We obtain non-trivial bounds for bilinear sums of trace functions below the Pólya-Vinogradov range assuming only that the geometric monodromy group of the underlying ell-adic sheaf satisfies certain simple structural properties, in contrast to previous works which handled only special cases of Kloosterman and hypergeometric sheaves. Our approach builds on a general "soft" stratification theorem for sums of products of trace functions, based on an idea of Junyan Xu, combined with a new robust version of the Goursat-Kolchin-Ribet criterion.

Neutrino oscillation parameters are subject to renormalization group (RG) evolution, just like all couplings and masses of Standard Model (SM) particles. Within the SM extended with three massive neutrinos, it is well known that RG running effects in the neutrino sector are small. However, the RG running of the elements of the leptonic mixing (PMNS) matrix below the electroweak symmetry breaking scale can be enhanced in the presence of light neutrinophilic new particles. In this work, using a particular low-scale neutrino mass model as an example, and by taking into account both atmospheric and astrophysical neutrino fluxes, we show that RG running of the PMNS matrix can lead to an increased number of high-energy tau neutrino events at IceCube. This excess manifests as an increased number of spatially displaced showers called ``double bangs". We find that the number of double bangs induced by new physics through RG effects can be comparable to that arising from SM interactions of astrophysical tau neutrinos.

The rapid advancement of AI-assisted software engineering has brought transformative potential to the field of software engineering, but existing tools and paradigms remain limited by cognitive overload, inefficient tool integration, and the narrow capabilities of AI copilots. In response, we propose Compiler.next, a novel search-based compiler designed to enable the seamless evolution of AI-native software systems as part of the emerging Software Engineering 3.0 era. Unlike traditional static compilers, Compiler.next takes human-written intents and automatically generates working software by searching for an optimal solution. This process involves dynamic optimization of cognitive architectures and their constituents (e.g., prompts, foundation model configurations, and system parameters) while finding the optimal trade-off between several objectives, such as accuracy, cost, and latency. This paper outlines the architecture of Compiler.next and positions it as a cornerstone in democratizing software development by lowering the technical barrier for non-experts, enabling scalable, adaptable, and reliable AI-powered software. We present a roadmap to address the core challenges in intent compilation, including developing quality programming constructs, effective search heuristics, reproducibility, and interoperability between compilers. Our vision lays the groundwork for fully automated, search-driven software development, fostering faster innovation and more efficient AI-driven systems.

This work introduces a distributed formation control strategy for multi-agent systems based solely on rotation symmetry constraints. We propose a potential function that enforces inter-agent \textbf{rotational} symmetries, whose gradient defines a control law that drives the agents toward a desired planar symmetric configuration. We show that only $n-1$ edges (the minimal connectivity requirement) are sufficient to implement the strategy, where $n$ is the number of agents. We further augment the design to address the \textbf{maneuvering problem}, enabling the formation to undergo coordinated translations, rotations, and scaling along a predefined virtual trajectory. Simulation examples are provided to validate the effectiveness of the proposed method.

We present a method to simulate non-coalescing impacts and rebounds of droplets onto the free surface of a liquid bath, together with new experimental data, focused on the low-speed impact of droplets. The method is derived from first principles and imposes only natural geometric and kinematic constraints on the motion of the impacting interfaces, yielding predictions for the evolution of the contact area, pressure distribution, and wave field generated on both impacting masses. This work generalises an existing kinematic-match method whose prior applications dealt with deformation of the surface of the bath only; i.e., neglecting that of the droplet. The method's extension to include droplet deformation gives predictions that compare favourably with existing experimental results and our new experiments conducted in the low-Weber-number regime.

We prove a few new properties of the $φ$-representation of integers, where $φ= (1+\sqrt{5})/2$. In particular, we prove a 2012 conjecture of Kimberling. As software assistants, we used the Walnut theorem-prover, and in one proof, ChatGPT 5.

Context: Laws and regulations increasingly shape software design, development, and quality assurance in regulated domains. Because legal provisions are written in technology-neutral language, deriving concrete specifications, requirements, and acceptance criteria to verify software compliance is difficult and error-prone. Recent advances in generative AI, especially large language models (LLMs), may help automate this process. Objective: We present the first systematic human-subject evaluation of LLMs' ability to derive Gherkin behavioural specifications from legal texts using a quasi-experimental design. Gherkin is a domain-specific language for scenario-based system behaviour descriptions in Given-When-Then form and is well suited to automation in software development. Methods: Ten participants evaluated 60 Gherkin specifications generated from food-safety regulations by Claude and Llama. Each participant assessed 12 specifications across five criteria: relevance, clarity, completeness, singularity, and time savings. Each specification was evaluated by two participants, yielding 120 assessments with quantitative ratings and qualitative feedback. Results: Ratings were uniformly high in the top two categories: relevance 95%, clarity 100%, completeness 94.2%, singularity 93.4%, and time savings 91.7%. No statistically reliable differences were found across participants or between LLMs. Qualitative feedback noted occasional omissions, hallucinations, and mixed intents, underscoring the need for human oversight, especially in safety-critical domains. Conclusion: In food safety, LLMs can assist in deriving Gherkin specifications from legal texts, but omissions and hallucinations require systematic human review.

The family of multi-plane phase retrieval sensors, such as the curvature and nonlinear curvature wavefront sensors (WFS), contain tip/tilt information embedded in their signals. We have built a nonlinear curvature WFS to study different wavefront reconstruction methods and test the ability to extract tip/tilt information. Using reliable and fast centroiding algorithms, combined with knowledge of the measured $z$-distance to each measurement plane, we demonstrate that image jitter may be sensed and compensated for using a fast steering mirror and the WFS in closed loop. This approach obviates the need for peripheral components such as quad-cells or access to a separate scientific imaging channel. Our laboratory experiments validate tip/tilt estimation and correction using nlCWFS data, achieving tip/tilt accuracy of +/-0.1, lambda/D for an unaberrated beam and better than ~+/-0.5, lambda/D in the presence of aberrations, consistent with prior numerical simulations. We further demonstrate a closed-loop tip/tilt control implementation and show a qualitative improvement in the stability and overall quality of multi-plane phase retrieval reconstructions.

In multi-center clinical research, privacy regulations often prohibit pooling individual-level records, complicating the analysis of time-to-event data. Current federated survival methods frequently require iterative communication or rely strictly on proportional hazards (PH) assumptions or require sensitive survival information. We propose a one-shot federated framework using pseudo-observations derived from a sequentially updated Kaplan-Meier estimator and fitted via a renewable generalized estimating equation. Unlike traditional methods, our approach allows flexible link functions tailored to the target estimand and accommodates non-proportional hazards. To address site-level heterogeneity, we introduce a covariate-wise debiasing procedure that shrinks noise-driven local deviations toward the global estimate while preserving genuine site-specific effects. Simulation studies demonstrate that our framework achieves inferential accuracy comparable to pooled Cox regression and the privacy-preserving One-shot Distributed Algorithm to fit a multicenter Cox proportional hazards model (ODAC) under PH assumptions, while recovering time-varying coefficient trajectories when PH is violated. Furthermore, simulations confirm that the debiasing procedure optimizes the bias-variance trade-off, adaptively balancing global stability with the preservation of genuine site-specific deviations. Applied to pediatric obesity data from the Chicago Area Patient-Centered Outcomes Research Network (CAPriCORN) network ($N=45,865$), the model produced robust estimates of time-invariant and time-varying hazard ratios, offering a flexible, privacy-preserving alternative for collaborative survival research.

The infamous weakness of molecular chiroptical responses challenges the all-optical realization of crucial applications such as enantio-selective sorting of chiral molecules, or biasing chiral chemical reactions. Chiral optical cavities are a natural choice for confronting this challenge. Ideally, the dissymmetry between the two helicities inside such cavities is maximized. In here, we propose a chiral infrared optical cavity formed by planar mirrors made of diffracting lattices of silver helices with almost maximum electromagnetic chirality. It combines the strong helicity selectivity of the helices with the helicity-preserving reflectivity that planar systems show at large incidence angles. For the manifold of cavity modes which have a component with zero in-plane momentum, we demonstrate an unprecedented dissymmetry of 95 % inside the cavity at the target frequency, making it a compelling candidate for enantio-selective applications.

Quantum speed limits are usually regarded as fundamental restrictions, constraining the amount of computation that can be achieved within some given time and energy. Complementary to this intuition, here we show that these limitations are also of operational value: they enable the secure generation of certified randomness. We consider a prepare-and-measure scenario with some (experimentally determined or promised) upper bound on the energy uncertainty of the average prepared quantum state, but without any further assumptions on the devices, Hilbert space or Hamiltonian. Given that we can freely choose the time at which to apply the untrusted preparation procedure, we show that this scenario admits the generation of randomness that is secure against adversaries with additional classical information. We show how to determine the amount of certified randomness given the observed correlations, discuss how interactions with the environment are taken into account, and sketch a conceivable experimental implementation. In particular, we show that single-mode coherent states admit this kind of certification of non-zero randomness in some parameter regimes, reinforcing existing demonstrations of nonclassicality in the simple harmonic oscillator. Our results extend earlier efforts to devise semi-device-independent protocols grounded in reasonable physical assumptions, and they contribute to the understanding of time-energy uncertainty relations via their operational consequences.

Power-law inflation has stood as a classical model in inflationary cosmology since the early 1980s, prized for its exact analytical solutions and ability to naturally resolve the Big Bang theory's horizon and flatness problems through exponential expansion. However, its simplest form appears incompatible with modern precision observations, motivating increasingly complex alternatives. In this work, we demonstrate how previous predictions with power-law inflation considered only a particular solution of the field equations, and derive the complete set of general analytical solutions that satisfy current theoretical and observational constraints. This finding revitalizes power-law inflation as a viable framework, offering new possibilities for cosmological model-building while preserving its original mathematical elegance.

With current and upcoming experiments on the horizon, the global 21-cm signal can open up new avenues for probing dark matter (DM) physics at redshifts that are otherwise inaccessible to other observables. This work investigates the effects of elastic scattering between DM and baryons on the global 21-cm signal in two distinct interacting DM (IDM) models: Coulomb-like and velocity-independent interactions. Our analysis incorporates key astrophysical parameters essential for accurately modeling the global signal, including star formation efficiency, escape fraction of ionizing photons, normalization of the X-ray luminosity, the number of Lyman-Werner photons emitted per stellar baryon, the minimum virial temperature of star-forming halos, as well as the IDM particle mass and cross section. We perform a Fisher analysis to forecast the sensitivity of four global 21-cm signal experimental scenarios as probes of DM-baryon scattering. We find that global signal experiments, even at the sensitivity of the current facilities such as EDGES and SARAS3, could improve existing cosmological and astrophysical constraints on DM-baryon scattering. Our results also highlight the degeneracies among the DM-baryon interaction cross section and astrophysical quantities. In particular, degeneracies between the IDM cross section and two astrophysical parameters, the minimum virial temperature, and Lyman-Werner photon production, can significantly impact the DM interaction inference. Conversely, the velocity-independent cross section is found to be insensitive to uncertainties in the X-ray luminosity. These findings underscore the necessity of accurately characterizing the uncertainties in astrophysical parameters to leverage the full potential of the 21-cm global signal experiments in probing IDM physics.

Graph-structured datasets are increasingly central to sensitive applications spanning social networks, biomedical research, and cryptographic systems. As organizations share these datasets with trusted parties for collaborative analysis, protecting against unauthorized redistribution becomes critical. Graph watermarking addresses this challenge by embedding detectable signatures that enable ownership verification and attribution of leaked data. However, despite advances in watermarking techniques, existing robustness evaluations remain limited to random edge perturbation attacks, overlooking more sophisticated adversaries who exploit community structure present in real-world graphs. We introduce the first systematic evaluation of cluster-aware attacks on graph watermarking schemes. We present a threat model in which adversaries leverage community detection algorithms to guide strategic edge modifications, targeting either intra-cluster densification with inter-cluster boundary removal, or intra-cluster sparsification with inter-cluster noise injection. Evaluating against the most comprehensively tested watermarking scheme, we demonstrate that cluster-aware attacks outperform random perturbations across real-world datasets and clustering algorithms. Our findings reveal that cluster-aware attacks reduce attribution accuracy while introducing comparable structural distortion to random attacks, demonstrating superior attack efficiency. These results establish that current watermarking schemes, evaluated solely against random perturbations, remain vulnerable to structure-aware adversarial behavior, highlighting the need for robust defenses that account for community-exploiting adversaries in graph-based privacy protection systems.

The current lunar exploration, particularly the regular large-scale cargo transportation in the Earth-Moon system proposes requirements for amount of low-energy lunar transfers. The conventional grid-search method to construct low-energy lunar transfers suffers from extensive computational effort and large-scale searches. To further improve the method, this paper focuses on one type of low-energy lunar transfers termed ballistic lunar transfers, and derives prior knowledge to narrow the scale of searches. The Sun-Earth/Moon planar bicircular restricted four-body problem (PBCR4BP) is adopted as the dynamical model to construct lunar transfers. First, the analytical conditions for ballistic capture are derived and summarized in form of exact ranges of the Jacobi energy at the lunar insertion point. Both sufficient and necessary condition and necessary condition are developed. These conditions suggest an important role of the Sun-Earth/Moon PBCR4BP rather than the Earth-Moon planar restricted three-body problem in achieving lunar ballistic capture. Then, a grid-search method combined with the analytical energy conditions is proposed to construct ballistic lunar transfers. Simulations shows that a high ballistic capture ratio is achieved by the proposed method (99.87% for direct insertion and 98.72% for retrograde insertion). Examining the obtained ballistic lunar transfers, the effectiveness of the analytical energy conditions is verified. Samples of our obtained lunar transfers achieves lower or comparable impulses compared to solutions obtained in the previous works. Solutions belonging to new or less-reported transfer families are also presented, and the potential engineering applications of these trajectories are briefly discussed.

We provide a method to obtain beyond-worst-case time complexity for any single-source-shortest-path (SSSP) algorithm by exploiting modular structures in graphs. The key novelty is a graph decomposition, called the acyclic-connected (A-C) tree, which breaks up a graph into a recursively nested sequence of strongly connected components in topological order. The A-C tree is optimal in the sense that it maximally decomposes the graph, formalised by a parameter called nesting width, measuring the extent to which a graph can be decomposed. We show how to compute the A-C tree in linear time, allowing it to be used as a preprocessing step for SSSP. Indeed, we transform any SSSP algorithm by first computing the A-C tree, and then running the SSSP algorithm in a careful recursive manner on the A-C tree. We illustrate this with two state-of-the-art algorithms: Dijkstra's algorithm and the recent sparse graph algorithm of Duan et al., obtaining improved time complexities of $O(m+n\log(\mathrm{nw}(G)))$ and $O(mα(n)+m\log^{2/3}(\mathrm{nw}(G)))$, respectively, where $\mathrm{nw}(G) \leq n$ is the nesting width of the graph $G$, and $α(n)$ is the extremely slow-growing inverse Ackermann function. Some classes of graphs, such as directed acyclic graphs, have bounded nesting width, and we obtain linear-time SSSP algorithms for these graphs.

Hibernation is an adaptation to extreme environmental seasonality that has been studied for almost 200 years, but our understanding of the underlying physiological system remains lacking due to the partially observed nature of the system. During hibernation, small mammals, such as the Arctic ground squirrel, exhibit dramatic oscillations in body temperature, typically one of the only physiological states measured, of up to 40 $^{\circ}$C. These spikes are known as interbout arousals and typically occur 10-20 times throughout hibernation. The physiological process that drives interbout arousals is unknown, but two distinct macro-scale mechanisms have been hypothesized. Using model selection for partially observed systems and classical dynamical systems theory, we are able to differentiate between these two hypotheses using only body temperature data recorded from a free-ranging Arctic ground squirrel, and show that our model can capture the broad features of the observed seasonal physiological transitions. We then modify our discovered physiological model of Arctic ground squirrel to include internally-encoded environmental information and find that we can qualitatively match body temperature data recorded from a wide range of species, including a bird, a shrew, and a bear, which also dynamically modulate body temperature. Our results suggest that a low-dimensional, environmentally sensitive core regulator could control body temperature across a diverse range of species -- a new understanding of the physiological organization across species. While the findings presented here are applicable to thermophysiology, the general modeling procedure is applicable to time series data collected from partially observed biological, chemical, physical, mechanical, and cosmic systems for which the goal is to elucidate the underlying mechanism or control structure.

With the increasing complexity of modern online service systems, understanding the state and behavior of the systems is essential for ensuring their reliability and stability. Therefore, metric monitoring systems are widely used and become an important infrastructure in online service systems. Engineers usually interact with metrics data by manually writing domain-specific language (DSL) queries to achieve various analysis objectives. However, writing these queries can be challenging and time-consuming, as it requires engineers to have high programming skills and understand the context of the system. In this paper, we focus on PromQL, which is the metric query DSL provided by the widely used metric monitoring system Prometheus. We aim to simplify metrics querying by enabling engineers to interact with metrics data in Prometheus through natural language, and we call this task text-to-PromQL. Building upon the insight, this paper proposes PromCopilot, a Large Language Model-based text-to-PromQL framework. PromCopilot first uses a knowledge graph to describe the complex context of a cloud native online service system. Then, through the synergistic reasoning of LLMs and the knowledge graph, PromCopilot transforms engineers' natural language questions into PromQL queries. To evaluate PromCopilot, we manually construct the first text-to-PromQL benchmark dataset which contains 280 metric query questions. The experiment results show that PromCopilot is effective in text-to-PromQL. When using GPT-4 as the backbone LLM, PromCopilot achieves an accuracy of 69.1\% in translating natural language questions to PromQL queries when using. To the best of our knowledge, this paper is the first study of text-to-PromQL, and PromCopilot pioneered the DSL generation framework for metric querying and analysis.

We prove an ambidexterity result for $\infty$-categories of $\infty$-categories admitting a collection of colimits. This unifies and extends two known phenomena: the identification of limits and colimits of presentable $\infty$-categories indexed by a space, and the $\infty$-semiadditivity of the $\infty$-category of $\infty$-categories with $π$-finite colimits proven by Harpaz. Our proof employs Stefanich's universal property for the higher category of iterated spans, which encodes ambidexterity phenomena in a coherent fashion.

Gaussian Processes (GPs) are widely used to model dependencies in spatial statistics and machine learning. However, exact inference is computationally intractable for GP regression, with a time complexity of $O(n^3)$. The Vecchia approximation scales up computation by introducing sparsity into the spatial dependency structure, represented by a directed acyclic graph (DAG). Despite its practical popularity, this approach lacks rigorous theoretical foundations, and the choice of DAG structure remains an open problem. In this paper, we systematically study the Vecchia approximation of the popular, isotropic Matérn GP as standalone stochastic process and uncover key probabilistic and statistical properties. We propose selecting parent sets as norming sets with fixed cardinality in the Vecchia approximation. On the probabilistic side, we show that the conditional distributions of Matérn GPs, as well as their Vecchia approximations, can be characterized by polynomial interpolations. This enables us to establish several results on small ball probabilities and the Reproducing Kernel Hilbert Spaces (RKHSs) of Vecchia GPs. Building on these probabilistic results, we prove that in the nonparametric regression model, the corresponding posterior contracts around the truth at the optimal minimax rate, both under oracle rescaling and hierarchical tuning of the prior. We illustrate the theoretical findings through numerical experiments on synthetic datasets. Our core algorithms are implemented in C++ with an R interface.

Self-interaction error (SIE), arising from the imperfect cancellation of the spurious classical Coulomb interaction between an electron and itself, is a persistent challenge in modern density functional approximations. This issue is illustrated using the prototypical one-electron system $H_2^+$. While significant efforts have been made to eliminate SIE through the development of computationally expensive nonlocal density functionals, it is equally important to explore whether SIE can be mitigated within the framework of more efficient semilocal density functionals. In this study, we present a non-empirical meta-generalized gradient approximation (meta-GGA) that incorporates the Laplacian of the electron density. Our results demonstrate that the meta-GGA significantly reduces SIE, yielding a binding energy curve for $H_2^+$ that matches the exact solution at equilibrium and improves across a broad range of bond lengths over those of the Perdew-Burke-Ernzerhof (PBE) and strongly-constrained and appropriately-normed (SCAN) semilocal density functionals. This advancement paves the way for further development within the realm of semilocal approximations.

In a multi-fidelity setting, data are available from two sources, high- and low-fidelity. Low-fidelity data has larger size and can be leveraged to make more efficient inference about quantities of interest, e.g. the mean, for high-fidelity variables. In this work, such multi-fidelity setting is studied when the goal is to fit more efficiently a parametric model to high-fidelity data. Three multi-fidelity parameter estimation methods are considered, joint maximum likelihood, (multi-fidelity) moment estimation and (multi-fidelity) marginal maximum likelihood, and are illustrated on several parametric models, with the focus on parametric families used in extreme value analysis. An application is also provided concerning quantification of occurrences of extreme ship motions generated by two computer codes of varying fidelity.

We present the basic properties of a new physical system: an individual V2+ ion embedded into an individual quantum dot. The system is realized utilizing molecular beam epitaxy and it is observed using a low-temperature polarization-resolved magneto-photoluminescence. The nature of the system is confirmed by observation of the excitonic lines split due to the interactions of a vanadium ion with carriers confined in a CdTe/ZnTe quantum dot. Observed data are explained by the numerical modeling which includes s,p-d exchange interaction, Zeeman splitting of the exciton and the ion, diamagnetic shift, and the presence of shear strain within the quantum dot. The fundamental state of vanadium exhibits a spin +/- 1/2 making this system a textbook localized qubit.

Magnetic solitons such as skyrmions and bimerons show great promise for both fundamental research and spintronic applications. Stabilizing and controlling topological spin textures in atomically thin van der Waals (vdW) materials has gained tremendous attention due to high tunability, enhanced functionality, and miniaturization. Here, we present an efficient spin-spiral approach based on first-principles, a method for mapping magnetic interactions from collective models onto arbitrary lattice symmetries, such as hexagonal and honeycomb lattices. Using atomistic spin models parametrized from first-principles, we predict the emergence of multiple topological spin textures in an all-magnetic vdW heterostructure Fe$_3$GeTe$_2$/Cr$_2$Ge$_2$Te$_6$ (FGT/CGT) -- an experimentally feasible system. Interestingly, the FGT layer favors out-of-plane magnetization, whereas the CGT layer prefers in-plane magnetocrystalline anisotropy. Néel-type nanoscale skyrmions are formed at zero field in the FGT layer due to interfacial Dzyaloshinskii-Moriya interaction (DMI), while nanoscale bimerons and antibimerons can co-exist in the CGT layer by the interplay between exchange frustration and DMI. Using the collective approach we apply, we reveal significant discretization effects in hexagonal and honeycomb geometries. In particular, we demonstrate that the lifting of geometric exchange frustration on the honeycomb significantly affects soliton barriers and pinning energetics. These fundamental results not only highlight the importance of spin simulations in discrete models for topological magnetism, especially in 2D materials, but may also help to pave the way for solitonic devices based on atomically thin vdW heterostructures.

We study a discrete and continuous version of the spectral Dirichlet problem in an open bounded connected set $Ω\subset \mathbb{R}^d$, in dimension $d\geq 2$. More precisely, consider the simple random walk on $\mathbb{Z}^d$ killed upon exiting the (large) bounded domain $Ω_N = (NΩ)\cap \mathbb{Z}^d$. We let $P_N$ its transition matrix and we study the properties of its ($L^2$-normalized) principal eigenvector $ϕ_N$, also known as ground state. Under mild assumptions on $Ω$, we give regularity estimates on $ϕ_N$, namely on its $k$-th order differences (or \(k\)-th order derivatives), with a uniform control inside $Ω_N$. We provide a completely probabilistic proof of these estimates: our starting point is a Feynman-Kac representation of $ϕ_N$, combined with gambler's ruin estimates and a new ``multi-mirror'' coupling, which may be of independent interest. We also obtain the same type of estimates for the first eigenfunction $φ_1$ of the corresponding continuous spectral Dirichlet problem, in relation with a Brownian motion killed upon exiting $Ω$. Finally, we take the opportunity to review (and slightly extend) some of the literature on the $L^2$ and uniform convergence of $ϕ_N$ to $φ_1$ in Lipschitz bounded domains of $\mathbb{R}^d$, which can be derived thanks to our estimates.

First principles real-time time dependent density functional theory (rt-TDDFT) calculations reveal the existence of ballistic photocurrents generated by Coulomb scattering, which has not previously been considered as a mechanism for the bulk photovoltaic effect. With monolayer GeS as an example, it is predicted that ballistic currents can be comparable to shift currents under experimentally accessible conditions.

In this article, we are concerned with characterising when solutions of perturbed linear stochastic Volterra summation equations are almost surely $p$-summable and when their continuous time counterparts, perturbed linear stochastic Volterra integro-differential equations, are almost surely $p$-integrable. In the discrete case, we find it necessary and sufficient that perturbing functions are $p$-summable in order to ensure paths of the discrete equation are almost surely $p$-summable, while in the continuous case, it transpires one can have almost surely $p$-integrable sample paths with non-integrable perturbation functions. For the continuous equation, the main converse is clinched by considering an appropriate discretisation and applying results from the discrete case. We also conduct a thorough study of the asymptotic behaviour of the trajectories of solutions to the continuous equation in the regime of $p$-integrable paths and provide a characterisation of almost sure convergence to zero in the case of diagonal noise. Additionally, we highlight how all proof methods can be applied to obtain stronger results for stochastic functional differential equations.

We describe the rings of invariants for the finite orthogonal groups of plus type in odd characteristic acting on the defining representations. We also describe the invariants of the corresponding Sylow subgroups in the defining characteristic. In both cases we construct minimal algebra generating sets and describe the relations among the generators. Both rings of invariants are shown to be complete intersections and thus are Cohen-Macaulay. We expect the techniques we use will generalise to give a systematic computation for rings of invariants for all of the finite classical groups in odd characteristic.

We investigate the low-energy effective theory of a Fermi surface coupled to an Ising-nematic quantum critical point in (2+1) spacetime dimensions with translation symmetry. We formulate the system using the large $N$ Yukawa-SYK model, whose saddle point is described by the Migdal-Eliashberg equations. The low-energy physics can be revealed by studying the Gaussian fluctuation spectrum around the saddle point, which is generated by the Bethe-Salpeter kernel $K_\text{BS}$. Based on the Ward identities, we propose an inner product on the space of two point functions, which reveals a large number of soft modes of $K_\text{BS}$. These soft modes parameterize deformation of the Fermi surface, and their fluctuation eigenvalues describe their decay rates. We analytically compute these eigenvalues for a circular Fermi surface, and we discover the odd-parity modes to be parametrically longer-lived than the even-parity modes, due to the kinematic constraint of fermions scattering on a convex FS. The sign of the eigenvalues signals an instability of the Ising-nematic quantum critical point at zero temperature for a convex Fermi surface. At finite temperature, the system can be stabilized by thermal fluctuations of the critical boson. We derive an effective action that describes the soft-mode dynamics, and it leads to a linearized Boltzmann equation, where the real part of the soft-mode eigenvalues can be interpreted as the collision rates. The structure of the effective action is similar to the theory of linear bosonization of a Fermi surface. As an application, we investigate the hydrodynamic transport of non-Fermi liquid. Analyzing the Boltzmann equation, we obtain a conventional hydrodynamic transport regime and a tomographic transport regime. In both regimes, the conductance of the system in finite geometry can be a sharp indicator for the soft-mode dynamics and non-Fermi liquid physics.

For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but it suffices that a semismooth derivative, i.e., a mapping satisfying a certain semismoothness property, is available. In this paper we consider not only semismooth derivatives of single-valued mappings, but also its interplay with the semismoothness$^*$ property for multifunctions. In particular, we are interested in semismooth derivatives of solution maps to parametric semismooth$^*$ inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. Further we show that semismooth derivatives coincide a.e. with generalized Jacobians and state some consequences concerning strict proto-differentiability for semismooth$^*$ multifunctions.