only necessary to solve elliptic equations with constant coefficients. We prove the high-order stability (high-order regularities of the solution) of the schemes, and establish an unconditional (without time step constraints) global optimal error estimate in two dimensions as well as a local error estimate in three dimensions for the first-order scheme. Furthermore, to more accurately approximate the incompressibility constraint without introducing extra stiffness into the system, the sequential regularization-SAV (SR-SAV) schemes are developed, and their error estimates are provided. In addition, we compare our proposed scheme with the classic linearized projection scheme to demonstrate its accuracy and efficiency.

A framework for probabilistic forecasting of vessel motion is developed and validated for a semisubmersible operating in long period swell. Bayesian statistical methods are applied to predictions of the heave response from a physics model using numerical wave spectra and measured motion data. Model diagnoses motivate an additional level of complexity required for the error structure in the Bayesian model, specifically to account for heteroskedasticity and time-correlated errors. The hybrid model forecasts were evaluated during periods where the heave resonance and cancellation frequencies were excited. The method is demonstrated to be effective for providing reliable quantification of uncertainty and correcting bias in the raw physics model predictions. This justifies its value for improving the efficiency and safety of offshore operations.

Proton exchange membrane fuel cells (PEMFCs) are a promising clean energy technology, offering high efficiency and near-zero operational emissions for stationery and automotive applications. However, their widespread adoption remains limited by insufficient durability, driven by the degradation of the catalyst layer and proton exchange membrane under realistic operating conditions. While the macroscopic consequences of degradation are well established experimentally, the atomistic and molecular mechanisms that initiate and propagate failure remain incompletely understood. This review synthesizes recent advances in computational modelling, spanning density functional theory, molecular dynamics, and emerging machine learning potentials, to examine how chemical, mechanical, electrochemical, and contamination driven degradation mechanisms operate across multiple length and time scales. Key topics include radical-induced membrane degradation, platinum dissolution and carbon support corrosion, mechanical fatigue under electrical and hygrothermal cycling, and the impact of ionic and gaseous contaminants. A central finding is that these degradation pathways are not independent, but form strongly coupled feedback loops that no existing computational framework has been designed to capture this coupling simultaneously. Future directions are proposed, with emphasis on multiscale modelling frameworks and the application of machine learning interatomic potentials to the electrified interface.

We introduce tropical singular intersection homologies (non-GM and GM) with the tropical coefficients on rational polyhedral spaces using their filtrations. We investigate their fundamental properties. In the non-GM case, we give a Poincaré duality and two bilinear pairings analogous to the cup and cap products under some assumptions. We compute the homologies in some cases.

We consider the initial-boundary value problem of a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities in a bounded domain $Ω\subset\mathbb{R}^3$. Our first aim is to study the energy equality for global weak solutions by establishing mixed $L_t^qL_x^r$-regularity conditions on the velocity field and its gradient, under which the global weak solution conserves its energy for all times. The proof is based on the propagation of regularity for weak solutions to the convective Cahn-Hilliard equation with a physically relevant Flory-Huggins-type potential, combined with global mollification and boundary cut-off techniques. Next, we prove the existence and uniqueness of global strong solutions in the general setting with non-constant gradient energy coefficient and non-degenerate mobility, provided that the initial velocity is sufficiently small and the initial phase-field variable is a sufficiently small perturbation of a local minimizer of the free energy. This yields Lyapunov stability for each steady state consisting of a zero velocity together with a local energy minimizer. The proof relies on the energy equality for (local) strong solutions and the Łojasiewicz-Simon approach.

RRAM-based in-memory computing (IMC) offers high energy efficiency but suffers from conductance drift that severely degrades long-term accuracy. Existing approaches including retraining, noise-aware training, and Batch Normalization (BN)-based calibration either require RRAM rewriting, demand large storage overhead, or rely on online correction. We propose VeRA+, a lightweight drift compensation framework that reuses shared projection matrices and introduces only two compact drift-specific vectors per drift level. A drift-aware scheduling algorithm offline-trains a small set of VeRA+ parameters and selects the appropriate set over time without any on-chip retraining or data replay. VeRA+ preserves up to 99.77% of the drift-free accuracy after ten years of simulated drift and reduces storage overhead by more than three orders of magnitude compared with BN-based calibration. To validate VeRA+ under realistic device behavior, we extract one-week drift statistics from measurements on our fabricated 1T1R RRAM devices and use them to simulate realistic drifted weights. Under these measured drift conditions, VeRA+ achieves accuracy close to the drift-free baseline, providing an efficient and practical solution for long-term drift resilience in RRAM-IMC.

Repairing faulty quantum circuits is challenging and requires automated solutions. We present QRep, an automated repair approach that iteratively identifies and repairs faults in a circuit. QRep uniformly applies patches across the circuit and assigns each gate a suspiciousness score, reflecting its likelihood of being faulty. It then narrows the search space by prioritising the most suspicious gates in subsequent iterations, increasing the repair efficiency. We evaluated QRep on 40 (real and synthetic) faulty circuits. QRep completely repaired 70% of them, and for the remaining circuits, the actual faulty gate was ranked within the top 44% most suspicious gates, demonstrating the effectiveness of QRep in fault localisation. Compared with two baseline approaches, QRep scales to larger and more complex circuits, up to 13 qubits.

Bitcoin already faces a quantum threat through Shor attacks on elliptic-curve signatures. This paper isolates the other component that public discussion often conflates with it: mining. Grover's algorithm halves the exponent of brute-force search, promising a quadratic edge to any quantum miner of Bitcoin. Exactly how large that edge grows depends on fault-tolerant hardware. No prior study has costed that hardware end to end. We build an open-source estimator that sweeps the full attack surface: reversible oracles for double-SHA-256 mining and RIPEMD-based address preimages, surface-code factory sizing, fleet logistics under Nakamoto-consensus timing, and Kardashev-scale energy accounting. A parametric sweep over difficulty bits b, runtime caps, and target success probabilities reveals a sharp transition. At the most favourable partial-preimage setting (b = 32, 2^224 marked states), a superconducting surface-code fleet still requires about 10^8 physical qubits and about 10^4 MW. That load is comparable to a large national grid. Tightening to Bitcoin's January 2025 mainnet difficulty (b about 79) explodes the bill to about 10^23 qubits and about 10^25 W, approaching the Kardashev Type II threshold. These numbers settle a narrower question than "Is Bitcoin quantum-secure?" Once Grover mining is lifted from asymptotic query counts to fault-tolerant physical cost, practical quantum mining collapses under oracle, distillation, and fleet overhead. To push mining into non-trivial consensus effects, one must invoke astronomical quantum fleets operating at energy scales that lie far above present-day civilization.

We show that any multiple-valued function can be represented by a linear lambda term typed in a second-order polymorphic type system, using two distinct styles. The first is a circuit style, which mimics combinational circuits in switching theory. The second is an inductive style, which follows a more traditional mathematical approach. We also discuss several optimizations for these representations. Furthermore, we present a case study that demonstrates the potential applications of our approach across various domains.

Sorkin's impossible operations demonstrate that causality of a quantum channel in QFT is an additional constraint on quantum operations above and beyond the locality of the channel. What has not been shown in the literature so far is how much of a constraint it is. Here we answer this question in perhaps the strongest possible terms: the set of causal channels is nowhere dense in the set of local channels. We connect this result to quantum information, showing that the set of causal unitaries has Haar measure $0$ in the set of all unitaries acting on a lattice. Finally, we close with discussion on the implications and connections to recent QFT measurement models.

Most Rate-Splitting Multiple Access (RSMA) implementations rely on successive interference cancellation (SIC) at the receiver, whose performance is inherently limited by error propagation during common-stream decoding. This paper addresses this issue by developing a SIC-free RSMA receiver based on joint demapping (JD), which directly evaluates bit vectors over a composite constellation. Using a two-user Multiple-Input Single-Output (MISO) prototype, we conduct over-the-air measurements to systematically compare SIC and JD-based receivers. The results show that the proposed SIC-free receiver provides stronger reliability and better practicality over a wider operating range, with all observations being consistent with theoretical expectations.

This paper presents a port-Hamiltonian formulation of hysteretic energy storage elements. First, we revisit the passivity property of backlash-driven storage elements by presenting a family of storage functions associated to the dissipativity property of such elements. We explicitly derive the corresponding available storage and required supply functions `a la Willems [1], and show the interlacing property of the aforementioned family of storage functions sandwiched between the available storage and required supply functions. Second, using the proposed family of storage functions, we present a port-Hamiltonian formulation of hysteretic inductors as prototypical storage elements in port-Hamiltonian systems. In particular, we show how a Hamiltonian function can be chosen from the family of storage functions and how the hysteretic elements can be expressed as port-Hamiltonian system with feedthrough term, where the feedthrough term represents energy dissipation. Correspondingly, we illustrate its applicability in describing an RLC circuit (in parallel and in series) containing a hysteretic inductor element.

This article studies randomization inference for treatment effects in randomized controlled trials with attrition, where outcomes are observed for only a subset of units. We assume monotonicity in reporting behavior as in \cite{lee2009training} and focus on the average treatment effect for always-reporters (AR-ATE), defined as units whose outcomes are observed under both treatment and control. Because always-reporter status is only partially revealed by observed assignment and response patterns, we propose a worst-case randomization test that maximizes the randomization p-value over all always-reporter configurations consistent with the data, with an optional pretest to prune implausible configurations. Using studentized Hajek- and chi-square-type statistics, we show the resulting procedure is finite-sample valid for the sharp null and asymptotically valid for the weak null. We also discuss computational implementations for discrete outcomes and integer-programming-based bounds for continuous outcomes.

Quantum optics provides a fundamental framework for understanding the interaction between light and matter at the quantum level. Recently, it has been shown that under incoherent pumping, the resonance fluorescence spectrum dramatically changes. Engineering the resonance fluorescence spectrum paves the way towards solid-state-based single-photon sources. In this paper, we start by reviewing and reproducing some of the results concerning the resonance fluorescence spectrum, single-photon sources, dressed-state lasers, and luminescence spectrum of a quantum dot in a microcavity. Photon correlations in quantum optical systems and spectral properties of radiation emitted by atomic and semiconductor systems interacting with external fields are investigated. The well known Mollow triplet structure of the emission spectrum is discussed, together with the role of dressed states in explaining the origin of the three spectral peaks. Furthermore, the luminescence spectra of quantum emitters coupled to microcavities are reviewed. The numerical results presented here contribute to the theoretical understanding of resonance fluorescence, photon correlations, and engineered emission in quantum optical systems. These studies highlight the rich physical properties arising from light matter interaction at the quantum level and demonstrate their relevance for emerging quantum technologies.

Missing data is a ubiquitous challenge in data analysis, often leading to biased and inaccurate results. Traditional imputation methods usually assume that the missingness mechanism is missing-at-random (MAR), where the missingness is independent of the missing values themselves. This assumption is frequently violated in real-world scenarios, prompted by recent advances in imputation methods using deep learning to address this challenge. However, these methods neglect the crucial issue of nonparametric identifiability in missing-not-at-random (MNAR) data, which can lead to biased and unreliable results. This paper seeks to bridge this gap by proposing a novel framework based on deep latent variable models for MNAR data. Building on the assumption of conditional no self-censoring given latent variables, we establish the identifiability of the data distribution. This crucial theoretical result guarantees the feasibility of our approach. To effectively estimate unknown parameters, we develop an efficient algorithm utilizing importance-weighted autoencoders. We demonstrate, both theoretically and empirically, that our estimation process accurately recovers the ground-truth joint distribution under specific regularity conditions. Extensive simulation studies and real-world data experiments showcase the advantages of our proposed method compared to various classical and state-of-the-art approaches to missing data imputation.

Classically scale-invariant (and perturbative) theories provide a way to understand large hierarchies, as scales are generated through dimensional transmutation. They always lead to first-order phase transitions, since symmetries are radiatively broken, and they generically feature quasi-flat potentials, which are suitable for inflation. We construct a simple but fully realistic model of this kind that accounts for all observational evidence of new physics and is remarkably compatible with the most recent constraints on inflationary observables from both the Planck/BICEP/Keck and the Atacama Cosmology Telescope (ACT) collaborations. This model illustrates how classical scale invariance generically leads to a non-standard cosmology in which inflation occurs in two stages: a slow-roll stage and a thermal stage, separated by a radiation-dominated era.

We introduce a hybrid spatiotemporal logic for automotive safety applications (HSTL), focused on highway driving. Spatiotemporal logic features specifications about vehicles throughout space and time, while hybrid logic enables precise references to individual vehicles and their historical positions. We define the semantics of HSTL and provide a baseline model-checking algorithm for it. We propose two optimized model-checking algorithms, which reduce the search space based on the reachable states and possible transitions from one state to another. All three model-checking algorithms are evaluated on a series of common driving scenarios such as safe following, safe crossings, overtaking, and platooning. An exponential performance improvement is observed for the optimized algorithms.

By exploiting small-scale structure formation probed by Lyman-$α$ forest observations, we study constraints on a model of dark matter from dark sector decay. We compute the phase space distribution of the dark matter and the linear matter power spectrum. We map the non-thermal dark matter distribution in this dark matter model to an approximate thermal warm dark matter distribution, and use this approximation to obtain a constraint from the Lyman-$α$ forest observation. We combine the latest Lyman-$α$ forest bounds with the constraint from the Big Bang Nucleosynthesis. As these two probes offer highly complementary constraints, we impose strong limits on sub-GeV dark matter. Consequently, masses lighter than $\sim 10^{-1}$ GeV are excluded, thereby significantly limiting the allowed parameter space. More broadly, our findings demonstrate the utility of small-scale structure observations in testing non-thermal dark matter paradigms, offering valuable insights for exploring a wider class of late-time decay models.

Spin-phonon interactions have a dual role in emerging spin-based quantum technologies. While they can be a limitation to device performance through decoherence, they also serve as a critical resource for coherent spin control, detection, and the realization of spin-based quantum networks. However, their direct characterization remains a challenge and is usually material-dependent. Here, we introduce a technique to probe spin-phonon coupling at millikelvin temperatures and gigahertz frequencies, using high-overtone bulk acoustic wave resonators (HBARs) integrated with arbitrary crystals via visco-elastic transfer of thin-film lithium niobate transducers. By tuning the Larmor frequency of dilute spin ensembles into resonance with HBAR modes, we extract the anisotropy and strength of spin-phonon interactions from acoustic dispersion and dissipation measurements. We demonstrate this approach in calcium tungstate (CaWO4) and yttrium orthosilicate (Y2SiO5), achieving cooperativities up to 0.5 for erbium dopant ensembles. Our method enables the study of spin-phonon interactions in complex crystalline materials, with minimal fabrication constraints. These results will facilitate the design of hybrid quantum systems and the quest for ion-matrix combination with enhanced spin-phonon coupling.

The first generation of students is learning to program alongside GenAI (Generative Artificial Intelligence) tools, raising questions about how young learners critically engage with them and perceive ethical responsibilities. While prior research has focused on university students or developers, little is known about secondary school novices, who represent the next cohort of software engineers. To address this gap, we conducted an exploratory study with 84 German secondary school students aged 16-19 attending software development workshops. We examined their critical thinking practices in AI-assisted programming, perceptions of AI ethics and responsibility, and gender-related differences in their views. Our results reveal an AI paradox: students demonstrate strong ethical reasoning and awareness about AI, yet many report integrating AI-generated code without a thorough understanding of it. The majority of our cohort attributed significant responsibility for AI practices to politics and corporations, potentially reflecting Germany's cultural context, with its strict regulations and data privacy discourse. Boys reported more frequent and experimental use of AI-assisted programming, whereas girls expressed greater scepticism and emphasised peer collaboration over GenAI assistance. Our findings highlight the importance of culturally responsive software engineering education that strengthens critical AI literacy in AI-assisted programming by linking ethics to concrete code artefacts and preparing young learners for this AI-driven software landscape.

We investigate spin-wave modes in confined ferromagnetic resonators with spherical and cylindrical geometries across the exchange-dominated, dipole-exchange, and dipolar interaction regimes. Starting from the linearized Landau-Lifshitz-Gilbert equation, we show that the projection of the total angular momentum and mirror parity are conserved quantities in the problem of axially symmetric resonators. These symmetries provide a natural classification of spin-wave modes and explain the degeneracy of exchange modes, as well as its lifting by dipolar interactions. Numerical analysis shows that the nonlocal dipolar interaction removes the exchange degeneracy and hybridizes modes, leading to avoided crossings between modes that belong to the same symmetry sector. To describe this behavior, we develop a coupled-mode theory formulated directly in terms of dynamical magnetization, which reduces the dipole-exchange problem to a finite system of interacting modes. The resulting framework provides a unified description of spin-wave spectra in confined magnetic particles from the exchange limit to the dipolar regime.

High-order interpolation on the Grassmann manifold $\Gr(n, p)$ is often hindered by the computational overhead and derivative instability of SVD-based geometric mappings. To solve the challenges, we propose a stabilized framework that combines Maximum-Volume (MV) local coordinates with Arnoldi-orthogonalized polynomial bases. First, manifold data are mapped to a well-conditioned Euclidean domain via MV coordinates. The approach bypasses the costly matrix factorizations inherent to traditional Riemannian normal coordinates. Within the coordinate space, we use the Vandermonde-with-Arnoldi (V+A) method for Lagrange interpolation and its confluent extension (CV+A) for derivative-enriched Hermite interpolation. By constructing discrete orthogonal bases directly from the parameter nodes, the solution of ill-conditioned linear system is avoided. Theoretical bounds are established to verify the stability of the geometric mapping and the polynomial approximation. Extensive numerical experiments demonstrate that the proposed MV-(C)V+A framework can produce highly accurate approximation in high-degree polynomial interpolation.

Although climate and nature related scenario analysis is increasingly important in finance, operational implementations remain limited for translating long horizon environmental scenarios into counterparty credit risk measures used in pricing and regulatory capital. We propose an environmental valuation adjustment framework for CVA with three components: (i) a scenario to credit translation that maps environmental scenario drivers into hazard rates; (ii) nature specific tail generators that quantify model risk in scenario generation; and (iii) a distributionally robust wrong way risk bound based on Kullback Leibler (KL) divergence. We compute climate CVAs using transition scenarios and nature CVAs using biodiversity indicators. Our results show that nature CVAs can vary materially across alternative ecosystem generators, highlighting an additional source of model uncertainty.