We revisit Royen's proof of the Gaussian correlation inequality from a supersymmetric point of view. Many key elements in Royen's proof of this inequality have natural geometric interpretations in terms of supersymmetric dimensional reduction from $\mathbb{R}^{3|2}$ to $\mathbb{R}^{1|0}$. In particular, the auxiliary multivariate Gamma distributions appearing in Royen's Laplace-transform argument arise naturally as the body of a supersymmetric radial variable on $\mathbb{R}^{3|2}$. The generalization to the half-integer multivariate Gamma case also follows naturally as a dimensional reduction from $\mathbb{R}^{k+2|2}$ to $\mathbb{R}^{k|0}$. This provides an example in which the supersymmetric localization method is applied to prove correlation inequalities with continuous parameters.

We establish a connection between policy evaluation in Markov decision processes and PageRank in network analysis. For a fixed policy, we show that the value function of a discounted Markov decision process can be obtained, up to an explicit rescaling, from the PageRank vector of a suitably defined time-reversed Markov chain. In this correspondence, the discount factor plays the role of the teleportation parameter, while rewards induce the restart distribution. Beyond the irreducible case, invoking quasi-stationary distributions and Doob $h$-transforms, we prove a general decomposition theorem showing that policy evaluation for arbitrary finite MDPs reduces to a collection of PageRank problems on the recurrent and transient components of the policy-induced Markov chain. This framework naturally extends to undiscounted MDPs with terminal states and to transition-dependent rewards. We conclude by showing efficiency of our approach on a numerical example of a sticky random walk on large deterministic and random graphs.

Autonomous driving systems (ADS) are increasingly deployed in real traffic, yet testing remains fundamentally challenging due to open environments, complex scenarios, and the lack of established processes and metrics. Despite extensive research, a gap persists between academic advances and their applicability in industrial practice. To address this, we conduct an interactive rapid review in collaboration with 21 practitioners from a leading automotive company. Practitioners identified 12 key challenges in ADS testing, and prioritised two as the most critical issues, namely approaches to and completeness of testing for End-to-End (E2E) ADS. We analyzed 17 research studies relevant to these two challenges, most of which focus on generating critical testing scenarios, and subsequently assessed their relevance and applicability in practice. Our study provides the first practitioner-driven review and evaluation of current ADS testing research, reveals practical challenges in ADS testing, offers rapid insights for practitioners, and highlights the need for more context-aware, industry-relevant solutions to bridge the gap between research and practice.

We prove an abstract theorem on keeping the compactness property of a linear operator after interpolation in Banach spaces. Our approach consists of two features. Applying the principle "reductio ad absurdum," we obtain a possibility to carry out all proofs only for some specially constructed subspaces of the given spaces, e.g., having a common Schauder basis. As a second feature, we consider in all assertions only embedding operators obtaining the full result just at the end of the paper. No analytical presentation of operators, spaces and interpolation functors is required and the complex method is admissible as a particular case.

The horocyclic evolutes of spacelike frontals in hyperbolic 2-space have already been defined. Using enveloid theorem, we now define the horocyclic parallel and involute of a spacelike frontal in hyperbolic 2-space as the normal envelopes of its normal and tangent horocycles, respectively. Meanwhile, we investigate the relations among horocyclic evolutes, parallels and involutes.

The inertia bound, introduced by Cvetković in 1971, is a fundamental result in spectral graph theory that provides an upper bound for the independence number of a graph in terms of spectral information about a weighted adjacency matrix of the graph. Recently, this bound has been extended to the socalled inertia-type bounds for estimating the independence and chromatic numbers of graph powers ($k$-independence number and distance-$k$ chromatic number of a graph). These bounds have recently found applications in coding theory and quantum information theory. The inertia-type bounds depend on the choice of a polynomial of degree $k$ and on the eigenvalues of the graph. Currently, optimizing these bounds requires solving several MILPs, which quickly becomes computationally demanding as the graph size or $k$ grows. This computational barrier is a major obstacle to the practical use of these bounds. Moreover, we have a limited theoretical understanding of their performance, even for small $k$. In this paper, we investigate their optimization and complexity. In particular, we improve the MILP formulations, reducing their computational burden and significantly decreasing the running time. Furthermore, we show that the optimization problems associated with the bounds are solvable in polynomial time for fixed $k$ and for small $k$.

We study an intuitionistic version of common knowledge logic (CK), called ICK, which was introduced by Jäger and Marti. ICK extends intuitionistic propositional logic (IPL) by multiple box modalities interpreted as knowledge operators for various agents and a common knowledge operator. Formulae are interpreted over birelational Kripke models satisfying a simple interaction principle between the intuitionistic order and the modal accessibility relations. Furthermore, we consider the restriction to reflexive, S4 and S5 models. We present axiomatizations as well as analytic cyclic sequent calculi for all considered logics and prove them to be sound and complete. Furthermore, we establish the finite model property and decidability, show that proof-search in the cyclic calculi can be automated, provide a translation of CK over S5 into ICK over S5 and establish that the proof-search and validity problems of all considered logics can be solved in exponential time.

This study investigates Chinese teachers' continuance intention to use generative artificial intelligence (AI) by integrating the Expectation-Confirmation Model with Institutional Theory. A sequential explanatory mixed-methods design was employed. Questionnaire data from 437 teachers were analysed using structural equation modelling, followed by semi-structured interviews with 15 teachers to further interpret the findings. The results indicate that confirmation, perceived usefulness, and satisfaction play important roles in shaping teachers' continuance intention, while institutional pressures, including coercive, normative, and mimetic influences, also contribute to continued use. Qualitative findings further reveal that teachers often use generative AI pragmatically to support tasks such as lesson preparation and idea generation, while simultaneously exercising caution and critically evaluating the reliability of AI-generated content. These findings highlight the combined influence of individual evaluations and institutional contexts on teachers' sustained engagement with generative AI in education.

We studied McMillan's free energy of the lock-in transition in charge-density waves. The wave profiles near the critical value were obtained by numerical annealing. First, we demonstrated that our method reproduces the previous studies. The obtained wave profiles include discommensurations near the critical value. Then, we calculated possible wave profiles in the commensurate state. We found that discommensurations are able to be excited in the commensurate state, leading the system to turn out to an incommensurate state. We proposed that these wave profiles result from topological invariants. Moreover, excitation of the discommensurations is favorable for the direction to the original wavelength of the incommensurate state. This is attributed to the nature of McMillan's free energy. The current-induced incommensurations, which we discovered with the diffraction study of $o$-TaS$_3$ [Inagaki \textit{et al.}, J. Phys. Soc. Jpn. 77, 093708 (2008)], is consistent with this study.

For quantum field theory on curved spacetimes, a critical role is played by their foliation into spacelike time-slices at each value $t$ of a coordinate time, with corresponding metric in ADM form. We provide a general construction for the spacetime quantum Levi-Civita connection when each spatial slice is replaced by a quantum Riemannian geometry. This is then fully solved for a class of spatial algebras including fuzzy spheres and for any time-dependent spatial quantum metric, shift 1-form and lapse function. The result takes a particularly simple form if the spatial metric evolves in time according to a first order ODE which, in the case of a fuzzy sphere, requires the spatial metric to rotate in time according to the value at each $t$ of the shift vector. As an application, our results provide in principle fuzzy versions of most (pseudo)-Riemannian manifolds. We also fully solve the case of rotationally invariant spacetimes with angular directions replaced by a discrete circle, including a new $\Bbb Z_n$-FLRW model.

In this paper, we characterize the existence of perfect state transfer (PST) and fractional revival in continuous-time quantum walks on the zero-divisor graph $Γ(\mathbb{Z}_n)$. By using the canonical equitable partition of $Γ(\mathbb{Z}_n)$ induced by the proper divisors of $n$, we derive a sufficient condition on $n$ for PST to occur between a pair of vertices. We show that fractional revival is restricted to cells of size $2$ within the equitable partition. Furthermore, assuming $-1$ is not an eigenvalue of the quotient spectrum, we establish that two vertices in $Γ(\mathbb{Z}_n)$ are strongly cospectral if and only if they form a cell of size $2$ within the equitable partition that is either a set of false twins or true twins. Finally, we provide a characterization of fractional revival on bipartite $Γ(\mathbb{Z}_n)$ and prove the non-existence of fractional revival on $Γ(\mathbb{Z}_{p^2q})$.

Given a polarised maximal degeneration of compact Calabi-Yau manifolds, assuming there exists a canonical basis of the section ring for the polarisation line bundle, satisfying the valuative independence condition, we will prove the metric SYZ conjecture.

In the article by Michael Chmutov, Max Glick and Pavel Pylyavskii \cite{Chmutov} the action of the cactus group $C_N$ on the set of semi-standard Young tableaux filled with the numbers from $1$ to $N$ was defined. Namely, they constructed the set of generators (we rightfully call them Bender-Knuth generators) of the cactus group and a group homomorphism from $C_N$ to Berenstein-Kirillov group $BK_N$ (cf. \cite{Berenstein_Kirillov}), which sends these generators to the Bender-Knuth involutions on the set of semi-standard Young tableaux. In \cite{Henriques_Kamnitzer} Andre Henriques and Joel Kamnitzer defined a natural action of cactus group $C_N$ on the tensor product of $N$ normal crystals via commutors. By applying their result I defined the action of cactus group $C_N$ on the set of short semi-standard Young tableaux filled with the numbers $1, 2, \ldots, N$ in \cite{Svyatnyy}. A semi-standard Young tableau is called \textit{short} if the number of cells in the first two columns with the numbers $\leqslant N$ is less than or equal to $N$. The set of short semi-standard Young tableaux obviously forms a subset inside the set of semi-standard Young tableaux. The purpose of this paper is to explicitly compute the action of Bender-Knuth generators of cactus group $C_N$ on the set of short semi-standard Young tableaux defined in \cite{Svyatnyy} and compare it with their action on the set of semi-standard Young tableaux defined in \cite{Chmutov}.

We summarise known algebraic and model theoretic results on the ring $\mathscr{A}$ of integers modulo infinitely large primes for number theorists, and share topics in transcendental number theory with algebraists and model theorists. In particular, we extend transcendence criteria by Anzawa--Funakura and Matsusaka--Seki in order to show application of non-standard analysis to the study of transcendence.

Purpose: As interest in CEST-MRI grows, particularly in the preclinical setting, the necessity for standardized and easy-to-use acquisition and data analysis pipelines has become apparent. While vendors have increasingly introduced support for CEST acquisitions on both clinical and preclinical hardware, image post-processing and analysis pipelines remain siloed based on privately developed code. We aim to develop an easy-to-use, open-source graphical user interface toolbox for preclinical CEST-MRI data analysis (Preclinical CEST-MRI Analysis Tool; Pre-CAT), supporting multiple acquisition types, organ systems, and CEST contrast mechanisms. Methods: Pre-CAT was developed in Python and utilizes the Numpy, Scipy, and Matplotlib libraries for data analysis and plotting. Inbuilt data processing steps include image loading, reconstruction, post-processing, and segmentation. Pre-CAT also supports data analysis for QUESP, CEST-MRF, and field mapping experiments using consensus protocols and methods. Pre-CAT's web interface and GUI were developed using Streamlit, an open-source Python framework. Pre-CAT is hosted and accessible online and can be downloaded for local installation to complete the data analysis pipeline in roughly one minute using modern hardware. Results: Pre-CAT analysis pipelines for Z-spectroscopy, CEST-MRF, and quantitative CEST (QUESP/QUEST) are demonstrated. Conclusion: With the introduction of Pre-CAT, we aim to standardize the preclinical CEST-MRI data analysis pipeline, fostering collaboration across research sites and reducing methodological redundancy. Pre-CAT is open-source and relatively modular, encouraging the addition of new methods and protocols.

The purpose of the current work is the development of an approach to account for quasi-static mechanical equilibrium in empirical (i.e., data-based) models for the stress field employing neural approximations (NAs), which include neural networks (NNs) and neural operators (NOs), in particular Fourier NOs (FNOs). Rather than including such constraints from physics in the loss function as done in the (now standard) physics-informed approach, the current approach incorporates or "encodes" such constraints directly into the architecture of the NA. As a result, both NA training and output are physically constrained in the physics-encoded approach, in contrast to the physics-informed approach, in which only training is physically constrained. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the current approach, a physics-encoded FNO (PeFNO) is developed for a heterogeneous polycrystalline material consisting of isotropic elastic grains and subject to uniaxial extension. Stress field data for this purpose are obtained from the numerical solution of corresponding boundary-value problems for quasi-static mechanical equilibrium. For comparison with the PeFNO, this data is also employed to develop an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO). As expected theoretically, and confirmed by this computational comparison, for comparable accuracy of the stress field itself as compared to the data, the stress field output by the trained and tested PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the PgFNO or the PiFNO.

In this paper, we prove a central limit theorem for inhomogeneous Diophantine approximation with a fixed shift, provided the shift is non-Liouville. This generalizes earlier work of Dolgopyat, Fayad, and Vinogradov~\cite{DFV}. This is achieved by translating the problem to one involving flows on homogeneous spaces. In this latter setting, we establish an effective multi-equidistribution result for diagonal translates of unipotent flows. This result is obtained by combining a recent result of Kim~\cite{Kim2024} with the height function construction of Shi~\cite{Shi20}. The central limit theorem is then deduced using the method of Björklund and Gorodnik~\cite{BG}.

Energy efficiency has emerged as a vital attribute of software quality, with significant implications for both environmental sustainability and operational costs. However, existing profiling tools operate only at runtime and coarse granularity, typically capturing energy at the process or method level. Such tools fail to expose how small code blocks, such as functions, loops, and conditionals, contribute to energy consumption, preventing developers from reasoning about and comparing the energy efficiency of programming constructs during design-time. To address this gap, we propose EnCoDe, a methodology for fine-grained, design-time energy estimation, with the following key contributions: (1) PowerLens, a novel measurement methodology that achieves reliable sub-millisecond energy readings for small code blocks; (2) Extensive empirical study on code blocks extracted from over 18,000 Python programs, uncovering linear and non-linear relationships between energy consumption and static code features such as structural, complexity, density, and contextual characteristics, resulting in a first-of-its-kind fine-grained dataset; and (3) Predictive modeling, in which machine learning models are trained on these features to accurately estimate and classify block-level energy consumption at design-time. Our results demonstrate stable, reproducible block-level estimations, with regressors achieving R^2 = 0.75 and classifiers achieving 80.6% accuracy in identifying energy hotspots, enabling developers to localize and address inefficient code regions early in the development process without execution.

Next-basket recommendation (NBR) is a type of recommendation that aims to predict a set of items a user will purchase based on their historical transaction basket sequences. It is governed by a dynamic interplay between two distinct user intents: habitual repurchase, which involves repeating past behaviors, and exploratory interest, which involves discovering new items. However, existing NBR methods generally suffer from two limitations: (1) they often entangle these conflicting motives within a single representation, causing habits to overshadow discovery, and (2) they rely on discrete sequential modeling that ignores continuous-time intervals and item-specific periodicities. In this paper, we propose a novel solution named Time-Interval Disentangled Experts (TIDE) to address these challenges. TIDE incorporates a Hawkes-enhanced Fourier Time Encoding to capture item-specific temporal periodicities and dynamic decay. To decouple user intentions, TIDE utilizes a dual-expert architecture that integrates a Habit Expert for recurring needs and a Pattern-Guided Exploration Expert for discovery. Combined with an item-aware gating mechanism, TIDE adaptively balances repurchase and exploration. Extensive experiments on four diverse real-world datasets demonstrate that TIDE consistently outperforms representative state-of-the-art NBR methods.

To address the limitations of existing Generative Fixed-Filter Active Noise Control (GFANC) methods, which rely on filter decomposition and recombination and require supervised learning with labeled data, this paper proposes a Transformer-based End-to-End Control-Filter Generation (E2E-CFG) framework. Unlike previous approaches that predict combination weights of sub control filters, the proposed method directly generates control filters in an unsupervised manner by integrating the co-processor and real-time controller into a fully differentiable ANC system, where the accumulated error signal is used as the training objective. By abandoning the decomposition--reconstruction process, the proposed design simplifies the control pipeline and avoids error accumulation, while the Transformer architecture effectively captures global and dynamic noise characteristics through its attention mechanism. Numerical simulations on real-recorded noises demonstrate that the proposed method achieves improved noise reduction performance and adaptability to different types of noises compared with the original GFANC framework.

For an integer $r \ge 2$ and an order $n \equiv 1, 3 \pmod{6}$, write $δ_r(n)$ for the minimum, over all $r$-colourings $χ: \binom{[n]}{3} \to [r]$, of $\max_{\mathcal{S}} \mathrm{disc}(\mathcal{S}, χ)$, where the maximum is over labelled Steiner triple systems $\mathcal{S}$ of order $n$ and $\mathrm{disc}(\mathcal{S}, χ) = \max_c |\#\{T \in \mathcal{S} : χ(T) = c\} - |\mathcal{S}|/r|$. Following Gishboliner, Glock, and Sgueglia \cite{GishbolinerGlockSgueglia2025}, the bulk of the recent work on this quantity has been on lower bounds for $r \ge 3$ (proving $δ_r(n) = Ω(n^2)$) and on structural characterisation of the low-discrepancy 2-colourings. We give three small computational contributions in the small-$n$ regime $n \in \{7, 9, 13, 15, 19, 21\}$: An exact value of $δ_2(n)$ for each such $n$, matching the formula $δ_2(n) = \min_{x \in [0, n] \cap \mathbb{Z}} |x(n-x)/2 - n(n-1)/12|$ obtained by optimising the GGS Example 1.1 family. Rigorous for $n \in \{7, 9\}$ via exhaustive search over labelled STSs ($30$ resp. $840$ systems) and over all $2$-colourings; computational for $n \in \{13, 15, 19, 21\}$ by simulated-annealing search; A wide near-optimal basin: at $n = 9$, every two-colour-flip neighbour of the optimal Example~1.1 colouring that maintains discrepancy $1.0$ exists; about $34\%$ of two-flip perturbations preserve optimality; Random-colouring statistics for $r \in \{2, 3, 4\}$: $\langle\max_{\mathcal{S}}\mathrm{disc}\rangle$ grows linearly in $n$, in agreement with a heuristic Gaussian estimate $n / \sqrt{6r} \cdot \sqrt{2 \log K}$ over $K$ sampled labellings; the typical-case discrepancy is far below the GGS worst-case $Ω(n^2)$. We additionally state a conjectural exact formula for $δ_2(n)$ that holds for every $n \equiv 1, 3 \pmod{6}$.

We report on the longitudinal magnetoresistance (MR) in thin metal films on an Ising-type antiferromagnetic insulator, Na5Co15.5Te6O36 (NCTO). Steep changes in the MR spectra with hysteresis were observed at spin-flip transitions driven by magnetic fields applied along the easy axis of the NCTO crystal. The MR jumps almost follow step-like changes in magnetization at the spin-flip transition. At very low temperatures where Co moments are partially frozen, the MR anomalies exhibit a tunnel-magnetoresistance-like shape. The observed MR anomalies at the spin-flip transition are attributed to strain effects via magnetostriction upon the magnetic-structure change of the Co nets in NCTO, because similar MR jumps are observed in both Pt/NCTO and Cu/NCTO. Interestingly, we found that the high-field slopes of the MR spectra show opposite signs between Pt/NCTO and Cu/NCTO at low temperatures. Because the opposite signs of the high-field MR are prominent below the antiferromagnetic transition temperature of NCTO, the interaction between the interface spin accumulation and magnetization is likely to contribute to the MR effect in the induced ferromagnetic state.

We introduce a technology to formally verify that a software system satisfies a temporal specification of functional correctness, without revealing the system itself. Our method combines a deductive approach to model checking to obtain a formal certificate of correctness for the system, with zero-knowledge proofs to convince an external verifier that the system -- kept secret -- complies with its specification of correctness -- made public. We consider proof certificates represented as ranking functions, and introduce both an explicit-state and a symbolic scheme for model checking in zero knowledge. Our explicit-state scheme assumes systems represented as transition graphs. We use polynomial commitments to convince the verifier that the public proof certificates correspond to the secret transition relation. Our symbolic scheme assumes systems specified as linear guarded commands and uses piecewise-linear ranking functions. We apply Farkas' lemma to obtain a witness for the validity of the ranking function with public and secret components, and employ sigma protocols for matrix multiplication and range proofs to convince the verifier of the witness's existence. We built a prototype to demonstrate the practical efficacy of our two schemes on linear temporal logic verification examples. Our technology enables formal verification in domains where both the safety and the confidentiality of the system under analysis are critical.

As global fossil fuel reserves diminish, there's a growing impetus for nations to transition towards renewable energy sources. Sri Lanka, for instance, aims to generate 70% of its electricity from renewable sources by 2030. Achieving this target requires optimal use of the existing power transmission infrastructure, as expanding the grid is both time-consuming and expensive. Traditionally, Static Line Ratings (SLRs) are used to define line capacity, often resulting in underutilization. Dynamic Line Rating (DLR), which estimates line capacity in real time based on weather conditions, offers a more efficient solution. However, DLR prediction is highly sensitive to environmental variability and forecasting complexity. This study proposes a novel multivariate Long Short-Term Memory (LSTM) model enhanced with an attention mechanism for improved DLR forecasting. Unlike traditional models that treat weather variables independently, the proposed approach captures nonlinear interdependencies among key environmental features such as ambient temperature, cable temperature, wind speed, humidity, and solar irradiance. The attention mechanism dynamically prioritizes the most relevant inputs during forecasting, leading to improved performance. Experimental evaluation on real-world DLR data demonstrates that the proposed model achieves a prediction accuracy of 95.84%, surpassing the conventional LSTM model's 94.62%. This improvement highlights the model's superior ability to deliver accurate and robust DLR forecasts. The findings confirm that incorporating multivariate features with attention enhances forecasting precision, supporting more efficient transmission line utilization and higher renewable energy integration.

The Von Neumann entropy of reduced states is a measure of bipartite entanglement. Despite its name, the entanglement entropy cannot by itself be used as a resource for creating thermodynamic heat flows. In order to extract heat from an entangled pure state, it first needs to be converted into a stochastically mixed state by a process of quantum state reduction. Here we show that even in a system with only two degrees of freedom, for which bipartite entanglement is the sole form of entanglement available, the entanglement entropy cannot be converted into thermodynamic entropy in a one-to-one fashion. Moreover, we show that the stochastic dynamics which is necessarily present in any realistic model of quantum state reduction, allows for multiple definitions of entropy. We indicate why quantum state reduction does not allow construction of a perpetuum mobile, despite some measures of entropy evolving non-monotonically during its dynamics. Finally, we relate the different measures of entropy to the information they contain about quantum entanglement and extractable heat, and show that models of quantum state reduction based on physical, correlated stochastic driving forces give rise to observable thermodynamic signatures of quantum state reduction that can be unambiguously distinguished from environment-induced dephasing.

We study extreme wave formation for the Korteweg-de Vries equation on the torus with random initial data of average size $ε$. We establish a large deviations principle for the supremum of the solution over arbitrarily long polynomial timescales $t \leq ε^{-n}$ for any fixed natural number $n$. This identifies the leading-order asymptotics of the probability of observing unusually large amplitudes. In this integrable setting, the dynamics evolves on invariant tori where Fourier moduli are almost conserved, ruling out mechanisms for extreme wave formation based on resonant energy exchange. As a result, large amplitudes can only arise through coherent structures or dispersive focusing, which corresponds to the quasi-synchronization of many phases. We show that the latter is dominant in the weakly nonlinear regime. Our approach combines a Birkhoff normal form analysis with probabilistic arguments, exploiting the stability of the integrable dynamics to control the probability of phase quasi-synchronization over long timescales.

We study the existence of Hamiltonian semisprays on Lie algebroids. This work is motivated by a problem studied by Vaisman for tangent bundles, and we extend this question to the setting of arbitrary Lie algebroids and provide a general solution. More precisely, given a Lie algebroid and a regular Lagrangian, we construct a family of Poisson brackets on the algebroid such that the Hamiltonian vector field associated with the corresponding energy function is a semispray. Our approach is based on the symplectic geometry of the prolongation of a Lie algebroid and a cohomological analysis of its vertical subbundle. The results provide a geometric framework for second-order Hamiltonian dynamics on Lie algebroids, extending some known facts in the classical tangent bundle case and revealing new interactions between Poisson geometry and algebroid structures.

Surface-bound electric charge on polymer materials can strongly influence droplet behaviour and solid-liquid charge transfer, but the mechanisms and the means to control these effects remain unclear. In this work, we systematically controlled the surface charge on polymer surfaces, including polytetrafluoroethylene (PTFE) and Nylon-66, by first neutralising the surfaces with an anti-static ion blower and then applying charge using an ion gun. We find that droplets pick up pre-deposited surface ions during the first wetting of the surface, and that the transferred charge directly correlates with the deposited charge encountered by the wetted area for moderate deposited densities (|σ_d |<40 μC/m2) independent of material properties. We also demonstrate that the deposited charge reduces contact angle and increases contact-line mobility in a manner consistent with an increase in effective solid surface energy. For higher surface charge densities, we observe instabilities such as droplet splitting or detachment. This work demonstrates an effective approach to control solid-liquid electrification, enabling amplification or suppression of surface charge and the directed manipulation of fluid motion on surfaces.

The stochastic simulation algorithm (SSA) is widely used to perform exact forward simulation of discrete stochastic processes in biology. However, the computational cost, driven by sequential event-by-event sampling across large ensembles, remains a computational barrier. We investigate whether reduced-precision floating-point arithmetic can accelerate SSA without degrading statistical fidelity, drawing on the success of reduced-precision methods in weather and climate modelling. We evaluate two strategies across five canonical models (birth--death, Schlögl, Telegraph, dimerisation, repressilator): (i) mixed precision, computing propensities in 16-bit while maintaining accumulators in 32-bit; and (ii) uniform precision, performing all arithmetic in 16-bit. Mixed-precision SSA produces ensemble statistics that closely match the 64-bit reference for all models, as measured by Kolmogorov--Smirnov tests and Wasserstein distances. Under uniform precision, deterministic rounding introduces systematic biases across several models, with catastrophic failures in some cases. Stochastic rounding (SR) and propensity normalisation eliminate these biases, restoring distributional fidelity across all models tested (KS $p > 0.05$). Our results establish mixed-precision SSA with SR as a viable acceleration strategy for mathematical biology: 16-bit formats shrink per-variable data size by $2$--$4\times$ relative to \texttt{fp32}/\texttt{fp64}, yielding comparable reductions in memory footprint and up to $\sim 1.5\times$ wall-clock speedup on CPU hardware that lacks native 16-bit arithmetic. As a hardware-level acceleration, mixed-precision SSA complements algorithmic methods such as tau-leaping and maps naturally onto modern GPU and TPU architectures with native 16-bit arithmetic.

We study the root-averaged density of states for the Anderson model on the Bethe lattice in the strong-disorder regime. Here the density of states means the root-averaged spectral measure, not a finite-volume eigenvalue counting limit. We assume that the single-site distribution has compact support and has a locally analytic density on an interval $I^\sharp$ containing a given interval $I$. Combining the random-walk expansion on the tree with a complex-analytic argument for the single-site Stieltjes transforms, we prove that the scaled averaged diagonal resolvent has a holomorphic continuation to a complex neighborhood of $I$ for all sufficiently large $λ$. By the Stieltjes inversion formula, the root-averaged density of states measure is absolutely continuous on the scaled energy window $λI$, and its density is real analytic and has a finite-order strong-disorder expansion there. In the scaled form $E=λξ$, the leading coefficient is the local density of the single-site distribution. All odd coefficients vanish, and the higher coefficients are finite sums determined by occupation profiles of short closed walks on the tree. For the uniform single-site distribution, we compute the first nonzero correction term explicitly.